Social-sparsity brain decoders: faster spatial sparsity
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This document presents a new method called social-sparsity for brain decoding from fMRI data that provides faster computation times compared to other spatial sparsity methods. Social-sparsity uses a heuristic that forgoes couplings between neighboring voxels during soft-thresholding, allowing it to run 10x faster than total variation regularization and 3x faster than graph-net methods while maintaining or improving prediction accuracy. Evaluation on visual recognition tasks found social-sparsity produced brain maps that segmented relevant regions well with improved run times over other methods.
![Brain decoder maps and prediction accuracy
Face vs house visual recognition [Haxby... 2001]
SVM
error: 26%
G Varoquaux 3](https://image.slidesharecdn.com/slides-160622152611/85/Social-sparsity-brain-decoders-faster-spatial-sparsity-4-320.jpg)
![Brain decoder maps and prediction accuracy
Face vs house visual recognition [Haxby... 2001]
Ridge
error: 15%
G Varoquaux 3](https://image.slidesharecdn.com/slides-160622152611/85/Social-sparsity-brain-decoders-faster-spatial-sparsity-5-320.jpg)
![Brain decoder maps and prediction accuracy
Face vs house visual recognition [Haxby... 2001]
Sparse model
error: 19%
Which decoder predicts best?
How to get good decoder maps?
G Varoquaux 3](https://image.slidesharecdn.com/slides-160622152611/85/Social-sparsity-brain-decoders-faster-spatial-sparsity-6-320.jpg)
![Sparse models
Ill-posed inverse problem
minw
l(y − Xw) + λ w 1
A priori:
A small fraction of the voxels are predictive
Sparse models to select relevant regions?
[Yamashita... 2008, Carroll... 2009]
G Varoquaux 4](https://image.slidesharecdn.com/slides-160622152611/85/Social-sparsity-brain-decoders-faster-spatial-sparsity-7-320.jpg)
![Sparse models
Ill-posed inverse problem ⇒ regularization
minw
l(y − Xw) + λ w 1
sparsity inducing norm 1 norm,
or elastic net
Elastic Net Can only select a
subset of relevant
voxels.
[Varoquaux... 2012]
G Varoquaux 4](https://image.slidesharecdn.com/slides-160622152611/85/Social-sparsity-brain-decoders-faster-spatial-sparsity-9-320.jpg)
![Spatial sparse penalties
Spatial regularization, total variation
minw
l(y − Xw) + λ
i
( w)i 2
12 norm: 1 norm of the
gradient magnitudePenalize the image gradient:
Shrinks jointly x , y , and z
Elastic Net TV + 1
[Gramfort... 2013]
G Varoquaux 5](https://image.slidesharecdn.com/slides-160622152611/85/Social-sparsity-brain-decoders-faster-spatial-sparsity-10-320.jpg)
![Spatial sparse penalties
Spatial regularization, total variation
minw
l(y − Xw) + λ
i
( w)i 2
More generally: analysis sparsity [Eickenberg... 2015]
Sparse in a transformation of the weights:
minw
l(y − Xw) + λ K w 21
For instance: overlapping blocks
(Kw)1 ↔ G1
(Kw)2 ↔ G2
... G2
G1
G Varoquaux 6](https://image.slidesharecdn.com/slides-160622152611/85/Social-sparsity-brain-decoders-faster-spatial-sparsity-11-320.jpg)
![Good convergence of solvers is important
Spatial regularization, total variation
minw
l(y − Xw) + λ
i
( w)i 2
x=17
L R
z=-17
Stopping: ∆E < 10−1
x=17
L R
z=-17
Stopping: ∆E < 10−5
[Dohmatob... 2014]
G Varoquaux 7](https://image.slidesharecdn.com/slides-160622152611/85/Social-sparsity-brain-decoders-faster-spatial-sparsity-12-320.jpg)
![Social sparsity: “soft-threshold” neighboring voxels
Sparsity must be combined with spatial structure
