Convex and Spectral Relaxations for Phase Retrieval, Seriation and Ranking. (Relaxations convexes et spectrales pour les problèmes de reconstruction de phase, seriation et classement)

Optimization is often the computational bottleneck in disciplines such as statistics, biology, physics, finance or economics. Many optimization problems can be directly cast in the wellstudied convex optimization framework. For non-convex problems, it is often possible to derive convex or spectral relaxations, i.e., derive approximations schemes using spectral or convex optimization tools. Convex and spectral relaxations usually provide guarantees on the quality of the retrieved solutions, which often transcribes in better performance and robustness in practical applications, compared to naive greedy schemes. In this thesis, we focus on the problems of phase retrieval, seriation and ranking from pairwise comparisons. For each of these combinatorial problems we formulate convex and spectral relaxations that are robust, flexible and scalable. • Phase retrieval seeks to reconstruct a complex signal, given a number of observations on the magnitude of linear measurements. In Chapter 2, we focus on problems arising in diffraction imaging, where various illuminations of a single object, e.g., a molecule, are performed through randomly coded masks. We show that exploiting structural assumptions on the signal and the observations, such as sparsity, smoothness or positivity, can significantly speed-up convergence and improve recovery performance. • The seriation problem seeks to reconstruct a linear ordering of items based on unsorted, possibly noisy, pairwise similarity information. The underlying assumption is that items can be ordered along a chain, where the similarity between items decreases with their distance within this chain. In Chapter 3, we first show that seriation can be formulated as a combinatorial minimization problem over the set of permutations, and then derive several convex relaxations that improve the robustness of seriation solutions in noisy settings compared to the spectral relaxation of Atkins et al. (1998). As an additional benefit, these convex relaxations allow to impose a priori constraints on the solution, hence solve semisupervised seriation problems. We establish new approximation bounds for some of these relaxations and present promising numerical experiments on archeological data, Markov chains and DNA assembly from shotgun gene sequencing data. • Given pairwise comparisons between n items, the ranking problem seeks to find the most consistent global order of these items, e.g., ranking players in a tournament. In practice, the information about pairwise comparisons is usually incomplete, especially when the set of items is very large, and the data may also be noisy, that is some pairwise comparisons could be incorrectly measured and inconsistent with a total order. In Chapter 4, we formulate this ranking problem as a seriation problem, by constructing an adequate similarity matrix from pairwise comparisons. Intuitively, ordering items based on this similarity