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LOW-RANK
MATRIX APPROXIMATION
WITH STABILITY
2019. 02. 19.
JoonyoungYi
joonyoung.yi@kaist.ac.kr
CONTENTS
1. Motivation
2. Stability of LRMA
3. SMA Algorithm
4. Experiments
5. Discussion
CONTENTS
1. Motivation
2. Stability of LRMA
3. SMA Algorithm
4. Experiments
5. Discussion
4
• Let are matrices.
• Low-rank Matrix Factorization Problem: 

We know only 1% of entries in R, how to complete low-rank matrix R?
• Optimization form:
LOW-RANK MATRIX FACTORIZATION (LRMF)
R U VT≒ xm m
n
n
r
r
R 2 Rm⇥n
, U 2 Rm⇥r
, V 2 Rn⇥r
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5
• In vanilla Low-rank Matrix Factorization Problem, 

a solution find one pair of U and V (single model).
• To enhance performance of single model, 

we can use ensemble methods.
• We can say some algorithm is low-rank matrix approximation

when it finds multiple low-rank matrices to complete final matrix.
• For example,
• [ICML’13] LLoRMA: Local Low-rank Matrix Approximation
• [NIPS’17] MRMA: Mixture Rank Matrix Approximation
• [AAAI’17] GLOMA: Global Information in Local Matrix Approximation
• [ICML’16] SMA: Low-rank Matrix Approximation with Stability
LOW-RANK MATRIX APPROXIMATION
6
• The Matrix R we have to complete is very sparse.
• We only know less than 1% of total entries.
• Thus, single out-of-distribution entry 

may significantly change the entire matrix.
• For example,
• Nowadays, due to the data poisoning attack, stability is becoming more
important.
• Data Poisoning Attack on Collaborative Filtering [NIPS’16, ’18]:

Promote and demote specific items by creating many accounts that.
MOTIVATION OF THIS PAPER
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2 8
7
• [NIPS’07] Probabilistic Matrix Factorization

• To handle noise by adding regularization terms
• Generalization error is too high.
• Koren handle stability issue by cross validation.
• Cross-validation technique have drawback that the amount of data available for
model learning is reduced.
• [ICML’13] LLoRMA suggested ensemble technique.
• Ensemble methods are computationally expensive due to the training sub models.
• →This paper will suggests stable ensemble method 

which is not computationally expensive.
EXISTING WORKS
ˆR = arg max
U,V
Loss(R, UV T
) + (kUk2
F + kV k2
F )
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CONTENTS
1. Motivation
2. Stability of LRMA
3. SMA Algorithm
4. Experiments
5. Discussion
9
• To suggest stable matrix approximation method: SMA
• First introduce the stability notion in LRMA.
• Develop theoretical guidelines for deriving LRMA solutions with high stability.
• Formulate a new optimization problem for achieving stable LRMA
• Develop a stochastic gradient descent method to solve the new optimization
problem.
• Show effectiveness in real datasets (Movielens, Netflix)
• The key contributions
• (1) First introduces the stability concept in LRMA.
• (2) Proposes a stable LRMA algorithm with high stability and high generalization
performance.
• (3) Makes significant improvement in prediction accuracy.
OVERVIEW OF THIS PAPER
10
• Replacing one element in the training set

does not result in significant change to the algorithm’s output

in a stable learning algorithm.
• Bousquet & Elisseeff (2001), Lan et al., (2008) London et al., (2013) demonstrated.
• Var[ Training error ] is small.
• Training error ≈ Test error
• This paper introduce this concepts to LRMA algorithm.
WHAT IS ALGORITHMIC STABILITY?
11
• Root Mean Square Error (RMSE) is a common evaluation metric for
recommendation tasks.
STABILITY W.R.T. MATRIX APPROXIMATION
Definition 1
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D⌦( ˆR)<latexit sha1_base64="dhb9scYYPkvAOc+mIZfYx7/K7+I=">AAACBnicbVBNS8NAEN34WetX1KMgi0Wol5KIoN4KevCkVYwtNKFMttt26eaD3Y1QQm5e/CtePKh49Td489+4aXPQ1gcDj/dmmJnnx5xJZVnfxtz8wuLScmmlvLq2vrFpbm3fyygRhDok4pFo+SApZyF1FFOctmJBIfA5bfrD89xvPlAhWRTeqVFMvQD6IesxAkpLHXPPDUANCPD0Iuuk7nVA+5DhqjsAld5mhx2zYtWsMfAssQtSQQUaHfPL7UYkCWioCAcp27YVKy8FoRjhNCu7iaQxkCH0aVvTEAIqvXT8R4YPtNLFvUjoChUeq78nUgikHAW+7syvltNeLv7ntRPVO/VSFsaJoiGZLOolHKsI56HgLhOUKD7SBIhg+lZMBiCAKB1dWYdgT788S5yj2lnNvjmu1K+KNEpoF+2jKrLRCaqjS9RADiLoET2jV/RmPBkvxrvxMWmdM4qZHfQHxucPLdKZGQ==</latexit><latexit sha1_base64="dhb9scYYPkvAOc+mIZfYx7/K7+I=">AAACBnicbVBNS8NAEN34WetX1KMgi0Wol5KIoN4KevCkVYwtNKFMttt26eaD3Y1QQm5e/CtePKh49Td489+4aXPQ1gcDj/dmmJnnx5xJZVnfxtz8wuLScmmlvLq2vrFpbm3fyygRhDok4pFo+SApZyF1FFOctmJBIfA5bfrD89xvPlAhWRTeqVFMvQD6IesxAkpLHXPPDUANCPD0Iuuk7nVA+5DhqjsAld5mhx2zYtWsMfAssQtSQQUaHfPL7UYkCWioCAcp27YVKy8FoRjhNCu7iaQxkCH0aVvTEAIqvXT8R4YPtNLFvUjoChUeq78nUgikHAW+7syvltNeLv7ntRPVO/VSFsaJoiGZLOolHKsI56HgLhOUKD7SBIhg+lZMBiCAKB1dWYdgT788S5yj2lnNvjmu1K+KNEpoF+2jKrLRCaqjS9RADiLoET2jV/RmPBkvxrvxMWmdM4qZHfQHxucPLdKZGQ==</latexit><latexit sha1_base64="dhb9scYYPkvAOc+mIZfYx7/K7+I=">AAACBnicbVBNS8NAEN34WetX1KMgi0Wol5KIoN4KevCkVYwtNKFMttt26eaD3Y1QQm5e/CtePKh49Td489+4aXPQ1gcDj/dmmJnnx5xJZVnfxtz8wuLScmmlvLq2vrFpbm3fyygRhDok4pFo+SApZyF1FFOctmJBIfA5bfrD89xvPlAhWRTeqVFMvQD6IesxAkpLHXPPDUANCPD0Iuuk7nVA+5DhqjsAld5mhx2zYtWsMfAssQtSQQUaHfPL7UYkCWioCAcp27YVKy8FoRjhNCu7iaQxkCH0aVvTEAIqvXT8R4YPtNLFvUjoChUeq78nUgikHAW+7syvltNeLv7ntRPVO/VSFsaJoiGZLOolHKsI56HgLhOUKD7SBIhg+lZMBiCAKB1dWYdgT788S5yj2lnNvjmu1K+KNEpoF+2jKrLRCaqjS9RADiLoET2jV/RmPBkvxrvxMWmdM4qZHfQHxucPLdKZGQ==</latexit><latexit sha1_base64="dhb9scYYPkvAOc+mIZfYx7/K7+I=">AAACBnicbVBNS8NAEN34WetX1KMgi0Wol5KIoN4KevCkVYwtNKFMttt26eaD3Y1QQm5e/CtePKh49Td489+4aXPQ1gcDj/dmmJnnx5xJZVnfxtz8wuLScmmlvLq2vrFpbm3fyygRhDok4pFo+SApZyF1FFOctmJBIfA5bfrD89xvPlAhWRTeqVFMvQD6IesxAkpLHXPPDUANCPD0Iuuk7nVA+5DhqjsAld5mhx2zYtWsMfAssQtSQQUaHfPL7UYkCWioCAcp27YVKy8FoRjhNCu7iaQxkCH0aVvTEAIqvXT8R4YPtNLFvUjoChUeq78nUgikHAW+7syvltNeLv7ntRPVO/VSFsaJoiGZLOolHKsI56HgLhOUKD7SBIhg+lZMBiCAKB1dWYdgT788S5yj2lnNvjmu1K+KNEpoF+2jKrLRCaqjS9RADiLoET2jV/RmPBkvxrvxMWmdM4qZHfQHxucPLdKZGQ==</latexit>
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⌦<latexit sha1_base64="agbmhSsAgboSTdDWRjG0AUj+MBI=">AAAB7HicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoN4CXjxpBDcJJEuYnUySMfNYZmaFsOQfvHhQ8eoHefNvnCR70MSChqKqm+6uOOHMWN//9gorq2vrG8XN0tb2zu5eef+gYVSqCQ2J4kq3YmwoZ5KGlllOW4mmWMScNuPR9dRvPlFtmJIPdpzQSOCBZH1GsHVSo3Mn6AB3yxW/6s+AlkmQkwrkqHfLX52eIqmg0hKOjWkHfmKjDGvLCKeTUic1NMFkhAe07ajEgpoom107QSdO6aG+0q6kRTP190SGhTFjEbtOge3QLHpT8T+vndr+ZZQxmaSWSjJf1E85sgpNX0c9pimxfOwIJpq5WxEZYo2JdQGVXAjB4svLJDyrXlWD+/NK7TZPowhHcAynEMAF1OAG6hACgUd4hld485T34r17H/PWgpfPHMIfeJ8/z1KO2w==</latexit><latexit sha1_base64="agbmhSsAgboSTdDWRjG0AUj+MBI=">AAAB7HicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoN4CXjxpBDcJJEuYnUySMfNYZmaFsOQfvHhQ8eoHefNvnCR70MSChqKqm+6uOOHMWN//9gorq2vrG8XN0tb2zu5eef+gYVSqCQ2J4kq3YmwoZ5KGlllOW4mmWMScNuPR9dRvPlFtmJIPdpzQSOCBZH1GsHVSo3Mn6AB3yxW/6s+AlkmQkwrkqHfLX52eIqmg0hKOjWkHfmKjDGvLCKeTUic1NMFkhAe07ajEgpoom107QSdO6aG+0q6kRTP190SGhTFjEbtOge3QLHpT8T+vndr+ZZQxmaSWSjJf1E85sgpNX0c9pimxfOwIJpq5WxEZYo2JdQGVXAjB4svLJDyrXlWD+/NK7TZPowhHcAynEMAF1OAG6hACgUd4hld485T34r17H/PWgpfPHMIfeJ8/z1KO2w==</latexit><latexit sha1_base64="agbmhSsAgboSTdDWRjG0AUj+MBI=">AAAB7HicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoN4CXjxpBDcJJEuYnUySMfNYZmaFsOQfvHhQ8eoHefNvnCR70MSChqKqm+6uOOHMWN//9gorq2vrG8XN0tb2zu5eef+gYVSqCQ2J4kq3YmwoZ5KGlllOW4mmWMScNuPR9dRvPlFtmJIPdpzQSOCBZH1GsHVSo3Mn6AB3yxW/6s+AlkmQkwrkqHfLX52eIqmg0hKOjWkHfmKjDGvLCKeTUic1NMFkhAe07ajEgpoom107QSdO6aG+0q6kRTP190SGhTFjEbtOge3QLHpT8T+vndr+ZZQxmaSWSjJf1E85sgpNX0c9pimxfOwIJpq5WxEZYo2JdQGVXAjB4svLJDyrXlWD+/NK7TZPowhHcAynEMAF1OAG6hACgUd4hld485T34r17H/PWgpfPHMIfeJ8/z1KO2w==</latexit><latexit sha1_base64="agbmhSsAgboSTdDWRjG0AUj+MBI=">AAAB7HicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoN4CXjxpBDcJJEuYnUySMfNYZmaFsOQfvHhQ8eoHefNvnCR70MSChqKqm+6uOOHMWN//9gorq2vrG8XN0tb2zu5eef+gYVSqCQ2J4kq3YmwoZ5KGlllOW4mmWMScNuPR9dRvPlFtmJIPdpzQSOCBZH1GsHVSo3Mn6AB3yxW/6s+AlkmQkwrkqHfLX52eIqmg0hKOjWkHfmKjDGvLCKeTUic1NMFkhAe07ajEgpoom107QSdO6aG+0q6kRTP190SGhTFjEbtOge3QLHpT8T+vndr+ZZQxmaSWSjJf1E85sgpNX0c9pimxfOwIJpq5WxEZYo2JdQGVXAjB4svLJDyrXlWD+/NK7TZPowhHcAynEMAF1OAG6hACgUd4hld485T34r17H/PWgpfPHMIfeJ8/z1KO2w==</latexit>
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• For any , Choose a subset of entries from R uniformly.
• For a given , is -stable if
D( ˆR) =
v
u
u
t 1
mn
mX
i=1
nX
j=1
(Ri,j
ˆRi,j)2 , D⌦( ˆR) =
v
u
u
t
1
|⌦|
X
(i,j)2⌦
(Ri,j
ˆRi,j)2
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• The generalization error is bounded with high probability.
• Minimizing the training error will induce minimizing test error.
• For any , Choose a subset of entries from R uniformly.
• For a given , is -stable if
12
• For example,
• Any two subsets of entries from R.
• Approximating R by is -stable.
• Approximating R by is -stable.
• is more stable than if .
• Minimizing leads higher generalization performance than .
• Smaller means more stable.
STABILITY W.R.T. MATRIX APPROXIMATION
Definition 1
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D⌦1
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D⌦2
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D⌦1
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D⌦2
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13
• Experiments on Movielens 1M dataset using RSVD (Patrek, 2007).
• In the existing LRMA method,
• The test error becomes lower as the rank increases.
• However, the stability becomes lower and the generalization error
increases as the rank increases.
• Can’t provide stable recommendation even the rank is as low as 20.
STABILITY VS. GENERALIZATION ERROR
Pr
h
|D(ˆR)D⌦(ˆR)|✏
i
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rank=5
rank=10
rank=15
rank=20
14
• remove a subset of easily predictable entries from Ω.
• Minimizing both and together will make more stable solution 

