Convex Relaxations for Permutation Problems

Part of Advances in Neural Information Processing Systems 26 (NIPS 2013)

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Authors

Fajwel Fogel, Rodolphe Jenatton, Francis Bach, Alexandre D'Aspremont

Abstract

<p>Seriation seeks to reconstruct a linear order between variables using unsorted similarity information. It has direct applications in archeology and shotgun gene sequencing for example. We prove the equivalence between the seriation and the combinatorial 2-sum problem (a quadratic minimization problem over permutations) over a class of similarity matrices. The seriation problem can be solved exactly by a spectral algorithm in the noiseless case and we produce a convex relaxation for the 2-sum problem to improve the robustness of solutions in a noisy setting. This relaxation also allows us to impose additional structural constraints on the solution, to solve semi-supervised seriation problems. We present numerical experiments on archeological data, Markov chains and gene sequences.</p>