A Comparison of Relaxations of Multiset Cannonical Correlation Analysis and Applications

Canonical correlation analysis is a statistical technique that is used to find relations between two sets of variables. An important extension in pattern analysis is to consider more than two sets of variables. This problem can be expressed as a quadratically constrained quadratic program (QCQP), commonly referred to Multi-set Canonical Correlation Analysis (MCCA). This is a non-convex problem and so greedy algorithms converge to local optima without any guarantees on global optimality. In this paper, we show that despite being highly structured, finding the optimal solution is NP-Hard. This motivates our relaxation of the QCQP to a semidefinite program (SDP). The SDP is convex, can be solved reasonably efficiently and comes with both absolute and output-sensitive approximation quality. In addition to theoretical guarantees, we do an extensive comparison of the QCQP method and the SDP relaxation on a variety of synthetic and real world data. Finally, we present two useful extensions: we incorporate kernel methods and computing multiple sets of canonical vectors

Singular Continuation: Generating Piece-wise Linear Approximations to Pareto Sets via Global Analysis

We propose a strategy for approximating Pareto optimal sets based on the global analysis framework proposed by Smale (Dynamical systems, New York, 1973, pp. 531-544). The method highlights and exploits the underlying manifold structure of the Pareto sets, approximating Pareto optima by means of simplicial complexes. The method distinguishes the hierarchy between singular set, Pareto critical set and stable Pareto critical set, and can handle the problem of superposition of local Pareto fronts, occurring in the general nonconvex case. Furthermore, a quadratic convergence result in a suitable set-wise sense is proven and tested in a number of numerical examples.Comment: 29 pages, 12 figure

Projection methods in conic optimization

There exist efficient algorithms to project a point onto the intersection of a convex cone and an affine subspace. Those conic projections are in turn the work-horse of a range of algorithms in conic optimization, having a variety of applications in science, finance and engineering. This chapter reviews some of these algorithms, emphasizing the so-called regularization algorithms for linear conic optimization, and applications in polynomial optimization. This is a presentation of the material of several recent research articles; we aim here at clarifying the ideas, presenting them in a general framework, and pointing out important techniques