than the solution of minimizing only.
• Formally,
KEY FINDING OF THIS PAPER
⌦0
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• Let (a set of easily predictable entries) which satisfies that
• Let (a set of entries hard to predict) .
• For any
• are -stable and -stable
respectively.
• Then,
Theorem 1
! ⇢ ⌦(|⌦| > 2)<latexit sha1_base64="ACA68GZluiyAFA/yKy15tzEFgIo=">AAACD3icbZDNSgMxFIUz9a/Wv6pLN8Ei1k2ZKYK6kYIbV1rBsYVOKZn0ThuayQxJRihtH8GNr+LGhYpbt+58G9N2BG09EPhy7r0k9/gxZ0rb9peVWVhcWl7JrubW1jc2t/LbO3cqSiQFl0Y8knWfKOBMgKuZ5lCPJZDQ51Dzexfjeu0epGKRuNX9GJoh6QgWMEq0sVr5Qy8KoUOwpxJfgcbe9eRaxMMfHOJzXD5q5Qt2yZ4Iz4OTQgGlqrbyn147okkIQlNOlGo4dqybAyI1oxxGOS9REBPaIx1oGBQkBNUcTBYa4QPjtHEQSXOExhP398SAhEr1Q990hkR31WxtbP5XayQ6OG0OmIgTDYJOHwoSjnWEx+ngNpNANe8bIFQy81dMu0QSqk2GOROCM7vyPLjl0lnJuTkuVK7SNLJoD+2jInLQCaqgS1RFLqLoAT2hF/RqPVrP1pv1Pm3NWOnMLvoj6+MbsJaaqw==</latexit><latexit sha1_base64="ACA68GZluiyAFA/yKy15tzEFgIo=">AAACD3icbZDNSgMxFIUz9a/Wv6pLN8Ei1k2ZKYK6kYIbV1rBsYVOKZn0ThuayQxJRihtH8GNr+LGhYpbt+58G9N2BG09EPhy7r0k9/gxZ0rb9peVWVhcWl7JrubW1jc2t/LbO3cqSiQFl0Y8knWfKOBMgKuZ5lCPJZDQ51Dzexfjeu0epGKRuNX9GJoh6QgWMEq0sVr5Qy8KoUOwpxJfgcbe9eRaxMMfHOJzXD5q5Qt2yZ4Iz4OTQgGlqrbyn147okkIQlNOlGo4dqybAyI1oxxGOS9REBPaIx1oGBQkBNUcTBYa4QPjtHEQSXOExhP398SAhEr1Q990hkR31WxtbP5XayQ6OG0OmIgTDYJOHwoSjnWEx+ngNpNANe8bIFQy81dMu0QSqk2GOROCM7vyPLjl0lnJuTkuVK7SNLJoD+2jInLQCaqgS1RFLqLoAT2hF/RqPVrP1pv1Pm3NWOnMLvoj6+MbsJaaqw==</latexit><latexit sha1_base64="ACA68GZluiyAFA/yKy15tzEFgIo=">AAACD3icbZDNSgMxFIUz9a/Wv6pLN8Ei1k2ZKYK6kYIbV1rBsYVOKZn0ThuayQxJRihtH8GNr+LGhYpbt+58G9N2BG09EPhy7r0k9/gxZ0rb9peVWVhcWl7JrubW1jc2t/LbO3cqSiQFl0Y8knWfKOBMgKuZ5lCPJZDQ51Dzexfjeu0epGKRuNX9GJoh6QgWMEq0sVr5Qy8KoUOwpxJfgcbe9eRaxMMfHOJzXD5q5Qt2yZ4Iz4OTQgGlqrbyn147okkIQlNOlGo4dqybAyI1oxxGOS9REBPaIx1oGBQkBNUcTBYa4QPjtHEQSXOExhP398SAhEr1Q990hkR31WxtbP5XayQ6OG0OmIgTDYJOHwoSjnWEx+ngNpNANe8bIFQy81dMu0QSqk2GOROCM7vyPLjl0lnJuTkuVK7SNLJoD+2jInLQCaqgS1RFLqLoAT2hF/RqPVrP1pv1Pm3NWOnMLvoj6+MbsJaaqw==</latexit><latexit sha1_base64="ACA68GZluiyAFA/yKy15tzEFgIo=">AAACD3icbZDNSgMxFIUz9a/Wv6pLN8Ei1k2ZKYK6kYIbV1rBsYVOKZn0ThuayQxJRihtH8GNr+LGhYpbt+58G9N2BG09EPhy7r0k9/gxZ0rb9peVWVhcWl7JrubW1jc2t/LbO3cqSiQFl0Y8knWfKOBMgKuZ5lCPJZDQ51Dzexfjeu0epGKRuNX9GJoh6QgWMEq0sVr5Qy8KoUOwpxJfgcbe9eRaxMMfHOJzXD5q5Qt2yZ4Iz4OTQgGlqrbyn147okkIQlNOlGo4dqybAyI1oxxGOS9REBPaIx1oGBQkBNUcTBYa4QPjtHEQSXOExhP398SAhEr1Q990hkR31WxtbP5XayQ6OG0OmIgTDYJOHwoSjnWEx+ngNpNANe8bIFQy81dMu0QSqk2GOROCM7vyPLjl0lnJuTkuVK7SNLJoD+2jInLQCaqgS1RFLqLoAT2hF/RqPVrP1pv1Pm3NWOnMLvoj6+MbsJaaqw==</latexit>
8(i, j) 2 !, |Ri,j
ˆRi,j|  D⌦( ˆR)<latexit sha1_base64="QKSqsgmGqwvE3kBA619AMM23eWA=">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</latexit><latexit sha1_base64="QKSqsgmGqwvE3kBA619AMM23eWA=">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</latexit><latexit sha1_base64="QKSqsgmGqwvE3kBA619AMM23eWA=">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</latexit><latexit sha1_base64="QKSqsgmGqwvE3kBA619AMM23eWA=">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</latexit>
⌦0
= ⌦ !<latexit sha1_base64="0HlFpR/iKUHIuv/mNYBPEb5sQhI=">AAACAXicbZBPS8MwGMbT+W/Of1VP4iU4RC+OVgT1IAy8eNIJ1g3WMtIs3cKStCSpMMrw4lfx4kHFq9/Cm9/GdOtBNx8I/Hje9+XN+4QJo0o7zrdVmptfWFwqL1dWVtfWN+zNrXsVpxITD8cslq0QKcKoIJ6mmpFWIgniISPNcHCZ15sPRCoaizs9TEjAUU/QiGKkjdWxd/wbTnroAF7ACcEj6Mc5dOyqU3PGgrPgFlAFhRod+8vvxjjlRGjMkFJt10l0kCGpKWZkVPFTRRKEB6hH2gYF4kQF2fiEEdw3ThdGsTRPaDh2f09kiCs15KHp5Ej31XQtN/+rtVMdnQUZFUmqicCTRVHKoI5hngfsUkmwZkMDCEtq/gpxH0mEtUmtYkJwp0+eBe+4dl5zb0+q9esijTLYBXvgELjgFNTBFWgAD2DwCJ7BK3iznqwX6936mLSWrGJmG/yR9fkDChSVhQ==</latexit><latexit sha1_base64="0HlFpR/iKUHIuv/mNYBPEb5sQhI=">AAACAXicbZBPS8MwGMbT+W/Of1VP4iU4RC+OVgT1IAy8eNIJ1g3WMtIs3cKStCSpMMrw4lfx4kHFq9/Cm9/GdOtBNx8I/Hje9+XN+4QJo0o7zrdVmptfWFwqL1dWVtfWN+zNrXsVpxITD8cslq0QKcKoIJ6mmpFWIgniISPNcHCZ15sPRCoaizs9TEjAUU/QiGKkjdWxd/wbTnroAF7ACcEj6Mc5dOyqU3PGgrPgFlAFhRod+8vvxjjlRGjMkFJt10l0kCGpKWZkVPFTRRKEB6hH2gYF4kQF2fiEEdw3ThdGsTRPaDh2f09kiCs15KHp5Ej31XQtN/+rtVMdnQUZFUmqicCTRVHKoI5hngfsUkmwZkMDCEtq/gpxH0mEtUmtYkJwp0+eBe+4dl5zb0+q9esijTLYBXvgELjgFNTBFWgAD2DwCJ7BK3iznqwX6936mLSWrGJmG/yR9fkDChSVhQ==</latexit><latexit sha1_base64="0HlFpR/iKUHIuv/mNYBPEb5sQhI=">AAACAXicbZBPS8MwGMbT+W/Of1VP4iU4RC+OVgT1IAy8eNIJ1g3WMtIs3cKStCSpMMrw4lfx4kHFq9/Cm9/GdOtBNx8I/Hje9+XN+4QJo0o7zrdVmptfWFwqL1dWVtfWN+zNrXsVpxITD8cslq0QKcKoIJ6mmpFWIgniISPNcHCZ15sPRCoaizs9TEjAUU/QiGKkjdWxd/wbTnroAF7ACcEj6Mc5dOyqU3PGgrPgFlAFhRod+8vvxjjlRGjMkFJt10l0kCGpKWZkVPFTRRKEB6hH2gYF4kQF2fiEEdw3ThdGsTRPaDh2f09kiCs15KHp5Ej31XQtN/+rtVMdnQUZFUmqicCTRVHKoI5hngfsUkmwZkMDCEtq/gpxH0mEtUmtYkJwp0+eBe+4dl5zb0+q9esijTLYBXvgELjgFNTBFWgAD2DwCJ7BK3iznqwX6936mLSWrGJmG/yR9fkDChSVhQ==</latexit><latexit sha1_base64="0HlFpR/iKUHIuv/mNYBPEb5sQhI=">AAACAXicbZBPS8MwGMbT+W/Of1VP4iU4RC+OVgT1IAy8eNIJ1g3WMtIs3cKStCSpMMrw4lfx4kHFq9/Cm9/GdOtBNx8I/Hje9+XN+4QJo0o7zrdVmptfWFwqL1dWVtfWN+zNrXsVpxITD8cslq0QKcKoIJ6mmpFWIgniISPNcHCZ15sPRCoaizs9TEjAUU/QiGKkjdWxd/wbTnroAF7ACcEj6Mc5dOyqU3PGgrPgFlAFhRod+8vvxjjlRGjMkFJt10l0kCGpKWZkVPFTRRKEB6hH2gYF4kQF2fiEEdw3ThdGsTRPaDh2f09kiCs15KHp5Ej31XQtN/+rtVMdnQUZFUmqicCTRVHKoI5hngfsUkmwZkMDCEtq/gpxH0mEtUmtYkJwp0+eBe+4dl5zb0+q9esijTLYBXvgELjgFNTBFWgAD2DwCJ7BK3iznqwX6936mLSWrGJmG/yR9fkDChSVhQ==</latexit>
1<latexit sha1_base64="r9UQgW05utQ0BxEYAZEYM48s6kw=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG8FL56kgrGFNpTNZtou3Wzi7kYooX/CiwcVr/4eb/4bt20O2vpg4PHeDDPzwlRwbVz32ymtrK6tb5Q3K1vbO7t71f2DB51kiqHPEpGodkg1Ci7RN9wIbKcKaRwKbIWj66nfekKleSLvzTjFIKYDyfucUWOldjdCYWjP61Vrbt2dgSwTryA1KNDsVb+6UcKyGKVhgmrd8dzUBDlVhjOBk0o305hSNqID7FgqaYw6yGf3TsiJVSLST5QtachM/T2R01jrcRzazpiaoV70puJ/Xicz/csg5zLNDEo2X9TPBDEJmT5PIq6QGTG2hDLF7a2EDamizNiIKjYEb/HlZeKf1a/q3t15rXFbpFGGIziGU/DgAhpwA03wgYGAZ3iFN+fReXHenY95a8kpZg7hD5zPHyphj6A=</latexit><latexit sha1_base64="r9UQgW05utQ0BxEYAZEYM48s6kw=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG8FL56kgrGFNpTNZtou3Wzi7kYooX/CiwcVr/4eb/4bt20O2vpg4PHeDDPzwlRwbVz32ymtrK6tb5Q3K1vbO7t71f2DB51kiqHPEpGodkg1Ci7RN9wIbKcKaRwKbIWj66nfekKleSLvzTjFIKYDyfucUWOldjdCYWjP61Vrbt2dgSwTryA1KNDsVb+6UcKyGKVhgmrd8dzUBDlVhjOBk0o305hSNqID7FgqaYw6yGf3TsiJVSLST5QtachM/T2R01jrcRzazpiaoV70puJ/Xicz/csg5zLNDEo2X9TPBDEJmT5PIq6QGTG2hDLF7a2EDamizNiIKjYEb/HlZeKf1a/q3t15rXFbpFGGIziGU/DgAhpwA03wgYGAZ3iFN+fReXHenY95a8kpZg7hD5zPHyphj6A=</latexit><latexit sha1_base64="r9UQgW05utQ0BxEYAZEYM48s6kw=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG8FL56kgrGFNpTNZtou3Wzi7kYooX/CiwcVr/4eb/4bt20O2vpg4PHeDDPzwlRwbVz32ymtrK6tb5Q3K1vbO7t71f2DB51kiqHPEpGodkg1Ci7RN9wIbKcKaRwKbIWj66nfekKleSLvzTjFIKYDyfucUWOldjdCYWjP61Vrbt2dgSwTryA1KNDsVb+6UcKyGKVhgmrd8dzUBDlVhjOBk0o305hSNqID7FgqaYw6yGf3TsiJVSLST5QtachM/T2R01jrcRzazpiaoV70puJ/Xicz/csg5zLNDEo2X9TPBDEJmT5PIq6QGTG2hDLF7a2EDamizNiIKjYEb/HlZeKf1a/q3t15rXFbpFGGIziGU/DgAhpwA03wgYGAZ3iFN+fReXHenY95a8kpZg7hD5zPHyphj6A=</latexit><latexit sha1_base64="r9UQgW05utQ0BxEYAZEYM48s6kw=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG8FL56kgrGFNpTNZtou3Wzi7kYooX/CiwcVr/4eb/4bt20O2vpg4PHeDDPzwlRwbVz32ymtrK6tb5Q3K1vbO7t71f2DB51kiqHPEpGodkg1Ci7RN9wIbKcKaRwKbIWj66nfekKleSLvzTjFIKYDyfucUWOldjdCYWjP61Vrbt2dgSwTryA1KNDsVb+6UcKyGKVhgmrd8dzUBDlVhjOBk0o305hSNqID7FgqaYw6yGf3TsiJVSLST5QtachM/T2R01jrcRzazpiaoV70puJ/Xicz/csg5zLNDEo2X9TPBDEJmT5PIq6QGTG2hDLF7a2EDamizNiIKjYEb/HlZeKf1a/q3t15rXFbpFGGIziGU/DgAhpwA03wgYGAZ3iFN+fReXHenY95a8kpZg7hD5zPHyphj6A=</latexit>
2<latexit sha1_base64="lbsE7NtCbH6B192W/4i6YwyXzIU=">AAAB7nicbVBNS8NAEJ34WetX1aOXxSJ4KkkR1FvBiyepYG2hDWWzmbRLN5u4uxFK6J/w4kHFq7/Hm//GbZuDtj4YeLw3w8y8IBVcG9f9dlZW19Y3Nktb5e2d3b39ysHhg04yxbDFEpGoTkA1Ci6xZbgR2EkV0jgQ2A5G11O//YRK80Tem3GKfkwHkkecUWOlTi9EYWi/3q9U3Zo7A1kmXkGqUKDZr3z1woRlMUrDBNW667mp8XOqDGcCJ+VepjGlbEQH2LVU0hi1n8/unZBTq4QkSpQtachM/T2R01jrcRzYzpiaoV70puJ/Xjcz0aWfc5lmBiWbL4oyQUxCps+TkCtkRowtoUxxeythQ6ooMzaisg3BW3x5mbTqtauad3debdwWaZTgGE7gDDy4gAbcQBNawEDAM7zCm/PovDjvzse8dcUpZo7gD5zPHyvkj6E=</latexit><latexit sha1_base64="lbsE7NtCbH6B192W/4i6YwyXzIU=">AAAB7nicbVBNS8NAEJ34WetX1aOXxSJ4KkkR1FvBiyepYG2hDWWzmbRLN5u4uxFK6J/w4kHFq7/Hm//GbZuDtj4YeLw3w8y8IBVcG9f9dlZW19Y3Nktb5e2d3b39ysHhg04yxbDFEpGoTkA1Ci6xZbgR2EkV0jgQ2A5G11O//YRK80Tem3GKfkwHkkecUWOlTi9EYWi/3q9U3Zo7A1kmXkGqUKDZr3z1woRlMUrDBNW667mp8XOqDGcCJ+VepjGlbEQH2LVU0hi1n8/unZBTq4QkSpQtachM/T2R01jrcRzYzpiaoV70puJ/Xjcz0aWfc5lmBiWbL4oyQUxCps+TkCtkRowtoUxxeythQ6ooMzaisg3BW3x5mbTqtauad3debdwWaZTgGE7gDDy4gAbcQBNawEDAM7zCm/PovDjvzse8dcUpZo7gD5zPHyvkj6E=</latexit><latexit sha1_base64="lbsE7NtCbH6B192W/4i6YwyXzIU=">AAAB7nicbVBNS8NAEJ34WetX1aOXxSJ4KkkR1FvBiyepYG2hDWWzmbRLN5u4uxFK6J/w4kHFq7/Hm//GbZuDtj4YeLw3w8y8IBVcG9f9dlZW19Y3Nktb5e2d3b39ysHhg04yxbDFEpGoTkA1Ci6xZbgR2EkV0jgQ2A5G11O//YRK80Tem3GKfkwHkkecUWOlTi9EYWi/3q9U3Zo7A1kmXkGqUKDZr3z1woRlMUrDBNW667mp8XOqDGcCJ+VepjGlbEQH2LVU0hi1n8/unZBTq4QkSpQtachM/T2R01jrcRzYzpiaoV70puJ/Xjcz0aWfc5lmBiWbL4oyQUxCps+TkCtkRowtoUxxeythQ6ooMzaisg3BW3x5mbTqtauad3debdwWaZTgGE7gDDy4gAbcQBNawEDAM7zCm/PovDjvzse8dcUpZo7gD5zPHyvkj6E=</latexit><latexit sha1_base64="lbsE7NtCbH6B192W/4i6YwyXzIU=">AAAB7nicbVBNS8NAEJ34WetX1aOXxSJ4KkkR1FvBiyepYG2hDWWzmbRLN5u4uxFK6J/w4kHFq7/Hm//GbZuDtj4YeLw3w8y8IBVcG9f9dlZW19Y3Nktb5e2d3b39ysHhg04yxbDFEpGoTkA1Ci6xZbgR2EkV0jgQ2A5G11O//YRK80Tem3GKfkwHkkecUWOlTi9EYWi/3q9U3Zo7A1kmXkGqUKDZr3z1woRlMUrDBNW667mp8XOqDGcCJ+VepjGlbEQH2LVU0hi1n8/unZBTq4QkSpQtachM/T2R01jrcRzYzpiaoV70puJ/Xjcz0aWfc5lmBiWbL4oyQUxCps+TkCtkRowtoUxxeythQ6ooMzaisg3BW3x5mbTqtauad3debdwWaZTgGE7gDDy4gAbcQBNawEDAM7zCm/PovDjvzse8dcUpZo7gD5zPHyvkj6E=</latexit>
15
• (Proof Sketch of Theorem 1)
• Let
• By Markov’s inequality
• Based on Hoeffding’s Lemma,
• Similar steps on
• Via simple calculations, we can prove Theorem 1.
KEY FINDING OF THIS PAPER
Hoeffding’s Lemma
• Let be random variable.
• E[X] = 0, and .
• Then, for any
Pr [X 2 [a, b]] = 1<latexit sha1_base64="vORvSu2v4VOymS2E5QVm5MFkhBw=">AAACCnicbVBNS8NAEN3Ur1q/oh69LC2CBymJCOpBKHjxJBWMLSShbLabdulmE3YnQim9e/GvePGg4tVf4M1/47bNQVsfDDzem2FmXpQJrsFxvq3S0vLK6lp5vbKxubW9Y+/u3es0V5R5NBWpakdEM8El84CDYO1MMZJEgrWiwdXEbz0wpXkq72CYsTAhPcljTgkYqWNXg6bCgWAx+LiNAy6xT46jEAeK9/oQ4kvsduyaU3emwIvELUgNFWh27K+gm9I8YRKoIFr7rpNBOCIKOBVsXAlyzTJCB6THfEMlSZgOR9NfxvjQKF0cp8qUBDxVf0+MSKL1MIlMZ0Kgr+e9ifif5+cQn4cjLrMcmKSzRXEuMKR4EgzucsUoiKEhhCpubsW0TxShYOKrmBDc+ZcXiXdSv6i7t6e1xk2RRhkdoCo6Qi46Qw10jZrIQxQ9omf0it6sJ+vFerc+Zq0lq5jZR39gff4AACaYtg==</latexit><latexit sha1_base64="vORvSu2v4VOymS2E5QVm5MFkhBw=">AAACCnicbVBNS8NAEN3Ur1q/oh69LC2CBymJCOpBKHjxJBWMLSShbLabdulmE3YnQim9e/GvePGg4tVf4M1/47bNQVsfDDzem2FmXpQJrsFxvq3S0vLK6lp5vbKxubW9Y+/u3es0V5R5NBWpakdEM8El84CDYO1MMZJEgrWiwdXEbz0wpXkq72CYsTAhPcljTgkYqWNXg6bCgWAx+LiNAy6xT46jEAeK9/oQ4kvsduyaU3emwIvELUgNFWh27K+gm9I8YRKoIFr7rpNBOCIKOBVsXAlyzTJCB6THfEMlSZgOR9NfxvjQKF0cp8qUBDxVf0+MSKL1MIlMZ0Kgr+e9ifif5+cQn4cjLrMcmKSzRXEuMKR4EgzucsUoiKEhhCpubsW0TxShYOKrmBDc+ZcXiXdSv6i7t6e1xk2RRhkdoCo6Qi46Qw10jZrIQxQ9omf0it6sJ+vFerc+Zq0lq5jZR39gff4AACaYtg==</latexit><latexit sha1_base64="vORvSu2v4VOymS2E5QVm5MFkhBw=">AAACCnicbVBNS8NAEN3Ur1q/oh69LC2CBymJCOpBKHjxJBWMLSShbLabdulmE3YnQim9e/GvePGg4tVf4M1/47bNQVsfDDzem2FmXpQJrsFxvq3S0vLK6lp5vbKxubW9Y+/u3es0V5R5NBWpakdEM8El84CDYO1MMZJEgrWiwdXEbz0wpXkq72CYsTAhPcljTgkYqWNXg6bCgWAx+LiNAy6xT46jEAeK9/oQ4kvsduyaU3emwIvELUgNFWh27K+gm9I8YRKoIFr7rpNBOCIKOBVsXAlyzTJCB6THfEMlSZgOR9NfxvjQKF0cp8qUBDxVf0+MSKL1MIlMZ0Kgr+e9ifif5+cQn4cjLrMcmKSzRXEuMKR4EgzucsUoiKEhhCpubsW0TxShYOKrmBDc+ZcXiXdSv6i7t6e1xk2RRhkdoCo6Qi46Qw10jZrIQxQ9omf0it6sJ+vFerc+Zq0lq5jZR39gff4AACaYtg==</latexit><latexit sha1_base64="vORvSu2v4VOymS2E5QVm5MFkhBw=">AAACCnicbVBNS8NAEN3Ur1q/oh69LC2CBymJCOpBKHjxJBWMLSShbLabdulmE3YnQim9e/GvePGg4tVf4M1/47bNQVsfDDzem2FmXpQJrsFxvq3S0vLK6lp5vbKxubW9Y+/u3es0V5R5NBWpakdEM8El84CDYO1MMZJEgrWiwdXEbz0wpXkq72CYsTAhPcljTgkYqWNXg6bCgWAx+LiNAy6xT46jEAeK9/oQ4kvsduyaU3emwIvELUgNFWh27K+gm9I8YRKoIFr7rpNBOCIKOBVsXAlyzTJCB6THfEMlSZgOR9NfxvjQKF0cp8qUBDxVf0+MSKL1MIlMZ0Kgr+e9ifif5+cQn4cjLrMcmKSzRXEuMKR4EgzucsUoiKEhhCpubsW0TxShYOKrmBDc+ZcXiXdSv6i7t6e1xk2RRhkdoCo6Qi46Qw10jZrIQxQ9omf0it6sJ+vFerc+Zq0lq5jZR39gff4AACaYtg==</latexit>
X 2 R<latexit sha1_base64="Up7QdIdytiotpNtWtEcWYrE7Zo4=">AAAB+HicbVBNS8NAFHypX7V+RT16WSyCp5KIUL0VvHiSKsYWmlA22027dLMJu5tCCf0nXjyoePWnePPfuGlz0NaBhWHmPd7shClnSjvOt1VZW9/Y3Kpu13Z29/YP7MOjJ5VkklCPJDyR3RArypmgnmaa024qKY5DTjvh+KbwOxMqFUvEo56mNIjxULCIEayN1LftLvKZQH6M9SgM84dZ3647DWcOtErcktShRLtvf/mDhGQxFZpwrFTPdVId5FhqRjid1fxM0RSTMR7SnqECx1QF+Tz5DJ0ZZYCiRJonNJqrvzdyHCs1jUMzWSRUy14h/uf1Mh1dBTkTaaapIItDUcaRTlBRAxowSYnmU0MwkcxkRWSEJSbalFUzJbjLX14l3kXjuuHeX9Zbd2UbVTiBUzgHF5rQgltogwcEJvAMr/Bm5daL9W59LEYrVrlzDH9gff4ATeSTCg==</latexit><latexit sha1_base64="Up7QdIdytiotpNtWtEcWYrE7Zo4=">AAAB+HicbVBNS8NAFHypX7V+RT16WSyCp5KIUL0VvHiSKsYWmlA22027dLMJu5tCCf0nXjyoePWnePPfuGlz0NaBhWHmPd7shClnSjvOt1VZW9/Y3Kpu13Z29/YP7MOjJ5VkklCPJDyR3RArypmgnmaa024qKY5DTjvh+KbwOxMqFUvEo56mNIjxULCIEayN1LftLvKZQH6M9SgM84dZ3647DWcOtErcktShRLtvf/mDhGQxFZpwrFTPdVId5FhqRjid1fxM0RSTMR7SnqECx1QF+Tz5DJ0ZZYCiRJonNJqrvzdyHCs1jUMzWSRUy14h/uf1Mh1dBTkTaaapIItDUcaRTlBRAxowSYnmU0MwkcxkRWSEJSbalFUzJbjLX14l3kXjuuHeX9Zbd2UbVTiBUzgHF5rQgltogwcEJvAMr/Bm5daL9W59LEYrVrlzDH9gff4ATeSTCg==</latexit><latexit sha1_base64="Up7QdIdytiotpNtWtEcWYrE7Zo4=">AAAB+HicbVBNS8NAFHypX7V+RT16WSyCp5KIUL0VvHiSKsYWmlA22027dLMJu5tCCf0nXjyoePWnePPfuGlz0NaBhWHmPd7shClnSjvOt1VZW9/Y3Kpu13Z29/YP7MOjJ5VkklCPJDyR3RArypmgnmaa024qKY5DTjvh+KbwOxMqFUvEo56mNIjxULCIEayN1LftLvKZQH6M9SgM84dZ3647DWcOtErcktShRLtvf/mDhGQxFZpwrFTPdVId5FhqRjid1fxM0RSTMR7SnqECx1QF+Tz5DJ0ZZYCiRJonNJqrvzdyHCs1jUMzWSRUy14h/uf1Mh1dBTkTaaapIItDUcaRTlBRAxowSYnmU0MwkcxkRWSEJSbalFUzJbjLX14l3kXjuuHeX9Zbd2UbVTiBUzgHF5rQgltogwcEJvAMr/Bm5daL9W59LEYrVrlzDH9gff4ATeSTCg==</latexit><latexit sha1_base64="Up7QdIdytiotpNtWtEcWYrE7Zo4=">AAAB+HicbVBNS8NAFHypX7V+RT16WSyCp5KIUL0VvHiSKsYWmlA22027dLMJu5tCCf0nXjyoePWnePPfuGlz0NaBhWHmPd7shClnSjvOt1VZW9/Y3Kpu13Z29/YP7MOjJ5VkklCPJDyR3RArypmgnmaa024qKY5DTjvh+KbwOxMqFUvEo56mNIjxULCIEayN1LftLvKZQH6M9SgM84dZ3647DWcOtErcktShRLtvf/mDhGQxFZpwrFTPdVId5FhqRjid1fxM0RSTMR7SnqECx1QF+Tz5DJ0ZZYCiRJonNJqrvzdyHCs1jUMzWSRUy14h/uf1Mh1dBTkTaaapIItDUcaRTlBRAxowSYnmU0MwkcxkRWSEJSbalFUzJbjLX14l3kXjuuHeX9Zbd2UbVTiBUzgHF5rQgltogwcEJvAMr/Bm5daL9W59LEYrVrlzDH9gff4ATeSTCg==</latexit>
16
• From Theorem 1, we know that minimizing 

is better than minimizing only.
• However, the condition of is 

not necessary.
• The conclusion will be the same if .
CONDITION RELAXATION
8(i, j) 2 !, |Ri,j
ˆRi,j|  D⌦( ˆR)<latexit sha1_base64="QKSqsgmGqwvE3kBA619AMM23eWA=">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</latexit><latexit sha1_base64="QKSqsgmGqwvE3kBA619AMM23eWA=">AAACPnicbVA9bxNBEN1LIARDghNKmhEWkiM51h2KFNJFSgoqCBFOInmt09x6bG+yt3fa3YtkXe6fpeEv0NHSUACipWTPuYJ8PGmlt2/maWZekitpXRh+C5aWHz1eebL6tPXs+dr6i/bG5onNCiNoIDKVmbMELSmpaeCkU3SWG8I0UXSaXBzU9dNLMlZm+rOb5zRKcarlRAp0XorbJ3ySGVQKurIH51vApQaepTTFHlzBcVzK3nkF28Bn6MrjqvlfAVcEPEU3E6jKwyrmH2tPt2nbitudsB8uAPdJ1JAOa3AUt7/ycSaKlLQTCq0dRmHuRiUaJ4WiqsULSzmKC5zS0FONKdlRubi/gjdeGYM/xD/tYKH+7ygxtXaeJr6zXtnerdXiQ7Vh4SbvRqXUeeFIi5tBk0KBy6AOE8bSkHBq7gkKI/2uIGZoUDgfecuHEN09+T4ZvO3v9aNPO539D00aq+wVe826LGK7bJ+9Z0dswAS7Zt/ZT/Yr+BL8CH4Hf25al4LG85LdQvD3H1bTrbQ=</latexit><latexit sha1_base64="QKSqsgmGqwvE3kBA619AMM23eWA=">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</latexit><latexit sha1_base64="QKSqsgmGqwvE3kBA619AMM23eWA=">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</latexit>
• Let (a set of easily predictable entries) which satisfies that
• Let (a set of entries hard to predict) .
• For any
• are -stable and -stable
respectively.
• Then,
Proposition 1
! ⇢ ⌦(|⌦| > 2)<latexit sha1_base64="ACA68GZluiyAFA/yKy15tzEFgIo=">AAACD3icbZDNSgMxFIUz9a/Wv6pLN8Ei1k2ZKYK6kYIbV1rBsYVOKZn0ThuayQxJRihtH8GNr+LGhYpbt+58G9N2BG09EPhy7r0k9/gxZ0rb9peVWVhcWl7JrubW1jc2t/LbO3cqSiQFl0Y8knWfKOBMgKuZ5lCPJZDQ51Dzexfjeu0epGKRuNX9GJoh6QgWMEq0sVr5Qy8KoUOwpxJfgcbe9eRaxMMfHOJzXD5q5Qt2yZ4Iz4OTQgGlqrbyn147okkIQlNOlGo4dqybAyI1oxxGOS9REBPaIx1oGBQkBNUcTBYa4QPjtHEQSXOExhP398SAhEr1Q990hkR31WxtbP5XayQ6OG0OmIgTDYJOHwoSjnWEx+ngNpNANe8bIFQy81dMu0QSqk2GOROCM7vyPLjl0lnJuTkuVK7SNLJoD+2jInLQCaqgS1RFLqLoAT2hF/RqPVrP1pv1Pm3NWOnMLvoj6+MbsJaaqw==</latexit><latexit sha1_base64="ACA68GZluiyAFA/yKy15tzEFgIo=">AAACD3icbZDNSgMxFIUz9a/Wv6pLN8Ei1k2ZKYK6kYIbV1rBsYVOKZn0ThuayQxJRihtH8GNr+LGhYpbt+58G9N2BG09EPhy7r0k9/gxZ0rb9peVWVhcWl7JrubW1jc2t/LbO3cqSiQFl0Y8knWfKOBMgKuZ5lCPJZDQ51Dzexfjeu0epGKRuNX9GJoh6QgWMEq0sVr5Qy8KoUOwpxJfgcbe9eRaxMMfHOJzXD5q5Qt2yZ4Iz4OTQgGlqrbyn147okkIQlNOlGo4dqybAyI1oxxGOS9REBPaIx1oGBQkBNUcTBYa4QPjtHEQSXOExhP398SAhEr1Q990hkR31WxtbP5XayQ6OG0OmIgTDYJOHwoSjnWEx+ngNpNANe8bIFQy81dMu0QSqk2GOROCM7vyPLjl0lnJuTkuVK7SNLJoD+2jInLQCaqgS1RFLqLoAT2hF/RqPVrP1pv1Pm3NWOnMLvoj6+MbsJaaqw==</latexit><latexit sha1_base64="ACA68GZluiyAFA/yKy15tzEFgIo=">AAACD3icbZDNSgMxFIUz9a/Wv6pLN8Ei1k2ZKYK6kYIbV1rBsYVOKZn0ThuayQxJRihtH8GNr+LGhYpbt+58G9N2BG09EPhy7r0k9/gxZ0rb9peVWVhcWl7JrubW1jc2t/LbO3cqSiQFl0Y8knWfKOBMgKuZ5lCPJZDQ51Dzexfjeu0epGKRuNX9GJoh6QgWMEq0sVr5Qy8KoUOwpxJfgcbe9eRaxMMfHOJzXD5q5Qt2yZ4Iz4OTQgGlqrbyn147okkIQlNOlGo4dqybAyI1oxxGOS9REBPaIx1oGBQkBNUcTBYa4QPjtHEQSXOExhP398SAhEr1Q990hkR31WxtbP5XayQ6OG0OmIgTDYJOHwoSjnWEx+ngNpNANe8bIFQy81dMu0QSqk2GOROCM7vyPLjl0lnJuTkuVK7SNLJoD+2jInLQCaqgS1RFLqLoAT2hF/RqPVrP1pv1Pm3NWOnMLvoj6+MbsJaaqw==</latexit><latexit sha1_base64="ACA68GZluiyAFA/yKy15tzEFgIo=">AAACD3icbZDNSgMxFIUz9a/Wv6pLN8Ei1k2ZKYK6kYIbV1rBsYVOKZn0ThuayQxJRihtH8GNr+LGhYpbt+58G9N2BG09EPhy7r0k9/gxZ0rb9peVWVhcWl7JrubW1jc2t/LbO3cqSiQFl0Y8knWfKOBMgKuZ5lCPJZDQ51Dzexfjeu0epGKRuNX9GJoh6QgWMEq0sVr5Qy8KoUOwpxJfgcbe9eRaxMMfHOJzXD5q5Qt2yZ4Iz4OTQgGlqrbyn147okkIQlNOlGo4dqybAyI1oxxGOS9REBPaIx1oGBQkBNUcTBYa4QPjtHEQSXOExhP398SAhEr1Q990hkR31WxtbP5XayQ6OG0OmIgTDYJOHwoSjnWEx+ngNpNANe8bIFQy81dMu0QSqk2GOROCM7vyPLjl0lnJuTkuVK7SNLJoD+2jInLQCaqgS1RFLqLoAT2hF/RqPVrP1pv1Pm3NWOnMLvoj6+MbsJaaqw==</latexit>
⌦0
= ⌦ !<latexit sha1_base64="0HlFpR/iKUHIuv/mNYBPEb5sQhI=">AAACAXicbZBPS8MwGMbT+W/Of1VP4iU4RC+OVgT1IAy8eNIJ1g3WMtIs3cKStCSpMMrw4lfx4kHFq9/Cm9/GdOtBNx8I/Hje9+XN+4QJo0o7zrdVmptfWFwqL1dWVtfWN+zNrXsVpxITD8cslq0QKcKoIJ6mmpFWIgniISPNcHCZ15sPRCoaizs9TEjAUU/QiGKkjdWxd/wbTnroAF7ACcEj6Mc5dOyqU3PGgrPgFlAFhRod+8vvxjjlRGjMkFJt10l0kCGpKWZkVPFTRRKEB6hH2gYF4kQF2fiEEdw3ThdGsTRPaDh2f09kiCs15KHp5Ej31XQtN/+rtVMdnQUZFUmqicCTRVHKoI5hngfsUkmwZkMDCEtq/gpxH0mEtUmtYkJwp0+eBe+4dl5zb0+q9esijTLYBXvgELjgFNTBFWgAD2DwCJ7BK3iznqwX6936mLSWrGJmG/yR9fkDChSVhQ==</latexit><latexit sha1_base64="0HlFpR/iKUHIuv/mNYBPEb5sQhI=">AAACAXicbZBPS8MwGMbT+W/Of1VP4iU4RC+OVgT1IAy8eNIJ1g3WMtIs3cKStCSpMMrw4lfx4kHFq9/Cm9/GdOtBNx8I/Hje9+XN+4QJo0o7zrdVmptfWFwqL1dWVtfWN+zNrXsVpxITD8cslq0QKcKoIJ6mmpFWIgniISPNcHCZ15sPRCoaizs9TEjAUU/QiGKkjdWxd/wbTnroAF7ACcEj6Mc5dOyqU3PGgrPgFlAFhRod+8vvxjjlRGjMkFJt10l0kCGpKWZkVPFTRRKEB6hH2gYF4kQF2fiEEdw3ThdGsTRPaDh2f09kiCs15KHp5Ej31XQtN/+rtVMdnQUZFUmqicCTRVHKoI5hngfsUkmwZkMDCEtq/gpxH0mEtUmtYkJwp0+eBe+4dl5zb0+q9esijTLYBXvgELjgFNTBFWgAD2DwCJ7BK3iznqwX6936mLSWrGJmG/yR9fkDChSVhQ==</latexit><latexit sha1_base64="0HlFpR/iKUHIuv/mNYBPEb5sQhI=">AAACAXicbZBPS8MwGMbT+W/Of1VP4iU4RC+OVgT1IAy8eNIJ1g3WMtIs3cKStCSpMMrw4lfx4kHFq9/Cm9/GdOtBNx8I/Hje9+XN+4QJo0o7zrdVmptfWFwqL1dWVtfWN+zNrXsVpxITD8cslq0QKcKoIJ6mmpFWIgniISPNcHCZ15sPRCoaizs9TEjAUU/QiGKkjdWxd/wbTnroAF7ACcEj6Mc5dOyqU3PGgrPgFlAFhRod+8vvxjjlRGjMkFJt10l0kCGpKWZkVPFTRRKEB6hH2gYF4kQF2fiEEdw3ThdGsTRPaDh2f09kiCs15KHp5Ej31XQtN/+rtVMdnQUZFUmqicCTRVHKoI5hngfsUkmwZkMDCEtq/gpxH0mEtUmtYkJwp0+eBe+4dl5zb0+q9esijTLYBXvgELjgFNTBFWgAD2DwCJ7BK3iznqwX6936mLSWrGJmG/yR9fkDChSVhQ==</latexit><latexit sha1_base64="0HlFpR/iKUHIuv/mNYBPEb5sQhI=">AAACAXicbZBPS8MwGMbT+W/Of1VP4iU4RC+OVgT1IAy8eNIJ1g3WMtIs3cKStCSpMMrw4lfx4kHFq9/Cm9/GdOtBNx8I/Hje9+XN+4QJo0o7zrdVmptfWFwqL1dWVtfWN+zNrXsVpxITD8cslq0QKcKoIJ6mmpFWIgniISPNcHCZ15sPRCoaizs9TEjAUU/QiGKkjdWxd/wbTnroAF7ACcEj6Mc5dOyqU3PGgrPgFlAFhRod+8vvxjjlRGjMkFJt10l0kCGpKWZkVPFTRRKEB6hH2gYF4kQF2fiEEdw3ThdGsTRPaDh2f09kiCs15KHp5Ej31XQtN/+rtVMdnQUZFUmqicCTRVHKoI5hngfsUkmwZkMDCEtq/gpxH0mEtUmtYkJwp0+eBe+4dl5zb0+q9esijTLYBXvgELjgFNTBFWgAD2DwCJ7BK3iznqwX6936mLSWrGJmG/yR9fkDChSVhQ==</latexit>
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← the only change from Theorem 1
17
• Theorem 1 and Proposition 1 only prove that 

