A graduated non-convexity relaxation for large scale seriation

Abstract

In this work we propose a highly scalable algorithm for solving the combinatorial data analysis problem of seriation. Seriation is a technique for optimizing a permutation of data instances, with respect to some proximity measure such that nearby instances in the linear arrangement are more similar. One consistent objective function for seriation is the 2-SUM minimization problem, which uses the 2-norm between instance locations to penalize non-zero similarity values, and can be written as a quadratic function of the permutation vector. Recently, two convex relaxations of the 2-SUM problem have been proposed, which can be solved as constrained quadratic programs using interior point methods; however, the interior point solvers become expensive when the problem size increases. In this paper we present a graduated non-convexity method for vector-based relaxations of the 2-SUM that yields better approximate solutions and scales to very large problem sizes. We conduct a number of experiments on real and synthetic datasets. The experimental results demonstrate that our proposed algorithm outperforms other approaches that solve the 2-SUM, and is the only competitive approach that can scale to large problem sizes.