Convex solvers for non-local sparsity are expensive
Not separable
Social sparsity:
forget the coupling between
soft thresholding
[Kowalski... 2013]
N1
x1
N2
x2
G Varoquaux 10](https://image.slidesharecdn.com/slides-160622152611/85/Social-sparsity-brain-decoders-faster-spatial-sparsity-18-320.jpg)
Social-sparsity brain decoders: faster spatial sparsity
- 1. Social-sparsity brain decoders: faster spatial sparsity G. Varoquaux, M. Kowalski, B. Thirion
- 2. Brain decoding with linear models Design matrix × Coefficients = Coefficients are brain maps Target G Varoquaux 2
- 3. Brain decoding with linear models Design matrix × Coefficients = Coefficients are brain maps Minimize the error: l(y − Xw) Target G Varoquaux 2
- 4. Brain decoder maps and prediction accuracy Face vs house visual recognition [Haxby... 2001] SVM error: 26% G Varoquaux 3
- 5. Brain decoder maps and prediction accuracy Face vs house visual recognition [Haxby... 2001] Ridge error: 15% G Varoquaux 3
- 6. Brain decoder maps and prediction accuracy Face vs house visual recognition [Haxby... 2001] Sparse model error: 19% Which decoder predicts best? How to get good decoder maps? G Varoquaux 3
- 7. Sparse models Ill-posed inverse problem minw l(y − Xw) + λ w 1 A priori: A small fraction of the voxels are predictive Sparse models to select relevant regions? [Yamashita... 2008, Carroll... 2009] G Varoquaux 4
- 8. Sparse models Ill-posed inverse problem ⇒ regularization minw l(y − Xw) + λ w 1 sparsity inducing norm 1 norm, or elastic net Elastic Net G Varoquaux 4
- 9. Sparse models Ill-posed inverse problem ⇒ regularization minw l(y − Xw) + λ w 1 sparsity inducing norm 1 norm, or elastic net Elastic Net Can only select a subset of relevant voxels. [Varoquaux... 2012] G Varoquaux 4
- 10. Spatial sparse penalties Spatial regularization, total variation minw l(y − Xw) + λ i ( w)i 2 12 norm: 1 norm of the gradient magnitudePenalize the image gradient: Shrinks jointly x , y , and z Elastic Net TV + 1 [Gramfort... 2013] G Varoquaux 5
- 11. Spatial sparse penalties Spatial regularization, total variation minw l(y − Xw) + λ i ( w)i 2 More generally: analysis sparsity [Eickenberg... 2015] Sparse in a transformation of the weights: minw l(y − Xw) + λ K w 21 For instance: overlapping blocks (Kw)1 ↔ G1 (Kw)2 ↔ G2 ... G2 G1 G Varoquaux 6
- 12. Good convergence of solvers is important Spatial regularization, total variation minw l(y − Xw) + λ i ( w)i 2 x=17 L R z=-17 Stopping: ∆E < 10−1 x=17 L R z=-17 Stopping: ∆E < 10−5 [Dohmatob... 2014] G Varoquaux 7
- 13. Sparse solvers Iterative Shrinkage-Thresholding Algorithm minw l(y − Xw) + λ i Kw 1 Settings: min l + p; l smooth, p non-smooth Minimize successively: (quadratic approx of l) + p 1. Gradient descent on smooth term FISTA loop 2. Proximal operator proxpx = miny 1 2 x − y 2 2 + p(y) G Varoquaux 8
- 14. Sparse solvers Iterative Shrinkage-Thresholding Algorithm minw l(y − Xw) + λ i w 1 Settings: min l + p; l smooth, p non-smooth Minimize successively: (quadratic approx of l) + p 1. Gradient descent on smooth term FISTA loop 2. Proximal operator proxpx = miny 1 2 x − y 2 2 + p(y) 1 penalty: “soft thresholding”: prox 1 : ∀i wi ← wi 1 − λ |wi| + G Varoquaux 8
- 15. Sparse solvers: proximals and co. 1 penalty: “soft thresholding”: prox 1 : ∀i wi ← wi 1 − λ |wi| + Group sparsity: prox 21 on G: ∀i ∈ G wi ← wi 1− λ j∈G w2 j + G2 G1 G Varoquaux 9