it is beneficial to remove easily predictable entries from Ω.
• However, does not show how many entries we should remove.
• Theorem 2 shows that removing more entries that satisfy 





can yield better .
HOW MANY ENTRIES SHOULD WE REMOVE?
⌦0
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• Let satisfying
• Let .
• For any
• 

are -stable and -stable respectively.
• Then,
Theorem 2
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2<latexit sha1_base64="lbsE7NtCbH6B192W/4i6YwyXzIU=">AAAB7nicbVBNS8NAEJ34WetX1aOXxSJ4KkkR1FvBiyepYG2hDWWzmbRLN5u4uxFK6J/w4kHFq7/Hm//GbZuDtj4YeLw3w8y8IBVcG9f9dlZW19Y3Nktb5e2d3b39ysHhg04yxbDFEpGoTkA1Ci6xZbgR2EkV0jgQ2A5G11O//YRK80Tem3GKfkwHkkecUWOlTi9EYWi/3q9U3Zo7A1kmXkGqUKDZr3z1woRlMUrDBNW667mp8XOqDGcCJ+VepjGlbEQH2LVU0hi1n8/unZBTq4QkSpQtachM/T2R01jrcRzYzpiaoV70puJ/Xjcz0aWfc5lmBiWbL4oyQUxCps+TkCtkRowtoUxxeythQ6ooMzaisg3BW3x5mbTqtauad3debdwWaZTgGE7gDDy4gAbcQBNawEDAM7zCm/PovDjvzse8dcUpZo7gD5zPHyvkj6E=</latexit><latexit sha1_base64="lbsE7NtCbH6B192W/4i6YwyXzIU=">AAAB7nicbVBNS8NAEJ34WetX1aOXxSJ4KkkR1FvBiyepYG2hDWWzmbRLN5u4uxFK6J/w4kHFq7/Hm//GbZuDtj4YeLw3w8y8IBVcG9f9dlZW19Y3Nktb5e2d3b39ysHhg04yxbDFEpGoTkA1Ci6xZbgR2EkV0jgQ2A5G11O//YRK80Tem3GKfkwHkkecUWOlTi9EYWi/3q9U3Zo7A1kmXkGqUKDZr3z1woRlMUrDBNW667mp8XOqDGcCJ+VepjGlbEQH2LVU0hi1n8/unZBTq4QkSpQtachM/T2R01jrcRzYzpiaoV70puJ/Xjcz0aWfc5lmBiWbL4oyQUxCps+TkCtkRowtoUxxeythQ6ooMzaisg3BW3x5mbTqtauad3debdwWaZTgGE7gDDy4gAbcQBNawEDAM7zCm/PovDjvzse8dcUpZo7gD5zPHyvkj6E=</latexit><latexit sha1_base64="lbsE7NtCbH6B192W/4i6YwyXzIU=">AAAB7nicbVBNS8NAEJ34WetX1aOXxSJ4KkkR1FvBiyepYG2hDWWzmbRLN5u4uxFK6J/w4kHFq7/Hm//GbZuDtj4YeLw3w8y8IBVcG9f9dlZW19Y3Nktb5e2d3b39ysHhg04yxbDFEpGoTkA1Ci6xZbgR2EkV0jgQ2A5G11O//YRK80Tem3GKfkwHkkecUWOlTi9EYWi/3q9U3Zo7A1kmXkGqUKDZr3z1woRlMUrDBNW667mp8XOqDGcCJ+VepjGlbEQH2LVU0hi1n8/unZBTq4QkSpQtachM/T2R01jrcRzYzpiaoV70puJ/Xjcz0aWfc5lmBiWbL4oyQUxCps+TkCtkRowtoUxxeythQ6ooMzaisg3BW3x5mbTqtauad3debdwWaZTgGE7gDDy4gAbcQBNawEDAM7zCm/PovDjvzse8dcUpZo7gD5zPHyvkj6E=</latexit><latexit sha1_base64="lbsE7NtCbH6B192W/4i6YwyXzIU=">AAAB7nicbVBNS8NAEJ34WetX1aOXxSJ4KkkR1FvBiyepYG2hDWWzmbRLN5u4uxFK6J/w4kHFq7/Hm//GbZuDtj4YeLw3w8y8IBVcG9f9dlZW19Y3Nktb5e2d3b39ysHhg04yxbDFEpGoTkA1Ci6xZbgR2EkV0jgQ2A5G11O//YRK80Tem3GKfkwHkkecUWOlTi9EYWi/3q9U3Zo7A1kmXkGqUKDZr3z1woRlMUrDBNW667mp8XOqDGcCJ+VepjGlbEQH2LVU0hi1n8/unZBTq4QkSpQtachM/T2R01jrcRzYzpiaoV70puJ/Xjcz0aWfc5lmBiWbL4oyQUxCps+TkCtkRowtoUxxeythQ6ooMzaisg3BW3x5mbTqtauad3debdwWaZTgGE7gDDy4gAbcQBNawEDAM7zCm/PovDjvzse8dcUpZo7gD5zPHyvkj6E=</latexit>
0D⌦( ˆR) + 1D⌦1
( ˆR) and 0D⌦( ˆR) + 1D⌦2
( ˆR)<latexit sha1_base64="kOdjHQRCJB+i2IYJSNyA74G44kg=">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</latexit><latexit sha1_base64="kOdjHQRCJB+i2IYJSNyA74G44kg=">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</latexit><latexit sha1_base64="kOdjHQRCJB+i2IYJSNyA74G44kg=">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</latexit><latexit sha1_base64="kOdjHQRCJB+i2IYJSNyA74G44kg=">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</latexit>
!2 ⇢ !1 ⇢ ⌦(|⌦| > 2)<latexit sha1_base64="g6OuGVQieyuSOkME+GdnCpGsQUk=">AAACInicbZDLSgMxFIYzXmu9VV26CRahbspMEbwspODGlVZwbKEdhkx62oYmM0OSEUrbd3Hjq7hx4W0l+DCm7Yja+kPg5zvncHL+IOZMadv+sObmFxaXljMr2dW19Y3N3Nb2rYoSScGlEY9kLSAKOAvB1UxzqMUSiAg4VIPu+ahevQOpWBTe6F4MniDtkLUYJdogP3faiAS0iV/CDZUECjROgfMDrkYAF/Dg2w7wGS4d+Lm8XbTHwrPGSU0epar4ubdGM6KJgFBTTpSqO3asvT6RmlEOw2wjURAT2iVtqBsbEgHK649vHOJ9Q5q4FUnzQo3H9PdEnwileiIwnYLojpqujeB/tXqiW8den4VxoiGkk0WthGMd4VFguMkkUM17xhAqmfkrph0iCdUm1qwJwZk+eda4peJJ0bk+zJcv0zQyaBftoQJy0BEqowtUQS6i6B49omf0Yj1YT9ar9T5pnbPSmR30R9bnF+Y+oiE=</latexit><latexit sha1_base64="g6OuGVQieyuSOkME+GdnCpGsQUk=">AAACInicbZDLSgMxFIYzXmu9VV26CRahbspMEbwspODGlVZwbKEdhkx62oYmM0OSEUrbd3Hjq7hx4W0l+DCm7Yja+kPg5zvncHL+IOZMadv+sObmFxaXljMr2dW19Y3N3Nb2rYoSScGlEY9kLSAKOAvB1UxzqMUSiAg4VIPu+ahevQOpWBTe6F4MniDtkLUYJdogP3faiAS0iV/CDZUECjROgfMDrkYAF/Dg2w7wGS4d+Lm8XbTHwrPGSU0epar4ubdGM6KJgFBTTpSqO3asvT6RmlEOw2wjURAT2iVtqBsbEgHK649vHOJ9Q5q4FUnzQo3H9PdEnwileiIwnYLojpqujeB/tXqiW8den4VxoiGkk0WthGMd4VFguMkkUM17xhAqmfkrph0iCdUm1qwJwZk+eda4peJJ0bk+zJcv0zQyaBftoQJy0BEqowtUQS6i6B49omf0Yj1YT9ar9T5pnbPSmR30R9bnF+Y+oiE=</latexit><latexit sha1_base64="g6OuGVQieyuSOkME+GdnCpGsQUk=">AAACInicbZDLSgMxFIYzXmu9VV26CRahbspMEbwspODGlVZwbKEdhkx62oYmM0OSEUrbd3Hjq7hx4W0l+DCm7Yja+kPg5zvncHL+IOZMadv+sObmFxaXljMr2dW19Y3N3Nb2rYoSScGlEY9kLSAKOAvB1UxzqMUSiAg4VIPu+ahevQOpWBTe6F4MniDtkLUYJdogP3faiAS0iV/CDZUECjROgfMDrkYAF/Dg2w7wGS4d+Lm8XbTHwrPGSU0epar4ubdGM6KJgFBTTpSqO3asvT6RmlEOw2wjURAT2iVtqBsbEgHK649vHOJ9Q5q4FUnzQo3H9PdEnwileiIwnYLojpqujeB/tXqiW8den4VxoiGkk0WthGMd4VFguMkkUM17xhAqmfkrph0iCdUm1qwJwZk+eda4peJJ0bk+zJcv0zQyaBftoQJy0BEqowtUQS6i6B49omf0Yj1YT9ar9T5pnbPSmR30R9bnF+Y+oiE=</latexit><latexit sha1_base64="g6OuGVQieyuSOkME+GdnCpGsQUk=">AAACInicbZDLSgMxFIYzXmu9VV26CRahbspMEbwspODGlVZwbKEdhkx62oYmM0OSEUrbd3Hjq7hx4W0l+DCm7Yja+kPg5zvncHL+IOZMadv+sObmFxaXljMr2dW19Y3N3Nb2rYoSScGlEY9kLSAKOAvB1UxzqMUSiAg4VIPu+ahevQOpWBTe6F4MniDtkLUYJdogP3faiAS0iV/CDZUECjROgfMDrkYAF/Dg2w7wGS4d+Lm8XbTHwrPGSU0epar4ubdGM6KJgFBTTpSqO3asvT6RmlEOw2wjURAT2iVtqBsbEgHK649vHOJ9Q5q4FUnzQo3H9PdEnwileiIwnYLojpqujeB/tXqiW8den4VxoiGkk0WthGMd4VFguMkkUM17xhAqmfkrph0iCdUm1qwJwZk+eda4peJJ0bk+zJcv0zQyaBftoQJy0BEqowtUQS6i6B49omf0Yj1YT9ar9T5pnbPSmR30R9bnF+Y+oiE=</latexit>
8(i, j) 2 !1, |Ri,j
ˆRi,j|  D⌦( ˆR)<latexit sha1_base64="gFltSJ3DgwCENwT3Lz9ebI4kOYs=">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</latexit><latexit sha1_base64="gFltSJ3DgwCENwT3Lz9ebI4kOYs=">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</latexit><latexit sha1_base64="gFltSJ3DgwCENwT3Lz9ebI4kOYs=">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</latexit><latexit sha1_base64="gFltSJ3DgwCENwT3Lz9ebI4kOYs=">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</latexit>
⌦1 = ⌦ !1 and ⌦2 = ⌦ !2<latexit sha1_base64="Z30TRLfrkJpbN38hWvhmtcZTKGk=">AAACLnicbVBNSwMxFMz6bf2qevQSLIIXy24R1IMgCOJJK1grdMuSTV/bYDa7JG/FsvQfefGv9CKi4tWfYdruQW0HAsPMPF7ehIkUBl33zZmZnZtfWFxaLqysrq1vFDe37kycag41HstY34fMgBQKaihQwn2igUWhhHr4cD7064+gjYjVLfYSaEaso0RbcIZWCooX/nUEHRZ49JSOKT2gfpxrPsITZpSpFu3ndlCZkqwExZJbdkegk8TLSYnkqAbFgd+KeRqBQi6ZMQ3PTbCZMY2CS+gX/NRAwvgD60DDUsUiMM1sdG+f7lmlRduxtk8hHam/JzIWGdOLQpuMGHbNf28oTvMaKbaPm5lQSYqg+HhRO5UUYzosj7aEBo6yZwnjWti/Ut5lmnG0FRdsCd7/kydJrVI+KXs3h6Wzq7yNJbJDdsk+8cgROSOXpEpqhJNnMiDv5MN5cV6dT+drHJ1x8plt8gfO9w+dxaYH</latexit><latexit sha1_base64="Z30TRLfrkJpbN38hWvhmtcZTKGk=">AAACLnicbVBNSwMxFMz6bf2qevQSLIIXy24R1IMgCOJJK1grdMuSTV/bYDa7JG/FsvQfefGv9CKi4tWfYdruQW0HAsPMPF7ehIkUBl33zZmZnZtfWFxaLqysrq1vFDe37kycag41HstY34fMgBQKaihQwn2igUWhhHr4cD7064+gjYjVLfYSaEaso0RbcIZWCooX/nUEHRZ49JSOKT2gfpxrPsITZpSpFu3ndlCZkqwExZJbdkegk8TLSYnkqAbFgd+KeRqBQi6ZMQ3PTbCZMY2CS+gX/NRAwvgD60DDUsUiMM1sdG+f7lmlRduxtk8hHam/JzIWGdOLQpuMGHbNf28oTvMaKbaPm5lQSYqg+HhRO5UUYzosj7aEBo6yZwnjWti/Ut5lmnG0FRdsCd7/kydJrVI+KXs3h6Wzq7yNJbJDdsk+8cgROSOXpEpqhJNnMiDv5MN5cV6dT+drHJ1x8plt8gfO9w+dxaYH</latexit><latexit sha1_base64="Z30TRLfrkJpbN38hWvhmtcZTKGk=">AAACLnicbVBNSwMxFMz6bf2qevQSLIIXy24R1IMgCOJJK1grdMuSTV/bYDa7JG/FsvQfefGv9CKi4tWfYdruQW0HAsPMPF7ehIkUBl33zZmZnZtfWFxaLqysrq1vFDe37kycag41HstY34fMgBQKaihQwn2igUWhhHr4cD7064+gjYjVLfYSaEaso0RbcIZWCooX/nUEHRZ49JSOKT2gfpxrPsITZpSpFu3ndlCZkqwExZJbdkegk8TLSYnkqAbFgd+KeRqBQi6ZMQ3PTbCZMY2CS+gX/NRAwvgD60DDUsUiMM1sdG+f7lmlRduxtk8hHam/JzIWGdOLQpuMGHbNf28oTvMaKbaPm5lQSYqg+HhRO5UUYzosj7aEBo6yZwnjWti/Ut5lmnG0FRdsCd7/kydJrVI+KXs3h6Wzq7yNJbJDdsk+8cgROSOXpEpqhJNnMiDv5MN5cV6dT+drHJ1x8plt8gfO9w+dxaYH</latexit><latexit sha1_base64="Z30TRLfrkJpbN38hWvhmtcZTKGk=">AAACLnicbVBNSwMxFMz6bf2qevQSLIIXy24R1IMgCOJJK1grdMuSTV/bYDa7JG/FsvQfefGv9CKi4tWfYdruQW0HAsPMPF7ehIkUBl33zZmZnZtfWFxaLqysrq1vFDe37kycag41HstY34fMgBQKaihQwn2igUWhhHr4cD7064+gjYjVLfYSaEaso0RbcIZWCooX/nUEHRZ49JSOKT2gfpxrPsITZpSpFu3ndlCZkqwExZJbdkegk8TLSYnkqAbFgd+KeRqBQi6ZMQ3PTbCZMY2CS+gX/NRAwvgD60DDUsUiMM1sdG+f7lmlRduxtk8hHam/JzIWGdOLQpuMGHbNf28oTvMaKbaPm5lQSYqg+HhRO5UUYzosj7aEBo6yZwnjWti/Ut5lmnG0FRdsCd7/kydJrVI+KXs3h6Wzq7yNJbJDdsk+8cgROSOXpEpqhJNnMiDv5MN5cV6dT+drHJ1x8plt8gfO9w+dxaYH</latexit>
• Therefore, it is desirable to choose Ω’ as the whole set of entries which are
harder to predict than average: 8(i, j) 2 ⌦0
, |Ri,j
ˆRi,j| D⌦( ˆR)<latexit sha1_base64="uTKvNUe/g/dCRGh3SrM5IrowvdY=">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</latexit><latexit sha1_base64="uTKvNUe/g/dCRGh3SrM5IrowvdY=">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</latexit><latexit sha1_base64="uTKvNUe/g/dCRGh3SrM5IrowvdY=">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</latexit><latexit sha1_base64="uTKvNUe/g/dCRGh3SrM5IrowvdY=">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</latexit>
18
• Theorem 1 and Theorem 2 only consider choosing one Ω’ 

to find stable matrix approximations.
• Is it possible to choose more than one Ω’ that satisfy stable condition,
and yield more stable solutions by minimizing them all together.
• The Theorem 3 shows that incorporating K such entry sets (Ω’) 

will be more stable than incorporating any (K − 1) out of the K entry sets.
BETTER STABILITY BY MULTIPLE SUBSETS
• Let satisfying
• Let .
• For any
• and 

