Section: New Results
Convex Relaxations for Permutation Problems
Participants : Fajwel Fogel [correspondent] , Rodolphe Jenatton, Francis Bach, Alexandre d'Aspremont.
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. In  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 performed numerical experiments on archeological data, Markov chains and gene sequences.