Corr clust-kiel
- 1. 11/5/2018 Mikael Kågebäck, Chalmers CSE 1 Credit: Aldebaran Correlation Clustering: A Tale of Two Cultures Erik Thiel, Morteza Chehreghani, Devdatt Dubhashi Chalmers University of Technology, Sweden
- 4. Colour Partitions • Ask humans subjects in a language group to label tiles with colour terms, then aggregate all results into a partition for that language. • Take the CIELAB colour coordinates to define a similarity between colour tiles and forma partition based on these similarities.
- 6. Regier T, Kemp C, Kay P. 2015. Word meanings across languages support efficient communication. In The Handbook of Language Emergence, ed. B MacWhinney, W O’Grady
- 7. Correlation Clustering • Input: Graph 𝐺 = 𝑉, 𝐸 and positive or negative weights 𝑤 𝑒 , 𝑒 ∈ 𝐸 • Output: A clustering of the vertices to maximize the sum of the weights of edges within each cluster.
- 9. Difference from usual Clustering • Weights can be positive or negative! • Contentious what’s ”good” quality clustering • But in correlation clustering there is unambiguous objective • The number of clusters need not be specified, will emerge from the optimizing the objective.
- 11. Approximation Algorithms • Bansal, Blum Chawla (2004): PTAS on complete graphs • Charikar Guruswami, Wirth (2005): APX hard on general graphs • Charikar et al (2005), Swamy (2004): 0.76 approximation • Guruswami-Giotis (2006): PTAS with fixed no of clusters
- 13. SDP Relaxation + Rounding
- 14. However … • No implementation, no code … • Doesn’t work in practice …
- 15. A Tale of Two Cultures • Deep elegant theory • “Polynomial time” • No implementation • No experiments on data sets • Does not work in practice or scale • Beamer/LaTeX • Sometimes theory • Linear or sub-linear • Well engineered implementation • Extensive testing on data sets • Must work in practice, scale to “Big Data” • Powerpoint Algorithms Theory Machine Learning
- 17. Tightness of Relaxation • The non-convex relaxation is tight: no gap between continuous and discrete problem, simple proof by randomized rounding. • In contrast SDP relaxation is not tight.
- 20. Non-convex Convergence Theory • For a differentiable (but not necessarily convex) function, the FW algorithm converges in 𝑂(1/ 𝑇) steps. • If the function is multilinear, then it converges in 𝑂(1/𝑇) steps. • Note that our correlation clustering objective is indeed multilinear!
- 22. Synthetic Data: Generative Model • Planted model with k clusters and noise p • With probability (1-p), high positive weight on edge within a cluster and high negative weight on edge across clusters, with probability p, arbitrary weight
- 32. Summary • Non-convex relaxation solved with Frank Wolfe yields an algorithm with guarantees that beats all other methods handily in both runtime and quality. • Combine theory and rigour of algorithms research with engineering good implementations and extensive testing on data.