are -stable and -stable respectively.
• Then,
Theorem 3
1<latexit sha1_base64="r9UQgW05utQ0BxEYAZEYM48s6kw=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG8FL56kgrGFNpTNZtou3Wzi7kYooX/CiwcVr/4eb/4bt20O2vpg4PHeDDPzwlRwbVz32ymtrK6tb5Q3K1vbO7t71f2DB51kiqHPEpGodkg1Ci7RN9wIbKcKaRwKbIWj66nfekKleSLvzTjFIKYDyfucUWOldjdCYWjP61Vrbt2dgSwTryA1KNDsVb+6UcKyGKVhgmrd8dzUBDlVhjOBk0o305hSNqID7FgqaYw6yGf3TsiJVSLST5QtachM/T2R01jrcRzazpiaoV70puJ/Xicz/csg5zLNDEo2X9TPBDEJmT5PIq6QGTG2hDLF7a2EDamizNiIKjYEb/HlZeKf1a/q3t15rXFbpFGGIziGU/DgAhpwA03wgYGAZ3iFN+fReXHenY95a8kpZg7hD5zPHyphj6A=</latexit><latexit sha1_base64="r9UQgW05utQ0BxEYAZEYM48s6kw=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG8FL56kgrGFNpTNZtou3Wzi7kYooX/CiwcVr/4eb/4bt20O2vpg4PHeDDPzwlRwbVz32ymtrK6tb5Q3K1vbO7t71f2DB51kiqHPEpGodkg1Ci7RN9wIbKcKaRwKbIWj66nfekKleSLvzTjFIKYDyfucUWOldjdCYWjP61Vrbt2dgSwTryA1KNDsVb+6UcKyGKVhgmrd8dzUBDlVhjOBk0o305hSNqID7FgqaYw6yGf3TsiJVSLST5QtachM/T2R01jrcRzazpiaoV70puJ/Xicz/csg5zLNDEo2X9TPBDEJmT5PIq6QGTG2hDLF7a2EDamizNiIKjYEb/HlZeKf1a/q3t15rXFbpFGGIziGU/DgAhpwA03wgYGAZ3iFN+fReXHenY95a8kpZg7hD5zPHyphj6A=</latexit><latexit sha1_base64="r9UQgW05utQ0BxEYAZEYM48s6kw=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG8FL56kgrGFNpTNZtou3Wzi7kYooX/CiwcVr/4eb/4bt20O2vpg4PHeDDPzwlRwbVz32ymtrK6tb5Q3K1vbO7t71f2DB51kiqHPEpGodkg1Ci7RN9wIbKcKaRwKbIWj66nfekKleSLvzTjFIKYDyfucUWOldjdCYWjP61Vrbt2dgSwTryA1KNDsVb+6UcKyGKVhgmrd8dzUBDlVhjOBk0o305hSNqID7FgqaYw6yGf3TsiJVSLST5QtachM/T2R01jrcRzazpiaoV70puJ/Xicz/csg5zLNDEo2X9TPBDEJmT5PIq6QGTG2hDLF7a2EDamizNiIKjYEb/HlZeKf1a/q3t15rXFbpFGGIziGU/DgAhpwA03wgYGAZ3iFN+fReXHenY95a8kpZg7hD5zPHyphj6A=</latexit><latexit sha1_base64="r9UQgW05utQ0BxEYAZEYM48s6kw=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG8FL56kgrGFNpTNZtou3Wzi7kYooX/CiwcVr/4eb/4bt20O2vpg4PHeDDPzwlRwbVz32ymtrK6tb5Q3K1vbO7t71f2DB51kiqHPEpGodkg1Ci7RN9wIbKcKaRwKbIWj66nfekKleSLvzTjFIKYDyfucUWOldjdCYWjP61Vrbt2dgSwTryA1KNDsVb+6UcKyGKVhgmrd8dzUBDlVhjOBk0o305hSNqID7FgqaYw6yGf3TsiJVSLST5QtachM/T2R01jrcRzazpiaoV70puJ/Xicz/csg5zLNDEo2X9TPBDEJmT5PIq6QGTG2hDLF7a2EDamizNiIKjYEb/HlZeKf1a/q3t15rXFbpFGGIziGU/DgAhpwA03wgYGAZ3iFN+fReXHenY95a8kpZg7hD5zPHyphj6A=</latexit>
2<latexit sha1_base64="lbsE7NtCbH6B192W/4i6YwyXzIU=">AAAB7nicbVBNS8NAEJ34WetX1aOXxSJ4KkkR1FvBiyepYG2hDWWzmbRLN5u4uxFK6J/w4kHFq7/Hm//GbZuDtj4YeLw3w8y8IBVcG9f9dlZW19Y3Nktb5e2d3b39ysHhg04yxbDFEpGoTkA1Ci6xZbgR2EkV0jgQ2A5G11O//YRK80Tem3GKfkwHkkecUWOlTi9EYWi/3q9U3Zo7A1kmXkGqUKDZr3z1woRlMUrDBNW667mp8XOqDGcCJ+VepjGlbEQH2LVU0hi1n8/unZBTq4QkSpQtachM/T2R01jrcRzYzpiaoV70puJ/Xjcz0aWfc5lmBiWbL4oyQUxCps+TkCtkRowtoUxxeythQ6ooMzaisg3BW3x5mbTqtauad3debdwWaZTgGE7gDDy4gAbcQBNawEDAM7zCm/PovDjvzse8dcUpZo7gD5zPHyvkj6E=</latexit><latexit sha1_base64="lbsE7NtCbH6B192W/4i6YwyXzIU=">AAAB7nicbVBNS8NAEJ34WetX1aOXxSJ4KkkR1FvBiyepYG2hDWWzmbRLN5u4uxFK6J/w4kHFq7/Hm//GbZuDtj4YeLw3w8y8IBVcG9f9dlZW19Y3Nktb5e2d3b39ysHhg04yxbDFEpGoTkA1Ci6xZbgR2EkV0jgQ2A5G11O//YRK80Tem3GKfkwHkkecUWOlTi9EYWi/3q9U3Zo7A1kmXkGqUKDZr3z1woRlMUrDBNW667mp8XOqDGcCJ+VepjGlbEQH2LVU0hi1n8/unZBTq4QkSpQtachM/T2R01jrcRzYzpiaoV70puJ/Xjcz0aWfc5lmBiWbL4oyQUxCps+TkCtkRowtoUxxeythQ6ooMzaisg3BW3x5mbTqtauad3debdwWaZTgGE7gDDy4gAbcQBNawEDAM7zCm/PovDjvzse8dcUpZo7gD5zPHyvkj6E=</latexit><latexit sha1_base64="lbsE7NtCbH6B192W/4i6YwyXzIU=">AAAB7nicbVBNS8NAEJ34WetX1aOXxSJ4KkkR1FvBiyepYG2hDWWzmbRLN5u4uxFK6J/w4kHFq7/Hm//GbZuDtj4YeLw3w8y8IBVcG9f9dlZW19Y3Nktb5e2d3b39ysHhg04yxbDFEpGoTkA1Ci6xZbgR2EkV0jgQ2A5G11O//YRK80Tem3GKfkwHkkecUWOlTi9EYWi/3q9U3Zo7A1kmXkGqUKDZr3z1woRlMUrDBNW667mp8XOqDGcCJ+VepjGlbEQH2LVU0hi1n8/unZBTq4QkSpQtachM/T2R01jrcRzYzpiaoV70puJ/Xjcz0aWfc5lmBiWbL4oyQUxCps+TkCtkRowtoUxxeythQ6ooMzaisg3BW3x5mbTqtauad3debdwWaZTgGE7gDDy4gAbcQBNawEDAM7zCm/PovDjvzse8dcUpZo7gD5zPHyvkj6E=</latexit><latexit sha1_base64="lbsE7NtCbH6B192W/4i6YwyXzIU=">AAAB7nicbVBNS8NAEJ34WetX1aOXxSJ4KkkR1FvBiyepYG2hDWWzmbRLN5u4uxFK6J/w4kHFq7/Hm//GbZuDtj4YeLw3w8y8IBVcG9f9dlZW19Y3Nktb5e2d3b39ysHhg04yxbDFEpGoTkA1Ci6xZbgR2EkV0jgQ2A5G11O//YRK80Tem3GKfkwHkkecUWOlTi9EYWi/3q9U3Zo7A1kmXkGqUKDZr3z1woRlMUrDBNW667mp8XOqDGcCJ+VepjGlbEQH2LVU0hi1n8/unZBTq4QkSpQtachM/T2R01jrcRzYzpiaoV70puJ/Xjcz0aWfc5lmBiWbL4oyQUxCps+TkCtkRowtoUxxeythQ6ooMzaisg3BW3x5mbTqtauad3debdwWaZTgGE7gDDy4gAbcQBNawEDAM7zCm/PovDjvzse8dcUpZo7gD5zPHyvkj6E=</latexit>
!1, · · · , !K 2 ⌦(K > 1, |⌦| > 2)<latexit sha1_base64="jaqMqkdMJ9Mi/9gSPy/BJZpHmgs=">AAACJnicbVDLSgMxFM34rPU16tJNsAgVpMwUQd2UghuhoBUcW+iUIZNJ29BMMiQZobT9Gzf+ihvBB+LOTzF9CNp6IOTcc88luSdMGFXacT6thcWl5ZXVzFp2fWNza9ve2b1TIpWYeFgwIeshUoRRTjxNNSP1RBIUh4zUwu7FqF+7J1JRwW91LyHNGLU5bVGMtJECu+SLmLRR4B5DH0dCK3NPlAqEPuXQvx5VMF+BJWhMgx9hYOriUWDnnIIzBpwn7pTkwBTVwH7xI4HTmHCNGVKq4TqJbvaR1BQzMsz6qSIJwl3UJg1DOYqJavbHew7hoVEi2BLSHK7hWP090UexUr04NM4Y6Y6a7Y3E/3qNVLfOmn3Kk1QTjicPtVIGtYCj0GBEJcGa9QxBWFLzV4g7SCKsTbRZE4I7u/I88YqF84J7c5IrX03TyIB9cADywAWnoAwuQRV4AIMH8ARewZv1aD1b79bHxLpgTWf2wB9YX9/UPKHI</latexit><latexit sha1_base64="jaqMqkdMJ9Mi/9gSPy/BJZpHmgs=">AAACJnicbVDLSgMxFM34rPU16tJNsAgVpMwUQd2UghuhoBUcW+iUIZNJ29BMMiQZobT9Gzf+ihvBB+LOTzF9CNp6IOTcc88luSdMGFXacT6thcWl5ZXVzFp2fWNza9ve2b1TIpWYeFgwIeshUoRRTjxNNSP1RBIUh4zUwu7FqF+7J1JRwW91LyHNGLU5bVGMtJECu+SLmLRR4B5DH0dCK3NPlAqEPuXQvx5VMF+BJWhMgx9hYOriUWDnnIIzBpwn7pTkwBTVwH7xI4HTmHCNGVKq4TqJbvaR1BQzMsz6qSIJwl3UJg1DOYqJavbHew7hoVEi2BLSHK7hWP090UexUr04NM4Y6Y6a7Y3E/3qNVLfOmn3Kk1QTjicPtVIGtYCj0GBEJcGa9QxBWFLzV4g7SCKsTbRZE4I7u/I88YqF84J7c5IrX03TyIB9cADywAWnoAwuQRV4AIMH8ARewZv1aD1b79bHxLpgTWf2wB9YX9/UPKHI</latexit><latexit sha1_base64="jaqMqkdMJ9Mi/9gSPy/BJZpHmgs=">AAACJnicbVDLSgMxFM34rPU16tJNsAgVpMwUQd2UghuhoBUcW+iUIZNJ29BMMiQZobT9Gzf+ihvBB+LOTzF9CNp6IOTcc88luSdMGFXacT6thcWl5ZXVzFp2fWNza9ve2b1TIpWYeFgwIeshUoRRTjxNNSP1RBIUh4zUwu7FqF+7J1JRwW91LyHNGLU5bVGMtJECu+SLmLRR4B5DH0dCK3NPlAqEPuXQvx5VMF+BJWhMgx9hYOriUWDnnIIzBpwn7pTkwBTVwH7xI4HTmHCNGVKq4TqJbvaR1BQzMsz6qSIJwl3UJg1DOYqJavbHew7hoVEi2BLSHK7hWP090UexUr04NM4Y6Y6a7Y3E/3qNVLfOmn3Kk1QTjicPtVIGtYCj0GBEJcGa9QxBWFLzV4g7SCKsTbRZE4I7u/I88YqF84J7c5IrX03TyIB9cADywAWnoAwuQRV4AIMH8ARewZv1aD1b79bHxLpgTWf2wB9YX9/UPKHI</latexit><latexit sha1_base64="jaqMqkdMJ9Mi/9gSPy/BJZpHmgs=">AAACJnicbVDLSgMxFM34rPU16tJNsAgVpMwUQd2UghuhoBUcW+iUIZNJ29BMMiQZobT9Gzf+ihvBB+LOTzF9CNp6IOTcc88luSdMGFXacT6thcWl5ZXVzFp2fWNza9ve2b1TIpWYeFgwIeshUoRRTjxNNSP1RBIUh4zUwu7FqF+7J1JRwW91LyHNGLU5bVGMtJECu+SLmLRR4B5DH0dCK3NPlAqEPuXQvx5VMF+BJWhMgx9hYOriUWDnnIIzBpwn7pTkwBTVwH7xI4HTmHCNGVKq4TqJbvaR1BQzMsz6qSIJwl3UJg1DOYqJavbHew7hoVEi2BLSHK7hWP090UexUr04NM4Y6Y6a7Y3E/3qNVLfOmn3Kk1QTjicPtVIGtYCj0GBEJcGa9QxBWFLzV4g7SCKsTbRZE4I7u/I88YqF84J7c5IrX03TyIB9cADywAWnoAwuQRV4AIMH8ARewZv1aD1b79bHxLpgTWf2wB9YX9/UPKHI</latexit>
8(i, j) 2 !k (1  k  K), |Ri,j
ˆRi,j|  D⌦( ˆR)<latexit sha1_base64="HIytyRzg3PvALpKL6Vm4n4dLyRc=">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</latexit><latexit sha1_base64="HIytyRzg3PvALpKL6Vm4n4dLyRc=">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</latexit><latexit sha1_base64="HIytyRzg3PvALpKL6Vm4n4dLyRc=">AAACUXicbVFNbxMxEJ0sFEpaIMCRy4gIKZFCtIuQgFslOFRCoqUitFIcrWYdJ3Hj9a5sL1K03d+IVHHhh3DhQPFu99APRrL1/OaNPfOc5EpaF4a/OsGdu1v37m8/6O7sPnz0uPfk6TebFYaLCc9UZk4SskJJLSZOOiVOciMoTZQ4TtYf6vzxd2GszPRXt8nFLKWllgvJyXkq7km2yAwphQM5wtMhMqmRZalYUrxmOIiQKYHrZv80HOEZHsWlHJ1W+ArZilx5VLXns0bDUnIrTqr8WMXsoL5m0MqGca8fjsMm8DaIWtCHNg7j3jmbZ7xIhXZckbXTKMzdrCTjJFei6rLCipz4mpZi6qGmVNhZ2VhS4UvPzNHP5pd22LBXK0pKrd2kiVfWLdubuZr8X25auMW7WSl1Xjih+eVDi0Khy7D2F+fSCO7UxgPiRvpeka/IEHf+F7rehOjmyLfB5PX4/Tj68qa/97l1YxuewwsYQARvYQ/24RAmwOEH/Ia/cNH52fkTQBBcSoNOW/MMrkWw8w8INrDN</latexit><latexit sha1_base64="HIytyRzg3PvALpKL6Vm4n4dLyRc=">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</latexit>
⌦k = ⌦ !k (81  k  K)<latexit sha1_base64="DetsFyQWCyItUB9gRmXqpNsHL2I=">AAACHXicbVDLSsNAFJ3UV62vqEs3g0WoC0siiroQCm4EQStYW2hCmExv2qGTBzMToYR+iRt/xY0LFRduxL9x2mahrQdmOJxzL/fe4yecSWVZ30Zhbn5hcam4XFpZXVvfMDe37mWcCgoNGvNYtHwigbMIGoopDq1EAAl9Dk2/fzHymw8gJIujOzVIwA1JN2IBo0RpyTOPnZsQusTr43M8ofgAO/FEc3DFCWJBOMc2djjg/vi/2vfMslW1xsCzxM5JGeWoe+an04lpGkKkKCdStm0rUW5GhGKUw7DkpBISQvukC21NIxKCdLPxeUO8p5UO1ovoFyk8Vn93ZCSUchD6ujIkqienvZH4n9dOVXDqZixKUgURnQwKUo5VjEdZ4Q4TQBUfaEKoYHpXTHtEEKp0oiUdgj198ixpHFbPqvbtUbl2nadRRDtoF1WQjU5QDV2iOmogih7RM3pFb8aT8WK8Gx+T0oKR92yjPzC+fgAV+Z+U</latexit><latexit sha1_base64="DetsFyQWCyItUB9gRmXqpNsHL2I=">AAACHXicbVDLSsNAFJ3UV62vqEs3g0WoC0siiroQCm4EQStYW2hCmExv2qGTBzMToYR+iRt/xY0LFRduxL9x2mahrQdmOJxzL/fe4yecSWVZ30Zhbn5hcam4XFpZXVvfMDe37mWcCgoNGvNYtHwigbMIGoopDq1EAAl9Dk2/fzHymw8gJIujOzVIwA1JN2IBo0RpyTOPnZsQusTr43M8ofgAO/FEc3DFCWJBOMc2djjg/vi/2vfMslW1xsCzxM5JGeWoe+an04lpGkKkKCdStm0rUW5GhGKUw7DkpBISQvukC21NIxKCdLPxeUO8p5UO1ovoFyk8Vn93ZCSUchD6ujIkqienvZH4n9dOVXDqZixKUgURnQwKUo5VjEdZ4Q4TQBUfaEKoYHpXTHtEEKp0oiUdgj198ixpHFbPqvbtUbl2nadRRDtoF1WQjU5QDV2iOmogih7RM3pFb8aT8WK8Gx+T0oKR92yjPzC+fgAV+Z+U</latexit><latexit sha1_base64="DetsFyQWCyItUB9gRmXqpNsHL2I=">AAACHXicbVDLSsNAFJ3UV62vqEs3g0WoC0siiroQCm4EQStYW2hCmExv2qGTBzMToYR+iRt/xY0LFRduxL9x2mahrQdmOJxzL/fe4yecSWVZ30Zhbn5hcam4XFpZXVvfMDe37mWcCgoNGvNYtHwigbMIGoopDq1EAAl9Dk2/fzHymw8gJIujOzVIwA1JN2IBo0RpyTOPnZsQusTr43M8ofgAO/FEc3DFCWJBOMc2djjg/vi/2vfMslW1xsCzxM5JGeWoe+an04lpGkKkKCdStm0rUW5GhGKUw7DkpBISQvukC21NIxKCdLPxeUO8p5UO1ovoFyk8Vn93ZCSUchD6ujIkqienvZH4n9dOVXDqZixKUgURnQwKUo5VjEdZ4Q4TQBUfaEKoYHpXTHtEEKp0oiUdgj198ixpHFbPqvbtUbl2nadRRDtoF1WQjU5QDV2iOmogih7RM3pFb8aT8WK8Gx+T0oKR92yjPzC+fgAV+Z+U</latexit><latexit sha1_base64="DetsFyQWCyItUB9gRmXqpNsHL2I=">AAACHXicbVDLSsNAFJ3UV62vqEs3g0WoC0siiroQCm4EQStYW2hCmExv2qGTBzMToYR+iRt/xY0LFRduxL9x2mahrQdmOJxzL/fe4yecSWVZ30Zhbn5hcam4XFpZXVvfMDe37mWcCgoNGvNYtHwigbMIGoopDq1EAAl9Dk2/fzHymw8gJIujOzVIwA1JN2IBo0RpyTOPnZsQusTr43M8ofgAO/FEc3DFCWJBOMc2djjg/vi/2vfMslW1xsCzxM5JGeWoe+an04lpGkKkKCdStm0rUW5GhGKUw7DkpBISQvukC21NIxKCdLPxeUO8p5UO1ovoFyk8Vn93ZCSUchD6ujIkqienvZH4n9dOVXDqZixKUgURnQwKUo5VjEdZ4Q4TQBUfaEKoYHpXTHtEEKp0oiUdgj198ixpHFbPqvbtUbl2nadRRDtoF1WQjU5QDV2iOmogih7RM3pFb8aT8WK8Gx+T0oKR92yjPzC+fgAV+Z+U</latexit>
✏ > 0 and 1 > 0, 1, · · · , K > 0 ⌃K
i=0 i = 1<latexit sha1_base64="flCvYaRbQuEbUHLoFYsJBrl9VjM=">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</latexit><latexit sha1_base64="flCvYaRbQuEbUHLoFYsJBrl9VjM=">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</latexit><latexit sha1_base64="flCvYaRbQuEbUHLoFYsJBrl9VjM=">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</latexit><latexit sha1_base64="flCvYaRbQuEbUHLoFYsJBrl9VjM=">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</latexit>
19
• One of the Guidelines to obtain hard predictable subsets of Ω?
• 1) Choose satisfying