- 16. Sparse solvers: proximals and co. 1 penalty: “soft thresholding”: prox 1 : ∀i wi ← wi 1 − λ |wi| + Group sparsity: prox 21 on G: ∀i ∈ G wi ← wi 1− λ j∈G w2 j + G2 G1 Overlapping groups, TV: Inner loop iterative solver G2 G1 G Varoquaux 9
- 17. Sparse solvers: proximals and co. Group sparsity: prox 21 on G: ∀i ∈ G wi ← wi 1− λ j∈G w2 j + G2 G1 Overlapping groups, TV: Inner loop iterative solver G2 G1 Social sparsity shrinkage: ∀i wi ← wi 1− λ j∈N(i) w2 j + N1 x1 N2 x2 G Varoquaux 9
- 18. Social sparsity: “soft-threshold” neighboring voxels Sparsity must be combined with spatial structure Convex solvers for non-local sparsity are expensive Not separable Social sparsity: forget the coupling between soft thresholding [Kowalski... 2013] N1 x1 N2 x2 G Varoquaux 10
- 19. Empirical evaluation for decoding 25% 10% 0% +10% TVl1 graph net social sparsity SVM + anova Prediction accuracy 1 20x 1 5x 1 2x 1x 2x 5x Run time bottle/scramble bottle/shoe cat/bottle cat/chair cat/face cat/house cat/scramble cat/shoe chair/scramble chair/shoe face/house face/scissors scissors/scramble shoe/scramble OASIS VBM male vs femaleG Varoquaux 11
- 20. Empirical evaluation for decoding 25% 10% 0% +10% TVl1 graph net social sparsity SVM + anova Prediction accuracy 1 20x 1 5x 1 2x 1x 2x 5x Run time bottle/scramble bottle/shoe cat/bottle cat/chair cat/face cat/house cat/scramble cat/shoe chair/scramble chair/shoe face/house face/scissors scissors/scramble shoe/scramble OASIS VBM male vs femaleG Varoquaux 11
- 21. Social sparsity maps L R z=16 y=34 face vs house TV- 1 Graph-net Social sparsity G Varoquaux 12
- 22. Social sparsity maps L R z=16 y=34 face vs house TV- 1 Graph-net Social sparsity G Varoquaux 12
- 23. @GaelVaroquaux Social-sparsity brain decoders: faster spatial sparsity Spatial sparsity improves prediction and denoises maps TV- 1 “space-net” very successful, but slow Social-sparsity: heuristic that forgoes couplings 10× faster than TV- 1 almost as accurate 3× faster than graph-net more accurate Maps segment well regions ni
- 24. References I M. K. Carroll, G. A. Cecchi, I. Rish, R. Garg, and A. R. Rao. Prediction and interpretation of distributed neural activity with sparse models. NeuroImage, 44(1):112 – 122, 2009. E. Dohmatob, A. Gramfort, B. Thirion, and G. Varoquaux. Benchmarking solvers for TV-l1 least-squares and logistic regression in brain imaging. PRNI, 2014. M. Eickenberg, E. Dohmatob, B. Thirion, and G. Varoquaux. Total variation meets sparsity: statistical learning with segmenting penalties. MICCAI, 2015. A. Gramfort, B. Thirion, and G. Varoquaux. Identifying predictive regions from fMRI with TV-L1 prior. In PRNI, pages 17–20, 2013. J. Haxby, I. Gobbini, M. Furey, ... Distributed and overlapping representations of faces and objects in ventral temporal cortex. Science, 293:2425, 2001.
- 25. References II M. Kowalski, K. Siedenburg, and M. Dorfler. Social sparsity! neighborhood systems enrich structured shrinkage operators. Transactions on Signal Processing, 61:2498, 2013. G. Varoquaux, A. Gramfort, and B. Thirion. Small-sample brain mapping: sparse recovery on spatially correlated designs with randomization and clustering. In ICML, page 1375, 2012. O. Yamashita, M. aki Sato, T. Yoshioka, F. Tong, and Y. Kamitani. Sparse estimation automatically selects voxels relevant for the decoding of fMRI activity patterns. NeuroImage, 42(4):1414 – 1429, 2008.