Let
• 2) Divide into K non-overlapping subsets with the condition that

, and let
• 3) Minimize
• Is it better than minimizing ?
• → Theorem 4
HOW TO FIND MULTIPLE SUBSETS?
8(i, j) 2 !, |Ri,j
ˆRi,j|  D⌦( ˆR)<latexit sha1_base64="QKSqsgmGqwvE3kBA619AMM23eWA=">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</latexit><latexit sha1_base64="QKSqsgmGqwvE3kBA619AMM23eWA=">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</latexit><latexit sha1_base64="QKSqsgmGqwvE3kBA619AMM23eWA=">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</latexit><latexit sha1_base64="QKSqsgmGqwvE3kBA619AMM23eWA=">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</latexit>
! ⇢ ⌦(|⌦| > 2)<latexit sha1_base64="ACA68GZluiyAFA/yKy15tzEFgIo=">AAACD3icbZDNSgMxFIUz9a/Wv6pLN8Ei1k2ZKYK6kYIbV1rBsYVOKZn0ThuayQxJRihtH8GNr+LGhYpbt+58G9N2BG09EPhy7r0k9/gxZ0rb9peVWVhcWl7JrubW1jc2t/LbO3cqSiQFl0Y8knWfKOBMgKuZ5lCPJZDQ51Dzexfjeu0epGKRuNX9GJoh6QgWMEq0sVr5Qy8KoUOwpxJfgcbe9eRaxMMfHOJzXD5q5Qt2yZ4Iz4OTQgGlqrbyn147okkIQlNOlGo4dqybAyI1oxxGOS9REBPaIx1oGBQkBNUcTBYa4QPjtHEQSXOExhP398SAhEr1Q990hkR31WxtbP5XayQ6OG0OmIgTDYJOHwoSjnWEx+ngNpNANe8bIFQy81dMu0QSqk2GOROCM7vyPLjl0lnJuTkuVK7SNLJoD+2jInLQCaqgS1RFLqLoAT2hF/RqPVrP1pv1Pm3NWOnMLvoj6+MbsJaaqw==</latexit><latexit sha1_base64="ACA68GZluiyAFA/yKy15tzEFgIo=">AAACD3icbZDNSgMxFIUz9a/Wv6pLN8Ei1k2ZKYK6kYIbV1rBsYVOKZn0ThuayQxJRihtH8GNr+LGhYpbt+58G9N2BG09EPhy7r0k9/gxZ0rb9peVWVhcWl7JrubW1jc2t/LbO3cqSiQFl0Y8knWfKOBMgKuZ5lCPJZDQ51Dzexfjeu0epGKRuNX9GJoh6QgWMEq0sVr5Qy8KoUOwpxJfgcbe9eRaxMMfHOJzXD5q5Qt2yZ4Iz4OTQgGlqrbyn147okkIQlNOlGo4dqybAyI1oxxGOS9REBPaIx1oGBQkBNUcTBYa4QPjtHEQSXOExhP398SAhEr1Q990hkR31WxtbP5XayQ6OG0OmIgTDYJOHwoSjnWEx+ngNpNANe8bIFQy81dMu0QSqk2GOROCM7vyPLjl0lnJuTkuVK7SNLJoD+2jInLQCaqgS1RFLqLoAT2hF/RqPVrP1pv1Pm3NWOnMLvoj6+MbsJaaqw==</latexit><latexit sha1_base64="ACA68GZluiyAFA/yKy15tzEFgIo=">AAACD3icbZDNSgMxFIUz9a/Wv6pLN8Ei1k2ZKYK6kYIbV1rBsYVOKZn0ThuayQxJRihtH8GNr+LGhYpbt+58G9N2BG09EPhy7r0k9/gxZ0rb9peVWVhcWl7JrubW1jc2t/LbO3cqSiQFl0Y8knWfKOBMgKuZ5lCPJZDQ51Dzexfjeu0epGKRuNX9GJoh6QgWMEq0sVr5Qy8KoUOwpxJfgcbe9eRaxMMfHOJzXD5q5Qt2yZ4Iz4OTQgGlqrbyn147okkIQlNOlGo4dqybAyI1oxxGOS9REBPaIx1oGBQkBNUcTBYa4QPjtHEQSXOExhP398SAhEr1Q990hkR31WxtbP5XayQ6OG0OmIgTDYJOHwoSjnWEx+ngNpNANe8bIFQy81dMu0QSqk2GOROCM7vyPLjl0lnJuTkuVK7SNLJoD+2jInLQCaqgS1RFLqLoAT2hF/RqPVrP1pv1Pm3NWOnMLvoj6+MbsJaaqw==</latexit><latexit sha1_base64="ACA68GZluiyAFA/yKy15tzEFgIo=">AAACD3icbZDNSgMxFIUz9a/Wv6pLN8Ei1k2ZKYK6kYIbV1rBsYVOKZn0ThuayQxJRihtH8GNr+LGhYpbt+58G9N2BG09EPhy7r0k9/gxZ0rb9peVWVhcWl7JrubW1jc2t/LbO3cqSiQFl0Y8knWfKOBMgKuZ5lCPJZDQ51Dzexfjeu0epGKRuNX9GJoh6QgWMEq0sVr5Qy8KoUOwpxJfgcbe9eRaxMMfHOJzXD5q5Qt2yZ4Iz4OTQgGlqrbyn147okkIQlNOlGo4dqybAyI1oxxGOS9REBPaIx1oGBQkBNUcTBYa4QPjtHEQSXOExhP398SAhEr1Q990hkR31WxtbP5XayQ6OG0OmIgTDYJOHwoSjnWEx+ngNpNANe8bIFQy81dMu0QSqk2GOROCM7vyPLjl0lnJuTkuVK7SNLJoD+2jInLQCaqgS1RFLqLoAT2hF/RqPVrP1pv1Pm3NWOnMLvoj6+MbsJaaqw==</latexit>
⌦0 = ⌦ !<latexit sha1_base64="pnDuJrZ2chHlkwbwzC/AgtPHcAo=">AAACAnicbZDLSsNAFIYnXmu9Rd3pZrAIbiyJCOpCKLhxpRWsLTShTKaTduhcwsxEKKHgxldx40LFrU/hzrdxmmahrT8MfPznHM6cP0oY1cbzvp25+YXFpeXSSnl1bX1j093avtcyVZg0sGRStSKkCaOCNAw1jLQSRRCPGGlGg8txvflAlKZS3JlhQkKOeoLGFCNjrY67G9xw0kMdD17ACcIjGMjccyte1csFZ8EvoAIK1TvuV9CVOOVEGMyQ1m3fS0yYIWUoZmRUDlJNEoQHqEfaFgXiRIdZfsMIHlinC2Op7BMG5u7viQxxrYc8sp0cmb6ero3N/2rt1MRnYUZFkhoi8GRRnDJoJBwHArtUEWzY0ALCitq/QtxHCmFjYyvbEPzpk2ehcVw9r/q3J5XadZFGCeyBfXAIfHAKauAK1EEDYPAInsEreHOenBfn3fmYtM45xcwO+CPn8wfWcpX3</latexit><latexit sha1_base64="pnDuJrZ2chHlkwbwzC/AgtPHcAo=">AAACAnicbZDLSsNAFIYnXmu9Rd3pZrAIbiyJCOpCKLhxpRWsLTShTKaTduhcwsxEKKHgxldx40LFrU/hzrdxmmahrT8MfPznHM6cP0oY1cbzvp25+YXFpeXSSnl1bX1j093avtcyVZg0sGRStSKkCaOCNAw1jLQSRRCPGGlGg8txvflAlKZS3JlhQkKOeoLGFCNjrY67G9xw0kMdD17ACcIjGMjccyte1csFZ8EvoAIK1TvuV9CVOOVEGMyQ1m3fS0yYIWUoZmRUDlJNEoQHqEfaFgXiRIdZfsMIHlinC2Op7BMG5u7viQxxrYc8sp0cmb6ero3N/2rt1MRnYUZFkhoi8GRRnDJoJBwHArtUEWzY0ALCitq/QtxHCmFjYyvbEPzpk2ehcVw9r/q3J5XadZFGCeyBfXAIfHAKauAK1EEDYPAInsEreHOenBfn3fmYtM45xcwO+CPn8wfWcpX3</latexit><latexit sha1_base64="pnDuJrZ2chHlkwbwzC/AgtPHcAo=">AAACAnicbZDLSsNAFIYnXmu9Rd3pZrAIbiyJCOpCKLhxpRWsLTShTKaTduhcwsxEKKHgxldx40LFrU/hzrdxmmahrT8MfPznHM6cP0oY1cbzvp25+YXFpeXSSnl1bX1j093avtcyVZg0sGRStSKkCaOCNAw1jLQSRRCPGGlGg8txvflAlKZS3JlhQkKOeoLGFCNjrY67G9xw0kMdD17ACcIjGMjccyte1csFZ8EvoAIK1TvuV9CVOOVEGMyQ1m3fS0yYIWUoZmRUDlJNEoQHqEfaFgXiRIdZfsMIHlinC2Op7BMG5u7viQxxrYc8sp0cmb6ero3N/2rt1MRnYUZFkhoi8GRRnDJoJBwHArtUEWzY0ALCitq/QtxHCmFjYyvbEPzpk2ehcVw9r/q3J5XadZFGCeyBfXAIfHAKauAK1EEDYPAInsEreHOenBfn3fmYtM45xcwO+CPn8wfWcpX3</latexit><latexit sha1_base64="pnDuJrZ2chHlkwbwzC/AgtPHcAo=">AAACAnicbZDLSsNAFIYnXmu9Rd3pZrAIbiyJCOpCKLhxpRWsLTShTKaTduhcwsxEKKHgxldx40LFrU/hzrdxmmahrT8MfPznHM6cP0oY1cbzvp25+YXFpeXSSnl1bX1j093avtcyVZg0sGRStSKkCaOCNAw1jLQSRRCPGGlGg8txvflAlKZS3JlhQkKOeoLGFCNjrY67G9xw0kMdD17ACcIjGMjccyte1csFZ8EvoAIK1TvuV9CVOOVEGMyQ1m3fS0yYIWUoZmRUDlJNEoQHqEfaFgXiRIdZfsMIHlinC2Op7BMG5u7viQxxrYc8sp0cmb6ero3N/2rt1MRnYUZFkhoi8GRRnDJoJBwHArtUEWzY0ALCitq/QtxHCmFjYyvbEPzpk2ehcVw9r/q3J5XadZFGCeyBfXAIfHAKauAK1EEDYPAInsEreHOenBfn3fmYtM45xcwO+CPn8wfWcpX3</latexit>
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!1, · · · , !K<latexit sha1_base64="Z+YX4EHR+yhemC3izIEkXju0pZI=">AAACAnicbVDLSsNAFJ3UV62vqDvdDBbBhZREBHVXcCMIUsHYQhPCZDJph85kwsxEKKHgxl9x40LFrV/hzr9x2mahrQcu93DOvczcE2WMKu0431ZlYXFpeaW6Wltb39jcsrd37pXIJSYeFkzIToQUYTQlnqaakU4mCeIRI+1ocDn22w9EKirSOz3MSMBRL6UJxUgbKbT3fMFJD4XuMfRxLLQyfapch3bdaTgTwHnilqQOSrRC+8uPBc45STVmSKmu62Q6KJDUFDMyqvm5IhnCA9QjXUNTxIkKiskNI3holBgmQppKNZyovzcKxJUa8shMcqT7atYbi/953Vwn50FB0yzXJMXTh5KcQS3gOBAYU0mwZkNDEJbU/BXiPpIIaxNbzYTgzp48T7yTxkXDvT2tN2/KNKpgHxyAI+CCM9AEV6AFPIDBI3gGr+DNerJerHfrYzpascqdXfAH1ucP5ZOWpA==</latexit><latexit sha1_base64="Z+YX4EHR+yhemC3izIEkXju0pZI=">AAACAnicbVDLSsNAFJ3UV62vqDvdDBbBhZREBHVXcCMIUsHYQhPCZDJph85kwsxEKKHgxl9x40LFrV/hzr9x2mahrQcu93DOvczcE2WMKu0431ZlYXFpeaW6Wltb39jcsrd37pXIJSYeFkzIToQUYTQlnqaakU4mCeIRI+1ocDn22w9EKirSOz3MSMBRL6UJxUgbKbT3fMFJD4XuMfRxLLQyfapch3bdaTgTwHnilqQOSrRC+8uPBc45STVmSKmu62Q6KJDUFDMyqvm5IhnCA9QjXUNTxIkKiskNI3holBgmQppKNZyovzcKxJUa8shMcqT7atYbi/953Vwn50FB0yzXJMXTh5KcQS3gOBAYU0mwZkNDEJbU/BXiPpIIaxNbzYTgzp48T7yTxkXDvT2tN2/KNKpgHxyAI+CCM9AEV6AFPIDBI3gGr+DNerJerHfrYzpascqdXfAH1ucP5ZOWpA==</latexit><latexit sha1_base64="Z+YX4EHR+yhemC3izIEkXju0pZI=">AAACAnicbVDLSsNAFJ3UV62vqDvdDBbBhZREBHVXcCMIUsHYQhPCZDJph85kwsxEKKHgxl9x40LFrV/hzr9x2mahrQcu93DOvczcE2WMKu0431ZlYXFpeaW6Wltb39jcsrd37pXIJSYeFkzIToQUYTQlnqaakU4mCeIRI+1ocDn22w9EKirSOz3MSMBRL6UJxUgbKbT3fMFJD4XuMfRxLLQyfapch3bdaTgTwHnilqQOSrRC+8uPBc45STVmSKmu62Q6KJDUFDMyqvm5IhnCA9QjXUNTxIkKiskNI3holBgmQppKNZyovzcKxJUa8shMcqT7atYbi/953Vwn50FB0yzXJMXTh5KcQS3gOBAYU0mwZkNDEJbU/BXiPpIIaxNbzYTgzp48T7yTxkXDvT2tN2/KNKpgHxyAI+CCM9AEV6AFPIDBI3gGr+DNerJerHfrYzpascqdXfAH1ucP5ZOWpA==</latexit><latexit sha1_base64="Z+YX4EHR+yhemC3izIEkXju0pZI=">AAACAnicbVDLSsNAFJ3UV62vqDvdDBbBhZREBHVXcCMIUsHYQhPCZDJph85kwsxEKKHgxl9x40LFrV/hzr9x2mahrQcu93DOvczcE2WMKu0431ZlYXFpeaW6Wltb39jcsrd37pXIJSYeFkzIToQUYTQlnqaakU4mCeIRI+1ocDn22w9EKirSOz3MSMBRL6UJxUgbKbT3fMFJD4XuMfRxLLQyfapch3bdaTgTwHnilqQOSrRC+8uPBc45STVmSKmu62Q6KJDUFDMyqvm5IhnCA9QjXUNTxIkKiskNI3holBgmQppKNZyovzcKxJUa8shMcqT7atYbi/953Vwn50FB0yzXJMXTh5KcQS3gOBAYU0mwZkNDEJbU/BXiPpIIaxNbzYTgzp48T7yTxkXDvT2tN2/KNKpgHxyAI+CCM9AEV6AFPIDBI3gGr+DNerJerHfrYzpascqdXfAH1ucP5ZOWpA==</latexit>
⌦k = ⌦ !k (81  k  K)<latexit sha1_base64="DetsFyQWCyItUB9gRmXqpNsHL2I=">AAACHXicbVDLSsNAFJ3UV62vqEs3g0WoC0siiroQCm4EQStYW2hCmExv2qGTBzMToYR+iRt/xY0LFRduxL9x2mahrQdmOJxzL/fe4yecSWVZ30Zhbn5hcam4XFpZXVvfMDe37mWcCgoNGvNYtHwigbMIGoopDq1EAAl9Dk2/fzHymw8gJIujOzVIwA1JN2IBo0RpyTOPnZsQusTr43M8ofgAO/FEc3DFCWJBOMc2djjg/vi/2vfMslW1xsCzxM5JGeWoe+an04lpGkKkKCdStm0rUW5GhGKUw7DkpBISQvukC21NIxKCdLPxeUO8p5UO1ovoFyk8Vn93ZCSUchD6ujIkqienvZH4n9dOVXDqZixKUgURnQwKUo5VjEdZ4Q4TQBUfaEKoYHpXTHtEEKp0oiUdgj198ixpHFbPqvbtUbl2nadRRDtoF1WQjU5QDV2iOmogih7RM3pFb8aT8WK8Gx+T0oKR92yjPzC+fgAV+Z+U</latexit><latexit sha1_base64="DetsFyQWCyItUB9gRmXqpNsHL2I=">AAACHXicbVDLSsNAFJ3UV62vqEs3g0WoC0siiroQCm4EQStYW2hCmExv2qGTBzMToYR+iRt/xY0LFRduxL9x2mahrQdmOJxzL/fe4yecSWVZ30Zhbn5hcam4XFpZXVvfMDe37mWcCgoNGvNYtHwigbMIGoopDq1EAAl9Dk2/fzHymw8gJIujOzVIwA1JN2IBo0RpyTOPnZsQusTr43M8ofgAO/FEc3DFCWJBOMc2djjg/vi/2vfMslW1xsCzxM5JGeWoe+an04lpGkKkKCdStm0rUW5GhGKUw7DkpBISQvukC21NIxKCdLPxeUO8p5UO1ovoFyk8Vn93ZCSUchD6ujIkqienvZH4n9dOVXDqZixKUgURnQwKUo5VjEdZ4Q4TQBUfaEKoYHpXTHtEEKp0oiUdgj198ixpHFbPqvbtUbl2nadRRDtoF1WQjU5QDV2iOmogih7RM3pFb8aT8WK8Gx+T0oKR92yjPzC+fgAV+Z+U</latexit><latexit sha1_base64="DetsFyQWCyItUB9gRmXqpNsHL2I=">AAACHXicbVDLSsNAFJ3UV62vqEs3g0WoC0siiroQCm4EQStYW2hCmExv2qGTBzMToYR+iRt/xY0LFRduxL9x2mahrQdmOJxzL/fe4yecSWVZ30Zhbn5hcam4XFpZXVvfMDe37mWcCgoNGvNYtHwigbMIGoopDq1EAAl9Dk2/fzHymw8gJIujOzVIwA1JN2IBo0RpyTOPnZsQusTr43M8ofgAO/FEc3DFCWJBOMc2djjg/vi/2vfMslW1xsCzxM5JGeWoe+an04lpGkKkKCdStm0rUW5GhGKUw7DkpBISQvukC21NIxKCdLPxeUO8p5UO1ovoFyk8Vn93ZCSUchD6ujIkqienvZH4n9dOVXDqZixKUgURnQwKUo5VjEdZ4Q4TQBUfaEKoYHpXTHtEEKp0oiUdgj198ixpHFbPqvbtUbl2nadRRDtoF1WQjU5QDV2iOmogih7RM3pFb8aT8WK8Gx+T0oKR92yjPzC+fgAV+Z+U</latexit><latexit sha1_base64="DetsFyQWCyItUB9gRmXqpNsHL2I=">AAACHXicbVDLSsNAFJ3UV62vqEs3g0WoC0siiroQCm4EQStYW2hCmExv2qGTBzMToYR+iRt/xY0LFRduxL9x2mahrQdmOJxzL/fe4yecSWVZ30Zhbn5hcam4XFpZXVvfMDe37mWcCgoNGvNYtHwigbMIGoopDq1EAAl9Dk2/fzHymw8gJIujOzVIwA1JN2IBo0RpyTOPnZsQusTr43M8ofgAO/FEc3DFCWJBOMc2djjg/vi/2vfMslW1xsCzxM5JGeWoe+an04lpGkKkKCdStm0rUW5GhGKUw7DkpBISQvukC21NIxKCdLPxeUO8p5UO1ovoFyk8Vn93ZCSUchD6ujIkqienvZH4n9dOVXDqZixKUgURnQwKUo5VjEdZ4Q4TQBUfaEKoYHpXTHtEEKp0oiUdgj198ixpHFbPqvbtUbl2nadRRDtoF1WQjU5QDV2iOmogih7RM3pFb8aT8WK8Gx+T0oKR92yjPzC+fgAV+Z+U</latexit>
20
HOW TO FIND MULTIPLE SUBSETS?
• Let satisfying
• Divide into K non-overlapping subsets 

with the conditions that
• Let and .
• For any
• and 

are -stable and -stable respectively.
• Then,
Theorem 4
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2<latexit sha1_base64="lbsE7NtCbH6B192W/4i6YwyXzIU=">AAAB7nicbVBNS8NAEJ34WetX1aOXxSJ4KkkR1FvBiyepYG2hDWWzmbRLN5u4uxFK6J/w4kHFq7/Hm//GbZuDtj4YeLw3w8y8IBVcG9f9dlZW19Y3Nktb5e2d3b39ysHhg04yxbDFEpGoTkA1Ci6xZbgR2EkV0jgQ2A5G11O//YRK80Tem3GKfkwHkkecUWOlTi9EYWi/3q9U3Zo7A1kmXkGqUKDZr3z1woRlMUrDBNW667mp8XOqDGcCJ+VepjGlbEQH2LVU0hi1n8/unZBTq4QkSpQtachM/T2R01jrcRzYzpiaoV70puJ/Xjcz0aWfc5lmBiWbL4oyQUxCps+TkCtkRowtoUxxeythQ6ooMzaisg3BW3x5mbTqtauad3debdwWaZTgGE7gDDy4gAbcQBNawEDAM7zCm/PovDjvzse8dcUpZo7gD5zPHyvkj6E=</latexit><latexit sha1_base64="lbsE7NtCbH6B192W/4i6YwyXzIU=">AAAB7nicbVBNS8NAEJ34WetX1aOXxSJ4KkkR1FvBiyepYG2hDWWzmbRLN5u4uxFK6J/w4kHFq7/Hm//GbZuDtj4YeLw3w8y8IBVcG9f9dlZW19Y3Nktb5e2d3b39ysHhg04yxbDFEpGoTkA1Ci6xZbgR2EkV0jgQ2A5G11O//YRK80Tem3GKfkwHkkecUWOlTi9EYWi/3q9U3Zo7A1kmXkGqUKDZr3z1woRlMUrDBNW667mp8XOqDGcCJ+VepjGlbEQH2LVU0hi1n8/unZBTq4QkSpQtachM/T2R01jrcRzYzpiaoV70puJ/Xjcz0aWfc5lmBiWbL4oyQUxCps+TkCtkRowtoUxxeythQ6ooMzaisg3BW3x5mbTqtauad3debdwWaZTgGE7gDDy4gAbcQBNawEDAM7zCm/PovDjvzse8dcUpZo7gD5zPHyvkj6E=</latexit><latexit sha1_base64="lbsE7NtCbH6B192W/4i6YwyXzIU=">AAAB7nicbVBNS8NAEJ34WetX1aOXxSJ4KkkR1FvBiyepYG2hDWWzmbRLN5u4uxFK6J/w4kHFq7/Hm//GbZuDtj4YeLw3w8y8IBVcG9f9dlZW19Y3Nktb5e2d3b39ysHhg04yxbDFEpGoTkA1Ci6xZbgR2EkV0jgQ2A5G11O//YRK80Tem3GKfkwHkkecUWOlTi9EYWi/3q9U3Zo7A1kmXkGqUKDZr3z1woRlMUrDBNW667mp8XOqDGcCJ+VepjGlbEQH2LVU0hi1n8/unZBTq4QkSpQtachM/T2R01jrcRzYzpiaoV70puJ/Xjcz0aWfc5lmBiWbL4oyQUxCps+TkCtkRowtoUxxeythQ6ooMzaisg3BW3x5mbTqtauad3debdwWaZTgGE7gDDy4gAbcQBNawEDAM7zCm/PovDjvzse8dcUpZo7gD5zPHyvkj6E=</latexit><latexit sha1_base64="lbsE7NtCbH6B192W/4i6YwyXzIU=">AAAB7nicbVBNS8NAEJ34WetX1aOXxSJ4KkkR1FvBiyepYG2hDWWzmbRLN5u4uxFK6J/w4kHFq7/Hm//GbZuDtj4YeLw3w8y8IBVcG9f9dlZW19Y3Nktb5e2d3b39ysHhg04yxbDFEpGoTkA1Ci6xZbgR2EkV0jgQ2A5G11O//YRK80Tem3GKfkwHkkecUWOlTi9EYWi/3q9U3Zo7A1kmXkGqUKDZr3z1woRlMUrDBNW667mp8XOqDGcCJ+VepjGlbEQH2LVU0hi1n8/unZBTq4QkSpQtachM/T2R01jrcRzYzpiaoV70puJ/Xjcz0aWfc5lmBiWbL4oyQUxCps+TkCtkRowtoUxxeythQ6ooMzaisg3BW3x5mbTqtauad3debdwWaZTgGE7gDDy4gAbcQBNawEDAM7zCm/PovDjvzse8dcUpZo7gD5zPHyvkj6E=</latexit>
⌦k = ⌦ !k (81  k  K)<latexit sha1_base64="DetsFyQWCyItUB9gRmXqpNsHL2I=">AAACHXicbVDLSsNAFJ3UV62vqEs3g0WoC0siiroQCm4EQStYW2hCmExv2qGTBzMToYR+iRt/xY0LFRduxL9x2mahrQdmOJxzL/fe4yecSWVZ30Zhbn5hcam4XFpZXVvfMDe37mWcCgoNGvNYtHwigbMIGoopDq1EAAl9Dk2/fzHymw8gJIujOzVIwA1JN2IBo0RpyTOPnZsQusTr43M8ofgAO/FEc3DFCWJBOMc2djjg/vi/2vfMslW1xsCzxM5JGeWoe+an04lpGkKkKCdStm0rUW5GhGKUw7DkpBISQvukC21NIxKCdLPxeUO8p5UO1ovoFyk8Vn93ZCSUchD6ujIkqienvZH4n9dOVXDqZixKUgURnQwKUo5VjEdZ4Q4TQBUfaEKoYHpXTHtEEKp0oiUdgj198ixpHFbPqvbtUbl2nadRRDtoF1WQjU5QDV2iOmogih7RM3pFb8aT8WK8Gx+T0oKR92yjPzC+fgAV+Z+U</latexit><latexit sha1_base64="DetsFyQWCyItUB9gRmXqpNsHL2I=">AAACHXicbVDLSsNAFJ3UV62vqEs3g0WoC0siiroQCm4EQStYW2hCmExv2qGTBzMToYR+iRt/xY0LFRduxL9x2mahrQdmOJxzL/fe4yecSWVZ30Zhbn5hcam4XFpZXVvfMDe37mWcCgoNGvNYtHwigbMIGoopDq1EAAl9Dk2/fzHymw8gJIujOzVIwA1JN2IBo0RpyTOPnZsQusTr43M8ofgAO/FEc3DFCWJBOMc2djjg/vi/2vfMslW1xsCzxM5JGeWoe+an04lpGkKkKCdStm0rUW5GhGKUw7DkpBISQvukC21NIxKCdLPxeUO8p5UO1ovoFyk8Vn93ZCSUchD6ujIkqienvZH4n9dOVXDqZixKUgURnQwKUo5VjEdZ4Q4TQBUfaEKoYHpXTHtEEKp0oiUdgj198ixpHFbPqvbtUbl2nadRRDtoF1WQjU5QDV2iOmogih7RM3pFb8aT8WK8Gx+T0oKR92yjPzC+fgAV+Z+U</latexit><latexit sha1_base64="DetsFyQWCyItUB9gRmXqpNsHL2I=">AAACHXicbVDLSsNAFJ3UV62vqEs3g0WoC0siiroQCm4EQStYW2hCmExv2qGTBzMToYR+iRt/xY0LFRduxL9x2mahrQdmOJxzL/fe4yecSWVZ30Zhbn5hcam4XFpZXVvfMDe37mWcCgoNGvNYtHwigbMIGoopDq1EAAl9Dk2/fzHymw8gJIujOzVIwA1JN2IBo0RpyTOPnZsQusTr43M8ofgAO/FEc3DFCWJBOMc2djjg/vi/2vfMslW1xsCzxM5JGeWoe+an04lpGkKkKCdStm0rUW5GhGKUw7DkpBISQvukC21NIxKCdLPxeUO8p5UO1ovoFyk8Vn93ZCSUchD6ujIkqienvZH4n9dOVXDqZixKUgURnQwKUo5VjEdZ4Q4TQBUfaEKoYHpXTHtEEKp0oiUdgj198ixpHFbPqvbtUbl2nadRRDtoF1WQjU5QDV2iOmogih7RM3pFb8aT8WK8Gx+T0oKR92yjPzC+fgAV+Z+U</latexit><latexit sha1_base64="DetsFyQWCyItUB9gRmXqpNsHL2I=">AAACHXicbVDLSsNAFJ3UV62vqEs3g0WoC0siiroQCm4EQStYW2hCmExv2qGTBzMToYR+iRt/xY0LFRduxL9x2mahrQdmOJxzL/fe4yecSWVZ30Zhbn5hcam4XFpZXVvfMDe37mWcCgoNGvNYtHwigbMIGoopDq1EAAl9Dk2/fzHymw8gJIujOzVIwA1JN2IBo0RpyTOPnZsQusTr43M8ofgAO/FEc3DFCWJBOMc2djjg/vi/2vfMslW1xsCzxM5JGeWoe+an04lpGkKkKCdStm0rUW5GhGKUw7DkpBISQvukC21NIxKCdLPxeUO8p5UO1ovoFyk8Vn93ZCSUchD6ujIkqienvZH4n9dOVXDqZixKUgURnQwKUo5VjEdZ4Q4TQBUfaEKoYHpXTHtEEKp0oiUdgj198ixpHFbPqvbtUbl2nadRRDtoF1WQjU5QDV2iOmogih7RM3pFb8aT8WK8Gx+T0oKR92yjPzC+fgAV+Z+U</latexit>
✏ > 0 and 1 > 0, 1, · · · , K > 0 ⌃K
i=0 i = 1<latexit sha1_base64="flCvYaRbQuEbUHLoFYsJBrl9VjM=">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</latexit><latexit sha1_base64="flCvYaRbQuEbUHLoFYsJBrl9VjM=">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</latexit><latexit sha1_base64="flCvYaRbQuEbUHLoFYsJBrl9VjM=">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</latexit><latexit sha1_base64="flCvYaRbQuEbUHLoFYsJBrl9VjM=">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</latexit>
⌦0 = ⌦ !<latexit sha1_base64="pnDuJrZ2chHlkwbwzC/AgtPHcAo=">AAACAnicbZDLSsNAFIYnXmu9Rd3pZrAIbiyJCOpCKLhxpRWsLTShTKaTduhcwsxEKKHgxldx40LFrU/hzrdxmmahrT8MfPznHM6cP0oY1cbzvp25+YXFpeXSSnl1bX1j093avtcyVZg0sGRStSKkCaOCNAw1jLQSRRCPGGlGg8txvflAlKZS3JlhQkKOeoLGFCNjrY67G9xw0kMdD17ACcIjGMjccyte1csFZ8EvoAIK1TvuV9CVOOVEGMyQ1m3fS0yYIWUoZmRUDlJNEoQHqEfaFgXiRIdZfsMIHlinC2Op7BMG5u7viQxxrYc8sp0cmb6ero3N/2rt1MRnYUZFkhoi8GRRnDJoJBwHArtUEWzY0ALCitq/QtxHCmFjYyvbEPzpk2ehcVw9r/q3J5XadZFGCeyBfXAIfHAKauAK1EEDYPAInsEreHOenBfn3fmYtM45xcwO+CPn8wfWcpX3</latexit><latexit sha1_base64="pnDuJrZ2chHlkwbwzC/AgtPHcAo=">AAACAnicbZDLSsNAFIYnXmu9Rd3pZrAIbiyJCOpCKLhxpRWsLTShTKaTduhcwsxEKKHgxldx40LFrU/hzrdxmmahrT8MfPznHM6cP0oY1cbzvp25+YXFpeXSSnl1bX1j093avtcyVZg0sGRStSKkCaOCNAw1jLQSRRCPGGlGg8txvflAlKZS3JlhQkKOeoLGFCNjrY67G9xw0kMdD17ACcIjGMjccyte1csFZ8EvoAIK1TvuV9CVOOVEGMyQ1m3fS0yYIWUoZmRUDlJNEoQHqEfaFgXiRIdZfsMIHlinC2Op7BMG5u7viQxxrYc8sp0cmb6ero3N/2rt1MRnYUZFkhoi8GRRnDJoJBwHArtUEWzY0ALCitq/QtxHCmFjYyvbEPzpk2ehcVw9r/q3J5XadZFGCeyBfXAIfHAKauAK1EEDYPAInsEreHOenBfn3fmYtM45xcwO+CPn8wfWcpX3</latexit><latexit sha1_base64="pnDuJrZ2chHlkwbwzC/AgtPHcAo=">AAACAnicbZDLSsNAFIYnXmu9Rd3pZrAIbiyJCOpCKLhxpRWsLTShTKaTduhcwsxEKKHgxldx40LFrU/hzrdxmmahrT8MfPznHM6cP0oY1cbzvp25+YXFpeXSSnl1bX1j093avtcyVZg0sGRStSKkCaOCNAw1jLQSRRCPGGlGg8txvflAlKZS3JlhQkKOeoLGFCNjrY67G9xw0kMdD17ACcIjGMjccyte1csFZ8EvoAIK1TvuV9CVOOVEGMyQ1m3fS0yYIWUoZmRUDlJNEoQHqEfaFgXiRIdZfsMIHlinC2Op7BMG5u7viQxxrYc8sp0cmb6ero3N/2rt1MRnYUZFkhoi8GRRnDJoJBwHArtUEWzY0ALCitq/QtxHCmFjYyvbEPzpk2ehcVw9r/q3J5XadZFGCeyBfXAIfHAKauAK1EEDYPAInsEreHOenBfn3fmYtM45xcwO+CPn8wfWcpX3</latexit><latexit sha1_base64="pnDuJrZ2chHlkwbwzC/AgtPHcAo=">AAACAnicbZDLSsNAFIYnXmu9Rd3pZrAIbiyJCOpCKLhxpRWsLTShTKaTduhcwsxEKKHgxldx40LFrU/hzrdxmmahrT8MfPznHM6cP0oY1cbzvp25+YXFpeXSSnl1bX1j093avtcyVZg0sGRStSKkCaOCNAw1jLQSRRCPGGlGg8txvflAlKZS3JlhQkKOeoLGFCNjrY67G9xw0kMdD17ACcIjGMjccyte1csFZ8EvoAIK1TvuV9CVOOVEGMyQ1m3fS0yYIWUoZmRUDlJNEoQHqEfaFgXiRIdZfsMIHlinC2Op7BMG5u7viQxxrYc8sp0cmb6ero3N/2rt1MRnYUZFkhoi8GRRnDJoJBwHArtUEWzY0ALCitq/QtxHCmFjYyvbEPzpk2ehcVw9r/q3J5XadZFGCeyBfXAIfHAKauAK1EEDYPAInsEreHOenBfn3fmYtM45xcwO+CPn8wfWcpX3</latexit>
! ⇢ ⌦(|⌦| > 2)<latexit sha1_base64="ACA68GZluiyAFA/yKy15tzEFgIo=">AAACD3icbZDNSgMxFIUz9a/Wv6pLN8Ei1k2ZKYK6kYIbV1rBsYVOKZn0ThuayQxJRihtH8GNr+LGhYpbt+58G9N2BG09EPhy7r0k9/gxZ0rb9peVWVhcWl7JrubW1jc2t/LbO3cqSiQFl0Y8knWfKOBMgKuZ5lCPJZDQ51Dzexfjeu0epGKRuNX9GJoh6QgWMEq0sVr5Qy8KoUOwpxJfgcbe9eRaxMMfHOJzXD5q5Qt2yZ4Iz4OTQgGlqrbyn147okkIQlNOlGo4dqybAyI1oxxGOS9REBPaIx1oGBQkBNUcTBYa4QPjtHEQSXOExhP398SAhEr1Q990hkR31WxtbP5XayQ6OG0OmIgTDYJOHwoSjnWEx+ngNpNANe8bIFQy81dMu0QSqk2GOROCM7vyPLjl0lnJuTkuVK7SNLJoD+2jInLQCaqgS1RFLqLoAT2hF/RqPVrP1pv1Pm3NWOnMLvoj6+MbsJaaqw==</latexit><latexit sha1_base64="ACA68GZluiyAFA/yKy15tzEFgIo=">AAACD3icbZDNSgMxFIUz9a/Wv6pLN8Ei1k2ZKYK6kYIbV1rBsYVOKZn0ThuayQxJRihtH8GNr+LGhYpbt+58G9N2BG09EPhy7r0k9/gxZ0rb9peVWVhcWl7JrubW1jc2t/LbO3cqSiQFl0Y8knWfKOBMgKuZ5lCPJZDQ51Dzexfjeu0epGKRuNX9GJoh6QgWMEq0sVr5Qy8KoUOwpxJfgcbe9eRaxMMfHOJzXD5q5Qt2yZ4Iz4OTQgGlqrbyn147okkIQlNOlGo4dqybAyI1oxxGOS9REBPaIx1oGBQkBNUcTBYa4QPjtHEQSXOExhP398SAhEr1Q990hkR31WxtbP5XayQ6OG0OmIgTDYJOHwoSjnWEx+ngNpNANe8bIFQy81dMu0QSqk2GOROCM7vyPLjl0lnJuTkuVK7SNLJoD+2jInLQCaqgS1RFLqLoAT2hF/RqPVrP1pv1Pm3NWOnMLvoj6+MbsJaaqw==</latexit><latexit sha1_base64="ACA68GZluiyAFA/yKy15tzEFgIo=">AAACD3icbZDNSgMxFIUz9a/Wv6pLN8Ei1k2ZKYK6kYIbV1rBsYVOKZn0ThuayQxJRihtH8GNr+LGhYpbt+58G9N2BG09EPhy7r0k9/gxZ0rb9peVWVhcWl7JrubW1jc2t/LbO3cqSiQFl0Y8knWfKOBMgKuZ5lCPJZDQ51Dzexfjeu0epGKRuNX9GJoh6QgWMEq0sVr5Qy8KoUOwpxJfgcbe9eRaxMMfHOJzXD5q5Qt2yZ4Iz4OTQgGlqrbyn147okkIQlNOlGo4dqybAyI1oxxGOS9REBPaIx1oGBQkBNUcTBYa4QPjtHEQSXOExhP398SAhEr1Q990hkR31WxtbP5XayQ6OG0OmIgTDYJOHwoSjnWEx+ngNpNANe8bIFQy81dMu0QSqk2GOROCM7vyPLjl0lnJuTkuVK7SNLJoD+2jInLQCaqgS1RFLqLoAT2hF/RqPVrP1pv1Pm3NWOnMLvoj6+MbsJaaqw==</latexit><latexit sha1_base64="ACA68GZluiyAFA/yKy15tzEFgIo=">AAACD3icbZDNSgMxFIUz9a/Wv6pLN8Ei1k2ZKYK6kYIbV1rBsYVOKZn0ThuayQxJRihtH8GNr+LGhYpbt+58G9N2BG09EPhy7r0k9/gxZ0rb9peVWVhcWl7JrubW1jc2t/LbO3cqSiQFl0Y8knWfKOBMgKuZ5lCPJZDQ51Dzexfjeu0epGKRuNX9GJoh6QgWMEq0sVr5Qy8KoUOwpxJfgcbe9eRaxMMfHOJzXD5q5Qt2yZ4Iz4OTQgGlqrbyn147okkIQlNOlGo4dqybAyI1oxxGOS9REBPaIx1oGBQkBNUcTBYa4QPjtHEQSXOExhP398SAhEr1Q990hkR31WxtbP5XayQ6OG0OmIgTDYJOHwoSjnWEx+ngNpNANe8bIFQy81dMu0QSqk2GOROCM7vyPLjl0lnJuTkuVK7SNLJoD+2jInLQCaqgS1RFLqLoAT2hF/RqPVrP1pv1Pm3NWOnMLvoj6+MbsJaaqw==</latexit>
8(i, j) 2 !, |Ri,j
ˆRi,j|  D⌦( ˆR)<latexit sha1_base64="QKSqsgmGqwvE3kBA619AMM23eWA=">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</latexit><latexit sha1_base64="QKSqsgmGqwvE3kBA619AMM23eWA=">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</latexit><latexit sha1_base64="QKSqsgmGqwvE3kBA619AMM23eWA=">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</latexit><latexit sha1_base64="QKSqsgmGqwvE3kBA619AMM23eWA=">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</latexit>
!1, · · · , !K<latexit sha1_base64="Z+YX4EHR+yhemC3izIEkXju0pZI=">AAACAnicbVDLSsNAFJ3UV62vqDvdDBbBhZREBHVXcCMIUsHYQhPCZDJph85kwsxEKKHgxl9x40LFrV/hzr9x2mahrQcu93DOvczcE2WMKu0431ZlYXFpeaW6Wltb39jcsrd37pXIJSYeFkzIToQUYTQlnqaakU4mCeIRI+1ocDn22w9EKirSOz3MSMBRL6UJxUgbKbT3fMFJD4XuMfRxLLQyfapch3bdaTgTwHnilqQOSrRC+8uPBc45STVmSKmu62Q6KJDUFDMyqvm5IhnCA9QjXUNTxIkKiskNI3holBgmQppKNZyovzcKxJUa8shMcqT7atYbi/953Vwn50FB0yzXJMXTh5KcQS3gOBAYU0mwZkNDEJbU/BXiPpIIaxNbzYTgzp48T7yTxkXDvT2tN2/KNKpgHxyAI+CCM9AEV6AFPIDBI3gGr+DNerJerHfrYzpascqdXfAH1ucP5ZOWpA==</latexit><latexit sha1_base64="Z+YX4EHR+yhemC3izIEkXju0pZI=">AAACAnicbVDLSsNAFJ3UV62vqDvdDBbBhZREBHVXcCMIUsHYQhPCZDJph85kwsxEKKHgxl9x40LFrV/hzr9x2mahrQcu93DOvczcE2WMKu0431ZlYXFpeaW6Wltb39jcsrd37pXIJSYeFkzIToQUYTQlnqaakU4mCeIRI+1ocDn22w9EKirSOz3MSMBRL6UJxUgbKbT3fMFJD4XuMfRxLLQyfapch3bdaTgTwHnilqQOSrRC+8uPBc45STVmSKmu62Q6KJDUFDMyqvm5IhnCA9QjXUNTxIkKiskNI3holBgmQppKNZyovzcKxJUa8shMcqT7atYbi/953Vwn50FB0yzXJMXTh5KcQS3gOBAYU0mwZkNDEJbU/BXiPpIIaxNbzYTgzp48T7yTxkXDvT2tN2/KNKpgHxyAI+CCM9AEV6AFPIDBI3gGr+DNerJerHfrYzpascqdXfAH1ucP5ZOWpA==</latexit><latexit sha1_base64="Z+YX4EHR+yhemC3izIEkXju0pZI=">AAACAnicbVDLSsNAFJ3UV62vqDvdDBbBhZREBHVXcCMIUsHYQhPCZDJph85kwsxEKKHgxl9x40LFrV/hzr9x2mahrQcu93DOvczcE2WMKu0431ZlYXFpeaW6Wltb39jcsrd37pXIJSYeFkzIToQUYTQlnqaakU4mCeIRI+1ocDn22w9EKirSOz3MSMBRL6UJxUgbKbT3fMFJD4XuMfRxLLQyfapch3bdaTgTwHnilqQOSrRC+8uPBc45STVmSKmu62Q6KJDUFDMyqvm5IhnCA9QjXUNTxIkKiskNI3holBgmQppKNZyovzcKxJUa8shMcqT7atYbi/953Vwn50FB0yzXJMXTh5KcQS3gOBAYU0mwZkNDEJbU/BXiPpIIaxNbzYTgzp48T7yTxkXDvT2tN2/KNKpgHxyAI+CCM9AEV6AFPIDBI3gGr+DNerJerHfrYzpascqdXfAH1ucP5ZOWpA==</latexit><latexit sha1_base64="Z+YX4EHR+yhemC3izIEkXju0pZI=">AAACAnicbVDLSsNAFJ3UV62vqDvdDBbBhZREBHVXcCMIUsHYQhPCZDJph85kwsxEKKHgxl9x40LFrV/hzr9x2mahrQcu93DOvczcE2WMKu0431ZlYXFpeaW6Wltb39jcsrd37pXIJSYeFkzIToQUYTQlnqaakU4mCeIRI+1ocDn22w9EKirSOz3MSMBRL6UJxUgbKbT3fMFJD4XuMfRxLLQyfapch3bdaTgTwHnilqQOSrRC+8uPBc45STVmSKmu62Q6KJDUFDMyqvm5IhnCA9QjXUNTxIkKiskNI3holBgmQppKNZyovzcKxJUa8shMcqT7atYbi/953Vwn50FB0yzXJMXTh5KcQS3gOBAYU0mwZkNDEJbU/BXiPpIIaxNbzYTgzp48T7yTxkXDvT2tN2/KNKpgHxyAI+CCM9AEV6AFPIDBI3gGr+DNerJerHfrYzpascqdXfAH1ucP5ZOWpA==</latexit>
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CONTENTS
1. Motivation
2. Stability of LRMA
3. SMA Algorithm
4. Experiments
5. Discussion
22
• Vanilla Low-rank Matrix Factorization:
• Model formulation of SMA:
• By Theorem 3, Multiple hard predictable subsets are better than single
predictable subsets. So, they make model can handle multiple subsets.
• Let
MODEL FORMULATION
ˆR = arg max
X
D⌦(X), rank(X) = r
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⌦1, · · · , ⌦K 2 ⌦, 8s 2 [1, K], D⌦x
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23
• The more hard-predictable subsets the better stability.
• However, it is expensive to find such “hard predictable subsets”. Why?
• Because we don’t know which subset of entries to choose 

without any prior knowledge.
• We can’t know final model without hard-predictable subsets.

Also, we can’t know hard-predictable subsets without final model.

(Which came first, the chicken or the egg?)
• In other words,
• Recall
• To obtain such is not trivial, because we can only check if the condition is
satisfied with the final model.
• But the final model cannot be known before we define and optimize a given loss
function.
DIFFICULTIES IN SELECTING HARD PREDICTABLE SUBSETS
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⌦1, · · · , ⌦K ⇢ ⌦, 8s 2 [1, K], D⌦s
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24
• To address this issue:
• 1) Approximate the targeted matrix R with existing LRMA solutions.
• 2) for each entry (i, j) ∈ Ω, it is chosen with large probability if 

and small probability otherwise
• 3) Obtain Ω’ by removing the chosen entries to satisfy the condition of
Proposition 1, or probe Ω’ to find hard predictable subsets that satisfy the
condition of Theorem 4.
• → In this paper they chose Theorem 4.
• Underlying Assumptions:
• LRMF methods will not dramatically differ from final model of SMA
• we can ensure that Ω’ will satisfy with high probability.
HARD PREDICTABLE SUBSETS SELECTION
25
THE SMA ALGORITHM
Pretraining Part:
Finding Ω’ for SMA algorithm
using Theorem 4
Core Training Part:
Add Regularizer to avoid overfitting
CONTENTS
1. Motivation
2. Stability of LRMA
3. SMA Algorithm
4. Experiments
5. Discussion
27
ACCURACY COMPARISONS
• Test RMSE on two benchmark datasets.
• Compared to other state-of-the-arts models, it showed good performance.
• SMA demonstrates similar stability to ensembles such as DFC, LLORMA, and
WEMAREC.
• Most importantly, other models can not guarantee a gap between training and test
errors, but SMA is possible.
28
• RMSE vs the number of subsets K
• As the number of subsets increases, the RMSE decreases.
• RMSE vs rank r
• As the rank r increases, the SMA model does not overfit and the RMSE decreases.
EFFECTS OF HYPER PARAMETERS
CONTENTS
1. Motivation
2. Stability of LRMA
3. SMA Algorithm
4. Experiments
5. Discussion
30
• Why did the authors choose Theorem 4 rather than Proposition 1 in the
final algorithm?
• I think we can make final algorithm with Proposition 1.
• The authors assumed that the model used in pretrain and the final model are
similar. Is the random sampling steps are necessary?
• I think they would better suggest reason why they use random sampling?
• Does this algorithm work for other fields or models?
• There is no assumption related to low-rank matrix factorization model in Theorem 1
or others. I think we can use this concept in other models.
• I think this method can be regarded as regularization.
• Do we really need Theorem 3?
• I think Theorem 4 is enough for final SMA algorithm.
DISCUSSION POINTS
ANY QUESTIONS?

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Low-rank Matrix Approximation with Stability

  • 1. LOW-RANK MATRIX APPROXIMATION WITH STABILITY 2019. 02. 19. JoonyoungYi joonyoung.yi@kaist.ac.kr
  • 2. CONTENTS 1. Motivation 2. Stability of LRMA 3. SMA Algorithm 4. Experiments 5. Discussion
  • 3. CONTENTS 1. Motivation 2. Stability of LRMA 3. SMA Algorithm 4. Experiments 5. Discussion
  • 4. 4 • Let are matrices. • Low-rank Matrix Factorization Problem: 
 We know only 1% of entries in R, how to complete low-rank matrix R? • Optimization form: LOW-RANK MATRIX FACTORIZATION (LRMF) R U VT≒ xm m n n r r R 2 Rm⇥n , U 2 Rm⇥r , V 2 Rn⇥r <latexit sha1_base64="WI/r+BC8V9iiHIKaNDQmKJSxYmg=">AAACQ3icdZDNSsNAFIUn9a/Wv6hLN4NFcCElEUHdFdy4klpMW0himUyn7dDJJMxMhBLycG58AHc+gRsXKm4FJ21AbfXCwOF893LvnCBmVCrLejJKC4tLyyvl1cra+sbmlrm905JRIjBxcMQi0QmQJIxy4iiqGOnEgqAwYKQdjC5y3r4jQtKI36hxTPwQDTjtU4yUtrqm24Qe5dALkRoGQdrMbtMQeoqGREKeHUHnfyw0bs1h/o27ZtWqWZOC88IuRBUU1eiaj14vwklIuMIMSenaVqz8FAlFMSNZxUskiREeoQFxteRI7/HTSQgZPNBOD/YjoR9XcOL+nEhRKOU4DHRnfq6cZbn5F3MT1T/zU8rjRBGOp4v6CYMqgnmisEcFwYqNtUBYUH0rxEMkEFY694oOwZ798rxwjmvnNfv6pFq/KtIogz2wDw6BDU5BHVyCBnAABvfgGbyCN+PBeDHejY9pa8koZnbBrzI+vwDdMbEZ</latexit><latexit sha1_base64="WI/r+BC8V9iiHIKaNDQmKJSxYmg=">AAACQ3icdZDNSsNAFIUn9a/Wv6hLN4NFcCElEUHdFdy4klpMW0himUyn7dDJJMxMhBLycG58AHc+gRsXKm4FJ21AbfXCwOF893LvnCBmVCrLejJKC4tLyyvl1cra+sbmlrm905JRIjBxcMQi0QmQJIxy4iiqGOnEgqAwYKQdjC5y3r4jQtKI36hxTPwQDTjtU4yUtrqm24Qe5dALkRoGQdrMbtMQeoqGREKeHUHnfyw0bs1h/o27ZtWqWZOC88IuRBUU1eiaj14vwklIuMIMSenaVqz8FAlFMSNZxUskiREeoQFxteRI7/HTSQgZPNBOD/YjoR9XcOL+nEhRKOU4DHRnfq6cZbn5F3MT1T/zU8rjRBGOp4v6CYMqgnmisEcFwYqNtUBYUH0rxEMkEFY694oOwZ798rxwjmvnNfv6pFq/KtIogz2wDw6BDU5BHVyCBnAABvfgGbyCN+PBeDHejY9pa8koZnbBrzI+vwDdMbEZ</latexit><latexit sha1_base64="WI/r+BC8V9iiHIKaNDQmKJSxYmg=">AAACQ3icdZDNSsNAFIUn9a/Wv6hLN4NFcCElEUHdFdy4klpMW0himUyn7dDJJMxMhBLycG58AHc+gRsXKm4FJ21AbfXCwOF893LvnCBmVCrLejJKC4tLyyvl1cra+sbmlrm905JRIjBxcMQi0QmQJIxy4iiqGOnEgqAwYKQdjC5y3r4jQtKI36hxTPwQDTjtU4yUtrqm24Qe5dALkRoGQdrMbtMQeoqGREKeHUHnfyw0bs1h/o27ZtWqWZOC88IuRBUU1eiaj14vwklIuMIMSenaVqz8FAlFMSNZxUskiREeoQFxteRI7/HTSQgZPNBOD/YjoR9XcOL+nEhRKOU4DHRnfq6cZbn5F3MT1T/zU8rjRBGOp4v6CYMqgnmisEcFwYqNtUBYUH0rxEMkEFY694oOwZ798rxwjmvnNfv6pFq/KtIogz2wDw6BDU5BHVyCBnAABvfgGbyCN+PBeDHejY9pa8koZnbBrzI+vwDdMbEZ</latexit><latexit sha1_base64="WI/r+BC8V9iiHIKaNDQmKJSxYmg=">AAACQ3icdZDNSsNAFIUn9a/Wv6hLN4NFcCElEUHdFdy4klpMW0himUyn7dDJJMxMhBLycG58AHc+gRsXKm4FJ21AbfXCwOF893LvnCBmVCrLejJKC4tLyyvl1cra+sbmlrm905JRIjBxcMQi0QmQJIxy4iiqGOnEgqAwYKQdjC5y3r4jQtKI36hxTPwQDTjtU4yUtrqm24Qe5dALkRoGQdrMbtMQeoqGREKeHUHnfyw0bs1h/o27ZtWqWZOC88IuRBUU1eiaj14vwklIuMIMSenaVqz8FAlFMSNZxUskiREeoQFxteRI7/HTSQgZPNBOD/YjoR9XcOL+nEhRKOU4DHRnfq6cZbn5F3MT1T/zU8rjRBGOp4v6CYMqgnmisEcFwYqNtUBYUH0rxEMkEFY694oOwZ798rxwjmvnNfv6pFq/KtIogz2wDw6BDU5BHVyCBnAABvfgGbyCN+PBeDHejY9pa8koZnbBrzI+vwDdMbEZ</latexit>
  • 5. 5 • In vanilla Low-rank Matrix Factorization Problem, 
 a solution find one pair of U and V (single model). • To enhance performance of single model, 
 we can use ensemble methods. • We can say some algorithm is low-rank matrix approximation
 when it finds multiple low-rank matrices to complete final matrix. • For example, • [ICML’13] LLoRMA: Local Low-rank Matrix Approximation • [NIPS’17] MRMA: Mixture Rank Matrix Approximation • [AAAI’17] GLOMA: Global Information in Local Matrix Approximation • [ICML’16] SMA: Low-rank Matrix Approximation with Stability LOW-RANK MATRIX APPROXIMATION
  • 6. 6 • The Matrix R we have to complete is very sparse. • We only know less than 1% of total entries. • Thus, single out-of-distribution entry 
 may significantly change the entire matrix. • For example, • Nowadays, due to the data poisoning attack, stability is becoming more important. • Data Poisoning Attack on Collaborative Filtering [NIPS’16, ’18]:
 Promote and demote specific items by creating many accounts that. MOTIVATION OF THIS PAPER  1 1 2 ? =  1 2 ⇥ ⇥ 1 1 ⇤ <latexit sha1_base64="8fmmBna+sU9of3gbsx7yA7IA1kw=">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</latexit><latexit sha1_base64="8fmmBna+sU9of3gbsx7yA7IA1kw=">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</latexit><latexit sha1_base64="8fmmBna+sU9of3gbsx7yA7IA1kw=">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</latexit><latexit sha1_base64="8fmmBna+sU9of3gbsx7yA7IA1kw=">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</latexit>  1 4 2 ? =  1 2 ⇥ ⇥ 1 4 ⇤ <latexit sha1_base64="hFXaPK74Z0k1muWfXYLuWvrtG5Q=">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</latexit><latexit sha1_base64="hFXaPK74Z0k1muWfXYLuWvrtG5Q=">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</latexit><latexit sha1_base64="hFXaPK74Z0k1muWfXYLuWvrtG5Q=">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</latexit><latexit sha1_base64="hFXaPK74Z0k1muWfXYLuWvrtG5Q=">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</latexit> 2 8
  • 7. 7 • [NIPS’07] Probabilistic Matrix Factorization
 • To handle noise by adding regularization terms • Generalization error is too high. • Koren handle stability issue by cross validation. • Cross-validation technique have drawback that the amount of data available for model learning is reduced. • [ICML’13] LLoRMA suggested ensemble technique. • Ensemble methods are computationally expensive due to the training sub models. • →This paper will suggests stable ensemble method 
 which is not computationally expensive. EXISTING WORKS ˆR = arg max U,V Loss(R, UV T ) + (kUk2 F + kV k2 F ) <latexit sha1_base64="Y0JKIfom9pegCScIoEXp7k/J3XQ=">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</latexit><latexit sha1_base64="Y0JKIfom9pegCScIoEXp7k/J3XQ=">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</latexit><latexit sha1_base64="Y0JKIfom9pegCScIoEXp7k/J3XQ=">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</latexit><latexit sha1_base64="Y0JKIfom9pegCScIoEXp7k/J3XQ=">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</latexit>
  • 8. CONTENTS 1. Motivation 2. Stability of LRMA 3. SMA Algorithm 4. Experiments 5. Discussion
  • 9. 9 • To suggest stable matrix approximation method: SMA • First introduce the stability notion in LRMA. • Develop theoretical guidelines for deriving LRMA solutions with high stability. • Formulate a new optimization problem for achieving stable LRMA • Develop a stochastic gradient descent method to solve the new optimization problem. • Show effectiveness in real datasets (Movielens, Netflix) • The key contributions • (1) First introduces the stability concept in LRMA. • (2) Proposes a stable LRMA algorithm with high stability and high generalization performance. • (3) Makes significant improvement in prediction accuracy. OVERVIEW OF THIS PAPER
  • 10. 10 • Replacing one element in the training set
 does not result in significant change to the algorithm’s output
 in a stable learning algorithm. • Bousquet & Elisseeff (2001), Lan et al., (2008) London et al., (2013) demonstrated. • Var[ Training error ] is small. • Training error ≈ Test error • This paper introduce this concepts to LRMA algorithm. WHAT IS ALGORITHMIC STABILITY?
  • 11. 11 • Root Mean Square Error (RMSE) is a common evaluation metric for recommendation tasks. STABILITY W.R.T. 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  • 12. • For any , Choose a subset of entries from R uniformly. • For a given , is -stable if 12 • For example, • Any two subsets of entries from R. • Approximating R by is -stable. • Approximating R by is -stable. • is more stable than if . • Minimizing leads higher generalization performance than . • Smaller means more stable. STABILITY W.R.T. 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  • 13. 13 • Experiments on Movielens 1M dataset using RSVD (Patrek, 2007). • In the existing LRMA method, • The test error becomes lower as the rank increases. • However, the stability becomes lower and the generalization error increases as the rank increases. • Can’t provide stable recommendation even the rank is as low as 20. STABILITY VS. GENERALIZATION ERROR Pr h |D(ˆR)D⌦(ˆR)|✏ i <latexitsha1_base64="6vGmXafZrEFLI7uLjbfnsPQBIWI=">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</latexit><latexitsha1_base64="6vGmXafZrEFLI7uLjbfnsPQBIWI=">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</latexit><latexitsha1_base64="6vGmXafZrEFLI7uLjbfnsPQBIWI=">AAACSHicbVBBb9MwGHXKBls3oMBxF2vVpHGgSiYk4DYJDpygm9ZtUh2VL+6X1prjRPYXpCrL3+PCjRv/gQsHmHab00Xa2HiSpaf33mf7e0mhlaMw/Bl0HqysPny0tt7d2Hz85Gnv2fNjl5dW4kjmOrenCTjUyuCIFGk8LSxClmg8Sc7eN/7JV7RO5eaIFgXGGcyMSpUE8tKk90UMLRcaUxrzcy4yoLkEXX2od8UcqDqsX/JXt+VJJT5nOIOa3wRanDf3cIGFUzo3XFg1m1M86fXDQbgEv0+ilvRZi+Gk90NMc1lmaEhqcG4chQXFFVhSUmPdFaXDAuQZzHDsqYEMXVwtm6j5jlemPM2tP4b4Ur09UUHm3CJLfLLZyd31GvF/3rik9G1cKVOUhEZeP5SWmlPOm1r5VFmUpBeegLTK/5XLOViQ5Mvv+hKiuyvfJ6O9wbtBdPC6v/+pbWONbbFttssi9obts49syEZMsm/sF/vD/gbfg9/BRXB5He0E7cwL9g86nSsfzLBT</latexit><latexitsha1_base64="6vGmXafZrEFLI7uLjbfnsPQBIWI=">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</latexit> rank=5 rank=10 rank=15 rank=20
  • 14. 14 • remove a subset of easily predictable entries from Ω. • Minimizing both and together will make more stable solution 
 than the solution of minimizing only. • Formally, KEY FINDING OF THIS PAPER ⌦0 <latexit sha1_base64="PQP4LdTFjijPYsqIUrCMjY8bFsk=">AAAB+nicbVBNS8NAEN3Ur1q/Yj16WSyip5KIoN4KXjxpBWMLTSib7aRdutmE3Y1aQv+KFw8qXv0l3vw3btsctPXBwOO9GWbmhSlnSjvOt1VaWl5ZXSuvVzY2t7Z37N3qvUoyScGjCU9kOyQKOBPgaaY5tFMJJA45tMLh5cRvPYBULBF3epRCEJO+YBGjRBupa1f9mxj65Aj7HCJNpEweu3bNqTtT4EXiFqSGCjS79pffS2gWg9CUE6U6rpPqICdSM8phXPEzBSmhQ9KHjqGCxKCCfHr7GB8apYejRJoSGk/V3xM5iZUaxaHpjIkeqHlvIv7ndTIdnQc5E2mmQdDZoijjWCd4EgTuMQlU85EhhEpmbsV0QCSh2sRVMSG48y8vEu+kflF3b09rjesijTLaRwfoGLnoDDXQFWoiD1H0hJ7RK3qzxtaL9W59zFpLVjGzh/7A+vwB9rST/Q==</latexit><latexit 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  • 15. 15 • (Proof Sketch of Theorem 1) • Let • By Markov’s inequality • Based on Hoeffding’s Lemma, • Similar steps on • Via simple calculations, we can prove Theorem 1. KEY FINDING OF THIS PAPER Hoeffding’s Lemma • Let be random variable. • E[X] = 0, and . • Then, for any Pr [X 2 [a, b]] = 1<latexit sha1_base64="vORvSu2v4VOymS2E5QVm5MFkhBw=">AAACCnicbVBNS8NAEN3Ur1q/oh69LC2CBymJCOpBKHjxJBWMLSShbLabdulmE3YnQim9e/GvePGg4tVf4M1/47bNQVsfDDzem2FmXpQJrsFxvq3S0vLK6lp5vbKxubW9Y+/u3es0V5R5NBWpakdEM8El84CDYO1MMZJEgrWiwdXEbz0wpXkq72CYsTAhPcljTgkYqWNXg6bCgWAx+LiNAy6xT46jEAeK9/oQ4kvsduyaU3emwIvELUgNFWh27K+gm9I8YRKoIFr7rpNBOCIKOBVsXAlyzTJCB6THfEMlSZgOR9NfxvjQKF0cp8qUBDxVf0+MSKL1MIlMZ0Kgr+e9ifif5+cQn4cjLrMcmKSzRXEuMKR4EgzucsUoiKEhhCpubsW0TxShYOKrmBDc+ZcXiXdSv6i7t6e1xk2RRhkdoCo6Qi46Qw10jZrIQxQ9omf0it6sJ+vFerc+Zq0lq5jZR39gff4AACaYtg==</latexit><latexit sha1_base64="vORvSu2v4VOymS2E5QVm5MFkhBw=">AAACCnicbVBNS8NAEN3Ur1q/oh69LC2CBymJCOpBKHjxJBWMLSShbLabdulmE3YnQim9e/GvePGg4tVf4M1/47bNQVsfDDzem2FmXpQJrsFxvq3S0vLK6lp5vbKxubW9Y+/u3es0V5R5NBWpakdEM8El84CDYO1MMZJEgrWiwdXEbz0wpXkq72CYsTAhPcljTgkYqWNXg6bCgWAx+LiNAy6xT46jEAeK9/oQ4kvsduyaU3emwIvELUgNFWh27K+gm9I8YRKoIFr7rpNBOCIKOBVsXAlyzTJCB6THfEMlSZgOR9NfxvjQKF0cp8qUBDxVf0+MSKL1MIlMZ0Kgr+e9ifif5+cQn4cjLrMcmKSzRXEuMKR4EgzucsUoiKEhhCpubsW0TxShYOKrmBDc+ZcXiXdSv6i7t6e1xk2RRhkdoCo6Qi46Qw10jZrIQxQ9omf0it6sJ+vFerc+Zq0lq5jZR39gff4AACaYtg==</latexit><latexit sha1_base64="vORvSu2v4VOymS2E5QVm5MFkhBw=">AAACCnicbVBNS8NAEN3Ur1q/oh69LC2CBymJCOpBKHjxJBWMLSShbLabdulmE3YnQim9e/GvePGg4tVf4M1/47bNQVsfDDzem2FmXpQJrsFxvq3S0vLK6lp5vbKxubW9Y+/u3es0V5R5NBWpakdEM8El84CDYO1MMZJEgrWiwdXEbz0wpXkq72CYsTAhPcljTgkYqWNXg6bCgWAx+LiNAy6xT46jEAeK9/oQ4kvsduyaU3emwIvELUgNFWh27K+gm9I8YRKoIFr7rpNBOCIKOBVsXAlyzTJCB6THfEMlSZgOR9NfxvjQKF0cp8qUBDxVf0+MSKL1MIlMZ0Kgr+e9ifif5+cQn4cjLrMcmKSzRXEuMKR4EgzucsUoiKEhhCpubsW0TxShYOKrmBDc+ZcXiXdSv6i7t6e1xk2RRhkdoCo6Qi46Qw10jZrIQxQ9omf0it6sJ+vFerc+Zq0lq5jZR39gff4AACaYtg==</latexit><latexit sha1_base64="vORvSu2v4VOymS2E5QVm5MFkhBw=">AAACCnicbVBNS8NAEN3Ur1q/oh69LC2CBymJCOpBKHjxJBWMLSShbLabdulmE3YnQim9e/GvePGg4tVf4M1/47bNQVsfDDzem2FmXpQJrsFxvq3S0vLK6lp5vbKxubW9Y+/u3es0V5R5NBWpakdEM8El84CDYO1MMZJEgrWiwdXEbz0wpXkq72CYsTAhPcljTgkYqWNXg6bCgWAx+LiNAy6xT46jEAeK9/oQ4kvsduyaU3emwIvELUgNFWh27K+gm9I8YRKoIFr7rpNBOCIKOBVsXAlyzTJCB6THfEMlSZgOR9NfxvjQKF0cp8qUBDxVf0+MSKL1MIlMZ0Kgr+e9ifif5+cQn4cjLrMcmKSzRXEuMKR4EgzucsUoiKEhhCpubsW0TxShYOKrmBDc+ZcXiXdSv6i7t6e1xk2RRhkdoCo6Qi46Qw10jZrIQxQ9omf0it6sJ+vFerc+Zq0lq5jZR39gff4AACaYtg==</latexit> X 2 R<latexit sha1_base64="Up7QdIdytiotpNtWtEcWYrE7Zo4=">AAAB+HicbVBNS8NAFHypX7V+RT16WSyCp5KIUL0VvHiSKsYWmlA22027dLMJu5tCCf0nXjyoePWnePPfuGlz0NaBhWHmPd7shClnSjvOt1VZW9/Y3Kpu13Z29/YP7MOjJ5VkklCPJDyR3RArypmgnmaa024qKY5DTjvh+KbwOxMqFUvEo56mNIjxULCIEayN1LftLvKZQH6M9SgM84dZ3647DWcOtErcktShRLtvf/mDhGQxFZpwrFTPdVId5FhqRjid1fxM0RSTMR7SnqECx1QF+Tz5DJ0ZZYCiRJonNJqrvzdyHCs1jUMzWSRUy14h/uf1Mh1dBTkTaaapIItDUcaRTlBRAxowSYnmU0MwkcxkRWSEJSbalFUzJbjLX14l3kXjuuHeX9Zbd2UbVTiBUzgHF5rQgltogwcEJvAMr/Bm5daL9W59LEYrVrlzDH9gff4ATeSTCg==</latexit><latexit sha1_base64="Up7QdIdytiotpNtWtEcWYrE7Zo4=">AAAB+HicbVBNS8NAFHypX7V+RT16WSyCp5KIUL0VvHiSKsYWmlA22027dLMJu5tCCf0nXjyoePWnePPfuGlz0NaBhWHmPd7shClnSjvOt1VZW9/Y3Kpu13Z29/YP7MOjJ5VkklCPJDyR3RArypmgnmaa024qKY5DTjvh+KbwOxMqFUvEo56mNIjxULCIEayN1LftLvKZQH6M9SgM84dZ3647DWcOtErcktShRLtvf/mDhGQxFZpwrFTPdVId5FhqRjid1fxM0RSTMR7SnqECx1QF+Tz5DJ0ZZYCiRJonNJqrvzdyHCs1jUMzWSRUy14h/uf1Mh1dBTkTaaapIItDUcaRTlBRAxowSYnmU0MwkcxkRWSEJSbalFUzJbjLX14l3kXjuuHeX9Zbd2UbVTiBUzgHF5rQgltogwcEJvAMr/Bm5daL9W59LEYrVrlzDH9gff4ATeSTCg==</latexit><latexit sha1_base64="Up7QdIdytiotpNtWtEcWYrE7Zo4=">AAAB+HicbVBNS8NAFHypX7V+RT16WSyCp5KIUL0VvHiSKsYWmlA22027dLMJu5tCCf0nXjyoePWnePPfuGlz0NaBhWHmPd7shClnSjvOt1VZW9/Y3Kpu13Z29/YP7MOjJ5VkklCPJDyR3RArypmgnmaa024qKY5DTjvh+KbwOxMqFUvEo56mNIjxULCIEayN1LftLvKZQH6M9SgM84dZ3647DWcOtErcktShRLtvf/mDhGQxFZpwrFTPdVId5FhqRjid1fxM0RSTMR7SnqECx1QF+Tz5DJ0ZZYCiRJonNJqrvzdyHCs1jUMzWSRUy14h/uf1Mh1dBTkTaaapIItDUcaRTlBRAxowSYnmU0MwkcxkRWSEJSbalFUzJbjLX14l3kXjuuHeX9Zbd2UbVTiBUzgHF5rQgltogwcEJvAMr/Bm5daL9W59LEYrVrlzDH9gff4ATeSTCg==</latexit><latexit sha1_base64="Up7QdIdytiotpNtWtEcWYrE7Zo4=">AAAB+HicbVBNS8NAFHypX7V+RT16WSyCp5KIUL0VvHiSKsYWmlA22027dLMJu5tCCf0nXjyoePWnePPfuGlz0NaBhWHmPd7shClnSjvOt1VZW9/Y3Kpu13Z29/YP7MOjJ5VkklCPJDyR3RArypmgnmaa024qKY5DTjvh+KbwOxMqFUvEo56mNIjxULCIEayN1LftLvKZQH6M9SgM84dZ3647DWcOtErcktShRLtvf/mDhGQxFZpwrFTPdVId5FhqRjid1fxM0RSTMR7SnqECx1QF+Tz5DJ0ZZYCiRJonNJqrvzdyHCs1jUMzWSRUy14h/uf1Mh1dBTkTaaapIItDUcaRTlBRAxowSYnmU0MwkcxkRWSEJSbalFUzJbjLX14l3kXjuuHeX9Zbd2UbVTiBUzgHF5rQgltogwcEJvAMr/Bm5daL9W59LEYrVrlzDH9gff4ATeSTCg==</latexit>
  • 16. 16 • From Theorem 1, we know that minimizing 
 is better than minimizing only. • However, the condition of is 
 not necessary. • The conclusion will be the same if . CONDITION RELAXATION 8(i, j) 2 !, |Ri,j ˆRi,j|  D⌦( ˆR)<latexit sha1_base64="QKSqsgmGqwvE3kBA619AMM23eWA=">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</latexit><latexit 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sha1_base64="QKSqsgmGqwvE3kBA619AMM23eWA=">AAACPnicbVA9bxNBEN1LIARDghNKmhEWkiM51h2KFNJFSgoqCBFOInmt09x6bG+yt3fa3YtkXe6fpeEv0NHSUACipWTPuYJ8PGmlt2/maWZekitpXRh+C5aWHz1eebL6tPXs+dr6i/bG5onNCiNoIDKVmbMELSmpaeCkU3SWG8I0UXSaXBzU9dNLMlZm+rOb5zRKcarlRAp0XorbJ3ySGVQKurIH51vApQaepTTFHlzBcVzK3nkF28Bn6MrjqvlfAVcEPEU3E6jKwyrmH2tPt2nbitudsB8uAPdJ1JAOa3AUt7/ycSaKlLQTCq0dRmHuRiUaJ4WiqsULSzmKC5zS0FONKdlRubi/gjdeGYM/xD/tYKH+7ygxtXaeJr6zXtnerdXiQ7Vh4SbvRqXUeeFIi5tBk0KBy6AOE8bSkHBq7gkKI/2uIGZoUDgfecuHEN09+T4ZvO3v9aNPO539D00aq+wVe826LGK7bJ+9Z0dswAS7Zt/ZT/Yr+BL8CH4Hf25al4LG85LdQvD3H1bTrbQ=</latexit> • Let (a set of easily predictable entries) which satisfies that • Let (a set of entries hard to predict) . • For any • are -stable and -stable respectively. • Then, Proposition 1 ! ⇢ ⌦(|⌦| > 2)<latexit 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sha1_base64="ACA68GZluiyAFA/yKy15tzEFgIo=">AAACD3icbZDNSgMxFIUz9a/Wv6pLN8Ei1k2ZKYK6kYIbV1rBsYVOKZn0ThuayQxJRihtH8GNr+LGhYpbt+58G9N2BG09EPhy7r0k9/gxZ0rb9peVWVhcWl7JrubW1jc2t/LbO3cqSiQFl0Y8knWfKOBMgKuZ5lCPJZDQ51Dzexfjeu0epGKRuNX9GJoh6QgWMEq0sVr5Qy8KoUOwpxJfgcbe9eRaxMMfHOJzXD5q5Qt2yZ4Iz4OTQgGlqrbyn147okkIQlNOlGo4dqybAyI1oxxGOS9REBPaIx1oGBQkBNUcTBYa4QPjtHEQSXOExhP398SAhEr1Q990hkR31WxtbP5XayQ6OG0OmIgTDYJOHwoSjnWEx+ngNpNANe8bIFQy81dMu0QSqk2GOROCM7vyPLjl0lnJuTkuVK7SNLJoD+2jInLQCaqgS1RFLqLoAT2hF/RqPVrP1pv1Pm3NWOnMLvoj6+MbsJaaqw==</latexit> ⌦0 = ⌦ !<latexit 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sha1_base64="r9UQgW05utQ0BxEYAZEYM48s6kw=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG8FL56kgrGFNpTNZtou3Wzi7kYooX/CiwcVr/4eb/4bt20O2vpg4PHeDDPzwlRwbVz32ymtrK6tb5Q3K1vbO7t71f2DB51kiqHPEpGodkg1Ci7RN9wIbKcKaRwKbIWj66nfekKleSLvzTjFIKYDyfucUWOldjdCYWjP61Vrbt2dgSwTryA1KNDsVb+6UcKyGKVhgmrd8dzUBDlVhjOBk0o305hSNqID7FgqaYw6yGf3TsiJVSLST5QtachM/T2R01jrcRzazpiaoV70puJ/Xicz/csg5zLNDEo2X9TPBDEJmT5PIq6QGTG2hDLF7a2EDamizNiIKjYEb/HlZeKf1a/q3t15rXFbpFGGIziGU/DgAhpwA03wgYGAZ3iFN+fReXHenY95a8kpZg7hD5zPHyphj6A=</latexit> 2<latexit 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sha1_base64="lbsE7NtCbH6B192W/4i6YwyXzIU=">AAAB7nicbVBNS8NAEJ34WetX1aOXxSJ4KkkR1FvBiyepYG2hDWWzmbRLN5u4uxFK6J/w4kHFq7/Hm//GbZuDtj4YeLw3w8y8IBVcG9f9dlZW19Y3Nktb5e2d3b39ysHhg04yxbDFEpGoTkA1Ci6xZbgR2EkV0jgQ2A5G11O//YRK80Tem3GKfkwHkkecUWOlTi9EYWi/3q9U3Zo7A1kmXkGqUKDZr3z1woRlMUrDBNW667mp8XOqDGcCJ+VepjGlbEQH2LVU0hi1n8/unZBTq4QkSpQtachM/T2R01jrcRzYzpiaoV70puJ/Xjcz0aWfc5lmBiWbL4oyQUxCps+TkCtkRowtoUxxeythQ6ooMzaisg3BW3x5mbTqtauad3debdwWaZTgGE7gDDy4gAbcQBNawEDAM7zCm/PovDjvzse8dcUpZo7gD5zPHyvkj6E=</latexit><latexit sha1_base64="lbsE7NtCbH6B192W/4i6YwyXzIU=">AAAB7nicbVBNS8NAEJ34WetX1aOXxSJ4KkkR1FvBiyepYG2hDWWzmbRLN5u4uxFK6J/w4kHFq7/Hm//GbZuDtj4YeLw3w8y8IBVcG9f9dlZW19Y3Nktb5e2d3b39ysHhg04yxbDFEpGoTkA1Ci6xZbgR2EkV0jgQ2A5G11O//YRK80Tem3GKfkwHkkecUWOlTi9EYWi/3q9U3Zo7A1kmXkGqUKDZr3z1woRlMUrDBNW667mp8XOqDGcCJ+VepjGlbEQH2LVU0hi1n8/unZBTq4QkSpQtachM/T2R01jrcRzYzpiaoV70puJ/Xjcz0aWfc5lmBiWbL4oyQUxCps+TkCtkRowtoUxxeythQ6ooMzaisg3BW3x5mbTqtauad3debdwWaZTgGE7gDDy4gAbcQBNawEDAM7zCm/PovDjvzse8dcUpZo7gD5zPHyvkj6E=</latexit> ← the only change from Theorem 1
  • 17. 17 • Theorem 1 and Proposition 1 only prove that 
 it is beneficial to remove easily predictable entries from Ω. • However, does not show how many entries we should remove. • Theorem 2 shows that removing more entries that satisfy 
 
 
 can yield better . HOW MANY ENTRIES SHOULD WE REMOVE? ⌦0 <latexit sha1_base64="/HK2LTN3a6E62rbq15wJMtnMcQM=">AAAB7XicbVBNS8NAEJ34WetX1aOXxSJ6KokI6q3gxZNWMLbQhrLZTtqlm03Y3Qgl9Ed48aDi1f/jzX/jts1BWx8MPN6bYWZemAqujet+O0vLK6tr66WN8ubW9s5uZW//USeZYuizRCSqFVKNgkv0DTcCW6lCGocCm+HweuI3n1BpnsgHM0oxiGlf8ogzaqzU7NzF2Kcn3UrVrblTkEXiFaQKBRrdylenl7AsRmmYoFq3PTc1QU6V4UzguNzJNKaUDWkf25ZKGqMO8um5Y3JslR6JEmVLGjJVf0/kNNZ6FIe2M6ZmoOe9ifif185MdBnkXKaZQclmi6JMEJOQye+kxxUyI0aWUKa4vZWwAVWUGZtQ2Ybgzb+8SPyz2lXNuz+v1m+LNEpwCEdwCh5cQB1uoAE+MBjCM7zCm5M6L8678zFrXXKKmQP4A+fzBzGIjww=</latexit><latexit sha1_base64="/HK2LTN3a6E62rbq15wJMtnMcQM=">AAAB7XicbVBNS8NAEJ34WetX1aOXxSJ6KokI6q3gxZNWMLbQhrLZTtqlm03Y3Qgl9Ed48aDi1f/jzX/jts1BWx8MPN6bYWZemAqujet+O0vLK6tr66WN8ubW9s5uZW//USeZYuizRCSqFVKNgkv0DTcCW6lCGocCm+HweuI3n1BpnsgHM0oxiGlf8ogzaqzU7NzF2Kcn3UrVrblTkEXiFaQKBRrdylenl7AsRmmYoFq3PTc1QU6V4UzguNzJNKaUDWkf25ZKGqMO8um5Y3JslR6JEmVLGjJVf0/kNNZ6FIe2M6ZmoOe9ifif185MdBnkXKaZQclmi6JMEJOQye+kxxUyI0aWUKa4vZWwAVWUGZtQ2Ybgzb+8SPyz2lXNuz+v1m+LNEpwCEdwCh5cQB1uoAE+MBjCM7zCm5M6L8678zFrXXKKmQP4A+fzBzGIjww=</latexit><latexit sha1_base64="/HK2LTN3a6E62rbq15wJMtnMcQM=">AAAB7XicbVBNS8NAEJ34WetX1aOXxSJ6KokI6q3gxZNWMLbQhrLZTtqlm03Y3Qgl9Ed48aDi1f/jzX/jts1BWx8MPN6bYWZemAqujet+O0vLK6tr66WN8ubW9s5uZW//USeZYuizRCSqFVKNgkv0DTcCW6lCGocCm+HweuI3n1BpnsgHM0oxiGlf8ogzaqzU7NzF2Kcn3UrVrblTkEXiFaQKBRrdylenl7AsRmmYoFq3PTc1QU6V4UzguNzJNKaUDWkf25ZKGqMO8um5Y3JslR6JEmVLGjJVf0/kNNZ6FIe2M6ZmoOe9ifif185MdBnkXKaZQclmi6JMEJOQye+kxxUyI0aWUKa4vZWwAVWUGZtQ2Ybgzb+8SPyz2lXNuz+v1m+LNEpwCEdwCh5cQB1uoAE+MBjCM7zCm5M6L8678zFrXXKKmQP4A+fzBzGIjww=</latexit><latexit sha1_base64="/HK2LTN3a6E62rbq15wJMtnMcQM=">AAAB7XicbVBNS8NAEJ34WetX1aOXxSJ6KokI6q3gxZNWMLbQhrLZTtqlm03Y3Qgl9Ed48aDi1f/jzX/jts1BWx8MPN6bYWZemAqujet+O0vLK6tr66WN8ubW9s5uZW//USeZYuizRCSqFVKNgkv0DTcCW6lCGocCm+HweuI3n1BpnsgHM0oxiGlf8ogzaqzU7NzF2Kcn3UrVrblTkEXiFaQKBRrdylenl7AsRmmYoFq3PTc1QU6V4UzguNzJNKaUDWkf25ZKGqMO8um5Y3JslR6JEmVLGjJVf0/kNNZ6FIe2M6ZmoOe9ifif185MdBnkXKaZQclmi6JMEJOQye+kxxUyI0aWUKa4vZWwAVWUGZtQ2Ybgzb+8SPyz2lXNuz+v1m+LNEpwCEdwCh5cQB1uoAE+MBjCM7zCm5M6L8678zFrXXKKmQP4A+fzBzGIjww=</latexit> • Let satisfying • Let . • For any • 
 are -stable and -stable respectively. • Then, Theorem 2 1<latexit sha1_base64="r9UQgW05utQ0BxEYAZEYM48s6kw=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG8FL56kgrGFNpTNZtou3Wzi7kYooX/CiwcVr/4eb/4bt20O2vpg4PHeDDPzwlRwbVz32ymtrK6tb5Q3K1vbO7t71f2DB51kiqHPEpGodkg1Ci7RN9wIbKcKaRwKbIWj66nfekKleSLvzTjFIKYDyfucUWOldjdCYWjP61Vrbt2dgSwTryA1KNDsVb+6UcKyGKVhgmrd8dzUBDlVhjOBk0o305hSNqID7FgqaYw6yGf3TsiJVSLST5QtachM/T2R01jrcRzazpiaoV70puJ/Xicz/csg5zLNDEo2X9TPBDEJmT5PIq6QGTG2hDLF7a2EDamizNiIKjYEb/HlZeKf1a/q3t15rXFbpFGGIziGU/DgAhpwA03wgYGAZ3iFN+fReXHenY95a8kpZg7hD5zPHyphj6A=</latexit><latexit 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  • 18. 18 • Theorem 1 and Theorem 2 only consider choosing one Ω’ 
 to find stable matrix approximations. • Is it possible to choose more than one Ω’ that satisfy stable condition, and yield more stable solutions by minimizing them all together. • The Theorem 3 shows that incorporating K such entry sets (Ω’) 
 will be more stable than incorporating any (K − 1) out of the K entry sets. BETTER STABILITY BY MULTIPLE SUBSETS • Let satisfying • Let . • For any • and 
 are -stable and -stable respectively. • Then, Theorem 3 1<latexit sha1_base64="r9UQgW05utQ0BxEYAZEYM48s6kw=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG8FL56kgrGFNpTNZtou3Wzi7kYooX/CiwcVr/4eb/4bt20O2vpg4PHeDDPzwlRwbVz32ymtrK6tb5Q3K1vbO7t71f2DB51kiqHPEpGodkg1Ci7RN9wIbKcKaRwKbIWj66nfekKleSLvzTjFIKYDyfucUWOldjdCYWjP61Vrbt2dgSwTryA1KNDsVb+6UcKyGKVhgmrd8dzUBDlVhjOBk0o305hSNqID7FgqaYw6yGf3TsiJVSLST5QtachM/T2R01jrcRzazpiaoV70puJ/Xicz/csg5zLNDEo2X9TPBDEJmT5PIq6QGTG2hDLF7a2EDamizNiIKjYEb/HlZeKf1a/q3t15rXFbpFGGIziGU/DgAhpwA03wgYGAZ3iFN+fReXHenY95a8kpZg7hD5zPHyphj6A=</latexit><latexit 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  • 19. 19 • One of the Guidelines to obtain hard predictable subsets of Ω? • 1) Choose satisfying
 Let • 2) Divide into K non-overlapping subsets with the condition that
 , and let • 3) Minimize • Is it better than minimizing ? • → Theorem 4 HOW TO FIND MULTIPLE SUBSETS? 8(i, j) 2 !, |Ri,j ˆRi,j|  D⌦( ˆR)<latexit sha1_base64="QKSqsgmGqwvE3kBA619AMM23eWA=">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</latexit><latexit 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sha1_base64="Z+YX4EHR+yhemC3izIEkXju0pZI=">AAACAnicbVDLSsNAFJ3UV62vqDvdDBbBhZREBHVXcCMIUsHYQhPCZDJph85kwsxEKKHgxl9x40LFrV/hzr9x2mahrQcu93DOvczcE2WMKu0431ZlYXFpeaW6Wltb39jcsrd37pXIJSYeFkzIToQUYTQlnqaakU4mCeIRI+1ocDn22w9EKirSOz3MSMBRL6UJxUgbKbT3fMFJD4XuMfRxLLQyfapch3bdaTgTwHnilqQOSrRC+8uPBc45STVmSKmu62Q6KJDUFDMyqvm5IhnCA9QjXUNTxIkKiskNI3holBgmQppKNZyovzcKxJUa8shMcqT7atYbi/953Vwn50FB0yzXJMXTh5KcQS3gOBAYU0mwZkNDEJbU/BXiPpIIaxNbzYTgzp48T7yTxkXDvT2tN2/KNKpgHxyAI+CCM9AEV6AFPIDBI3gGr+DNerJerHfrYzpascqdXfAH1ucP5ZOWpA==</latexit><latexit sha1_base64="Z+YX4EHR+yhemC3izIEkXju0pZI=">AAACAnicbVDLSsNAFJ3UV62vqDvdDBbBhZREBHVXcCMIUsHYQhPCZDJph85kwsxEKKHgxl9x40LFrV/hzr9x2mahrQcu93DOvczcE2WMKu0431ZlYXFpeaW6Wltb39jcsrd37pXIJSYeFkzIToQUYTQlnqaakU4mCeIRI+1ocDn22w9EKirSOz3MSMBRL6UJxUgbKbT3fMFJD4XuMfRxLLQyfapch3bdaTgTwHnilqQOSrRC+8uPBc45STVmSKmu62Q6KJDUFDMyqvm5IhnCA9QjXUNTxIkKiskNI3holBgmQppKNZyovzcKxJUa8shMcqT7atYbi/953Vwn50FB0yzXJMXTh5KcQS3gOBAYU0mwZkNDEJbU/BXiPpIIaxNbzYTgzp48T7yTxkXDvT2tN2/KNKpgHxyAI+CCM9AEV6AFPIDBI3gGr+DNerJerHfrYzpascqdXfAH1ucP5ZOWpA==</latexit> ⌦k = ⌦ !k (81  k  K)<latexit 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  • 20. 20 HOW TO FIND MULTIPLE SUBSETS? • Let satisfying • Divide into K non-overlapping subsets 
 with the conditions that • Let and . • For any • and 
 are -stable and -stable respectively. • Then, Theorem 4 1<latexit sha1_base64="r9UQgW05utQ0BxEYAZEYM48s6kw=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG8FL56kgrGFNpTNZtou3Wzi7kYooX/CiwcVr/4eb/4bt20O2vpg4PHeDDPzwlRwbVz32ymtrK6tb5Q3K1vbO7t71f2DB51kiqHPEpGodkg1Ci7RN9wIbKcKaRwKbIWj66nfekKleSLvzTjFIKYDyfucUWOldjdCYWjP61Vrbt2dgSwTryA1KNDsVb+6UcKyGKVhgmrd8dzUBDlVhjOBk0o305hSNqID7FgqaYw6yGf3TsiJVSLST5QtachM/T2R01jrcRzazpiaoV70puJ/Xicz/csg5zLNDEo2X9TPBDEJmT5PIq6QGTG2hDLF7a2EDamizNiIKjYEb/HlZeKf1a/q3t15rXFbpFGGIziGU/DgAhpwA03wgYGAZ3iFN+fReXHenY95a8kpZg7hD5zPHyphj6A=</latexit><latexit 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sha1_base64="r9UQgW05utQ0BxEYAZEYM48s6kw=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG8FL56kgrGFNpTNZtou3Wzi7kYooX/CiwcVr/4eb/4bt20O2vpg4PHeDDPzwlRwbVz32ymtrK6tb5Q3K1vbO7t71f2DB51kiqHPEpGodkg1Ci7RN9wIbKcKaRwKbIWj66nfekKleSLvzTjFIKYDyfucUWOldjdCYWjP61Vrbt2dgSwTryA1KNDsVb+6UcKyGKVhgmrd8dzUBDlVhjOBk0o305hSNqID7FgqaYw6yGf3TsiJVSLST5QtachM/T2R01jrcRzazpiaoV70puJ/Xicz/csg5zLNDEo2X9TPBDEJmT5PIq6QGTG2hDLF7a2EDamizNiIKjYEb/HlZeKf1a/q3t15rXFbpFGGIziGU/DgAhpwA03wgYGAZ3iFN+fReXHenY95a8kpZg7hD5zPHyphj6A=</latexit> 2<latexit sha1_base64="lbsE7NtCbH6B192W/4i6YwyXzIU=">AAAB7nicbVBNS8NAEJ34WetX1aOXxSJ4KkkR1FvBiyepYG2hDWWzmbRLN5u4uxFK6J/w4kHFq7/Hm//GbZuDtj4YeLw3w8y8IBVcG9f9dlZW19Y3Nktb5e2d3b39ysHhg04yxbDFEpGoTkA1Ci6xZbgR2EkV0jgQ2A5G11O//YRK80Tem3GKfkwHkkecUWOlTi9EYWi/3q9U3Zo7A1kmXkGqUKDZr3z1woRlMUrDBNW667mp8XOqDGcCJ+VepjGlbEQH2LVU0hi1n8/unZBTq4QkSpQtachM/T2R01jrcRzYzpiaoV70puJ/Xjcz0aWfc5lmBiWbL4oyQUxCps+TkCtkRowtoUxxeythQ6ooMzaisg3BW3x5mbTqtauad3debdwWaZTgGE7gDDy4gAbcQBNawEDAM7zCm/PovDjvzse8dcUpZo7gD5zPHyvkj6E=</latexit><latexit sha1_base64="lbsE7NtCbH6B192W/4i6YwyXzIU=">AAAB7nicbVBNS8NAEJ34WetX1aOXxSJ4KkkR1FvBiyepYG2hDWWzmbRLN5u4uxFK6J/w4kHFq7/Hm//GbZuDtj4YeLw3w8y8IBVcG9f9dlZW19Y3Nktb5e2d3b39ysHhg04yxbDFEpGoTkA1Ci6xZbgR2EkV0jgQ2A5G11O//YRK80Tem3GKfkwHkkecUWOlTi9EYWi/3q9U3Zo7A1kmXkGqUKDZr3z1woRlMUrDBNW667mp8XOqDGcCJ+VepjGlbEQH2LVU0hi1n8/unZBTq4QkSpQtachM/T2R01jrcRzYzpiaoV70puJ/Xjcz0aWfc5lmBiWbL4oyQUxCps+TkCtkRowtoUxxeythQ6ooMzaisg3BW3x5mbTqtauad3debdwWaZTgGE7gDDy4gAbcQBNawEDAM7zCm/PovDjvzse8dcUpZo7gD5zPHyvkj6E=</latexit><latexit sha1_base64="lbsE7NtCbH6B192W/4i6YwyXzIU=">AAAB7nicbVBNS8NAEJ34WetX1aOXxSJ4KkkR1FvBiyepYG2hDWWzmbRLN5u4uxFK6J/w4kHFq7/Hm//GbZuDtj4YeLw3w8y8IBVcG9f9dlZW19Y3Nktb5e2d3b39ysHhg04yxbDFEpGoTkA1Ci6xZbgR2EkV0jgQ2A5G11O//YRK80Tem3GKfkwHkkecUWOlTi9EYWi/3q9U3Zo7A1kmXkGqUKDZr3z1woRlMUrDBNW667mp8XOqDGcCJ+VepjGlbEQH2LVU0hi1n8/unZBTq4QkSpQtachM/T2R01jrcRzYzpiaoV70puJ/Xjcz0aWfc5lmBiWbL4oyQUxCps+TkCtkRowtoUxxeythQ6ooMzaisg3BW3x5mbTqtauad3debdwWaZTgGE7gDDy4gAbcQBNawEDAM7zCm/PovDjvzse8dcUpZo7gD5zPHyvkj6E=</latexit><latexit sha1_base64="lbsE7NtCbH6B192W/4i6YwyXzIU=">AAAB7nicbVBNS8NAEJ34WetX1aOXxSJ4KkkR1FvBiyepYG2hDWWzmbRLN5u4uxFK6J/w4kHFq7/Hm//GbZuDtj4YeLw3w8y8IBVcG9f9dlZW19Y3Nktb5e2d3b39ysHhg04yxbDFEpGoTkA1Ci6xZbgR2EkV0jgQ2A5G11O//YRK80Tem3GKfkwHkkecUWOlTi9EYWi/3q9U3Zo7A1kmXkGqUKDZr3z1woRlMUrDBNW667mp8XOqDGcCJ+VepjGlbEQH2LVU0hi1n8/unZBTq4QkSpQtachM/T2R01jrcRzYzpiaoV70puJ/Xjcz0aWfc5lmBiWbL4oyQUxCps+TkCtkRowtoUxxeythQ6ooMzaisg3BW3x5mbTqtauad3debdwWaZTgGE7gDDy4gAbcQBNawEDAM7zCm/PovDjvzse8dcUpZo7gD5zPHyvkj6E=</latexit> ⌦k = ⌦ !k (81  k  K)<latexit sha1_base64="DetsFyQWCyItUB9gRmXqpNsHL2I=">AAACHXicbVDLSsNAFJ3UV62vqEs3g0WoC0siiroQCm4EQStYW2hCmExv2qGTBzMToYR+iRt/xY0LFRduxL9x2mahrQdmOJxzL/fe4yecSWVZ30Zhbn5hcam4XFpZXVvfMDe37mWcCgoNGvNYtHwigbMIGoopDq1EAAl9Dk2/fzHymw8gJIujOzVIwA1JN2IBo0RpyTOPnZsQusTr43M8ofgAO/FEc3DFCWJBOMc2djjg/vi/2vfMslW1xsCzxM5JGeWoe+an04lpGkKkKCdStm0rUW5GhGKUw7DkpBISQvukC21NIxKCdLPxeUO8p5UO1ovoFyk8Vn93ZCSUchD6ujIkqienvZH4n9dOVXDqZixKUgURnQwKUo5VjEdZ4Q4TQBUfaEKoYHpXTHtEEKp0oiUdgj198ixpHFbPqvbtUbl2nadRRDtoF1WQjU5QDV2iOmogih7RM3pFb8aT8WK8Gx+T0oKR92yjPzC+fgAV+Z+U</latexit><latexit 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  • 21. CONTENTS 1. Motivation 2. Stability of LRMA 3. SMA Algorithm 4. Experiments 5. Discussion
  • 22. 22 • Vanilla Low-rank Matrix Factorization: • Model formulation of SMA: • By Theorem 3, Multiple hard predictable subsets are better than single predictable subsets. So, they make model can handle multiple subsets. • Let MODEL FORMULATION ˆR = arg max X D⌦(X), rank(X) = r <latexit sha1_base64="psT5t07Pum7RxcISoUh4ShU0u6I=">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</latexit><latexit sha1_base64="psT5t07Pum7RxcISoUh4ShU0u6I=">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</latexit><latexit sha1_base64="psT5t07Pum7RxcISoUh4ShU0u6I=">AAACI3icbZDLSsNAFIYnXmu9RV26GSyCgpREBBURCrpw5Q2rgSaEk+m0HTqZhJmJWEIexo2v4saFihsXvouT2oW3HwY+/nMOc84fpZwp7Tjv1tj4xOTUdGWmOjs3v7BoLy1fqySThDZJwhPpRaAoZ4I2NdOceqmkEEec3kT9o7J+c0ulYom40oOUBjF0BeswAtpYoX3g90DnlwU+xD7ILvZjuAtzryhB9wjw/LgI/bOYdgFveJtbWILoGzD9MrRrTt0ZCv8FdwQ1NNJ5aL/67YRkMRWacFCq5TqpDnKQmhFOi6qfKZoC6UOXtgwKiKkK8uGRBV43Tht3Emme0Hjofp/IIVZqEEems1xd/a6V5n+1VqY7e0HORJppKsjXR52MY53gMjHcZpISzQcGgEhmdsWkBxKINrlWTQju75P/QnO7vl93L3ZqjdNRGhW0itbQBnLRLmqgE3SOmoige/SIntGL9WA9Wa/W21frmDWaWUE/ZH18Ahrlos8=</latexit><latexit sha1_base64="psT5t07Pum7RxcISoUh4ShU0u6I=">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</latexit> ⌦1, · · · , ⌦K 2 ⌦, 8s 2 [1, K], D⌦x D⌦<latexit sha1_base64="SXP3WFNFTVFQngBeeQU/9jO5SS4=">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</latexit><latexit sha1_base64="SXP3WFNFTVFQngBeeQU/9jO5SS4=">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</latexit><latexit sha1_base64="SXP3WFNFTVFQngBeeQU/9jO5SS4=">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</latexit><latexit sha1_base64="SXP3WFNFTVFQngBeeQU/9jO5SS4=">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</latexit>
  • 23. 23 • The more hard-predictable subsets the better stability. • However, it is expensive to find such “hard predictable subsets”. Why? • Because we don’t know which subset of entries to choose 
 without any prior knowledge. • We can’t know final model without hard-predictable subsets.
 Also, we can’t know hard-predictable subsets without final model.
 (Which came first, the chicken or the egg?) • In other words, • Recall • To obtain such is not trivial, because we can only check if the condition is satisfied with the final model. • But the final model cannot be known before we define and optimize a given loss function. DIFFICULTIES IN SELECTING HARD PREDICTABLE SUBSETS ⌦s<latexit sha1_base64="NbxRFS1uoRou7bg3C7Pnj6iBJyw=">AAAB7nicbVBNS8NAEJ34WetX1aOXYBE8lUQE9Vbw4kkrGFtoQ9lsJ+3S3U3c3Qgl9E948aDi1d/jzX/jts1BWx8MPN6bYWZelHKmjed9O0vLK6tr66WN8ubW9s5uZW//QSeZohjQhCeqFRGNnEkMDDMcW6lCIiKOzWh4NfGbT6g0S+S9GaUYCtKXLGaUGCu1OrcC+6Sru5WqV/OmcBeJX5AqFGh0K1+dXkIzgdJQTrRu+15qwpwowyjHcbmTaUwJHZI+ti2VRKAO8+m9Y/fYKj03TpQtadyp+nsiJ0LrkYhspyBmoOe9ifif185MfBHmTKaZQUlni+KMuyZxJ8+7PaaQGj6yhFDF7K0uHRBFqLERlW0I/vzLiyQ4rV3W/Luzav2mSKMEh3AEJ+DDOdThGhoQAAUOz/AKb86j8+K8Ox+z1iWnmDmAP3A+fwBbq4/B</latexit><latexit sha1_base64="NbxRFS1uoRou7bg3C7Pnj6iBJyw=">AAAB7nicbVBNS8NAEJ34WetX1aOXYBE8lUQE9Vbw4kkrGFtoQ9lsJ+3S3U3c3Qgl9E948aDi1d/jzX/jts1BWx8MPN6bYWZelHKmjed9O0vLK6tr66WN8ubW9s5uZW//QSeZohjQhCeqFRGNnEkMDDMcW6lCIiKOzWh4NfGbT6g0S+S9GaUYCtKXLGaUGCu1OrcC+6Sru5WqV/OmcBeJX5AqFGh0K1+dXkIzgdJQTrRu+15qwpwowyjHcbmTaUwJHZI+ti2VRKAO8+m9Y/fYKj03TpQtadyp+nsiJ0LrkYhspyBmoOe9ifif185MfBHmTKaZQUlni+KMuyZxJ8+7PaaQGj6yhFDF7K0uHRBFqLERlW0I/vzLiyQ4rV3W/Luzav2mSKMEh3AEJ+DDOdThGhoQAAUOz/AKb86j8+K8Ox+z1iWnmDmAP3A+fwBbq4/B</latexit><latexit sha1_base64="NbxRFS1uoRou7bg3C7Pnj6iBJyw=">AAAB7nicbVBNS8NAEJ34WetX1aOXYBE8lUQE9Vbw4kkrGFtoQ9lsJ+3S3U3c3Qgl9E948aDi1d/jzX/jts1BWx8MPN6bYWZelHKmjed9O0vLK6tr66WN8ubW9s5uZW//QSeZohjQhCeqFRGNnEkMDDMcW6lCIiKOzWh4NfGbT6g0S+S9GaUYCtKXLGaUGCu1OrcC+6Sru5WqV/OmcBeJX5AqFGh0K1+dXkIzgdJQTrRu+15qwpwowyjHcbmTaUwJHZI+ti2VRKAO8+m9Y/fYKj03TpQtadyp+nsiJ0LrkYhspyBmoOe9ifif185MfBHmTKaZQUlni+KMuyZxJ8+7PaaQGj6yhFDF7K0uHRBFqLERlW0I/vzLiyQ4rV3W/Luzav2mSKMEh3AEJ+DDOdThGhoQAAUOz/AKb86j8+K8Ox+z1iWnmDmAP3A+fwBbq4/B</latexit><latexit sha1_base64="NbxRFS1uoRou7bg3C7Pnj6iBJyw=">AAAB7nicbVBNS8NAEJ34WetX1aOXYBE8lUQE9Vbw4kkrGFtoQ9lsJ+3S3U3c3Qgl9E948aDi1d/jzX/jts1BWx8MPN6bYWZelHKmjed9O0vLK6tr66WN8ubW9s5uZW//QSeZohjQhCeqFRGNnEkMDDMcW6lCIiKOzWh4NfGbT6g0S+S9GaUYCtKXLGaUGCu1OrcC+6Sru5WqV/OmcBeJX5AqFGh0K1+dXkIzgdJQTrRu+15qwpwowyjHcbmTaUwJHZI+ti2VRKAO8+m9Y/fYKj03TpQtadyp+nsiJ0LrkYhspyBmoOe9ifif185MfBHmTKaZQUlni+KMuyZxJ8+7PaaQGj6yhFDF7K0uHRBFqLERlW0I/vzLiyQ4rV3W/Luzav2mSKMEh3AEJ+DDOdThGhoQAAUOz/AKb86j8+K8Ox+z1iWnmDmAP3A+fwBbq4/B</latexit> ⌦1, · · · , ⌦K ⇢ ⌦, 8s 2 [1, K], D⌦s D⌦<latexit sha1_base64="ZfIrYCjAEya1Nq46ClKLphyJYcQ=">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</latexit><latexit sha1_base64="ZfIrYCjAEya1Nq46ClKLphyJYcQ=">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</latexit><latexit sha1_base64="ZfIrYCjAEya1Nq46ClKLphyJYcQ=">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</latexit><latexit sha1_base64="ZfIrYCjAEya1Nq46ClKLphyJYcQ=">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</latexit>
  • 24. 24 • To address this issue: • 1) Approximate the targeted matrix R with existing LRMA solutions. • 2) for each entry (i, j) ∈ Ω, it is chosen with large probability if 
 and small probability otherwise • 3) Obtain Ω’ by removing the chosen entries to satisfy the condition of Proposition 1, or probe Ω’ to find hard predictable subsets that satisfy the condition of Theorem 4. • → In this paper they chose Theorem 4. • Underlying Assumptions: • LRMF methods will not dramatically differ from final model of SMA • we can ensure that Ω’ will satisfy with high probability. HARD PREDICTABLE SUBSETS SELECTION
  • 25. 25 THE SMA ALGORITHM Pretraining Part: Finding Ω’ for SMA algorithm using Theorem 4 Core Training Part: Add Regularizer to avoid overfitting
  • 26. CONTENTS 1. Motivation 2. Stability of LRMA 3. SMA Algorithm 4. Experiments 5. Discussion
  • 27. 27 ACCURACY COMPARISONS • Test RMSE on two benchmark datasets. • Compared to other state-of-the-arts models, it showed good performance. • SMA demonstrates similar stability to ensembles such as DFC, LLORMA, and WEMAREC. • Most importantly, other models can not guarantee a gap between training and test errors, but SMA is possible.
  • 28. 28 • RMSE vs the number of subsets K • As the number of subsets increases, the RMSE decreases. • RMSE vs rank r • As the rank r increases, the SMA model does not overfit and the RMSE decreases. EFFECTS OF HYPER PARAMETERS
  • 29. CONTENTS 1. Motivation 2. Stability of LRMA 3. SMA Algorithm 4. Experiments 5. Discussion
  • 30. 30 • Why did the authors choose Theorem 4 rather than Proposition 1 in the final algorithm? • I think we can make final algorithm with Proposition 1. • The authors assumed that the model used in pretrain and the final model are similar. Is the random sampling steps are necessary? • I think they would better suggest reason why they use random sampling? • Does this algorithm work for other fields or models? • There is no assumption related to low-rank matrix factorization model in Theorem 1 or others. I think we can use this concept in other models. • I think this method can be regarded as regularization. • Do we really need Theorem 3? • I think Theorem 4 is enough for final SMA algorithm. DISCUSSION POINTS