Raphael Hauser

A DC-programming algorithm for kernel selection

We address the problem of learning a kernel for a given supervised learning task. Our approach consists in searching within the convex hull of a prescribed set of basic kernels for one which minimizes a convex regularization functional. A unique feature of this approach compared to others in the literature is that the number of basic kernels can be infinite. We only require that they are continuously parameterized. For example, the basic kernels could be isotropic Gaussians with variance in a prescribed interval or even Gaussians parameterized by multiple continuous parameters. Our work builds upon a formulation involving a minimax optimization problem and a recently proposed greedy algorithm for learning the kernel. Although this optimization problem is not convex, it belongs to the larger class of DC (difference of convex functions) programs. Therefore, we apply recent results from DC optimization theory to create a new algorithm for learning the kernel. Our experimental results on benchmark data sets show that this algorithm outperforms a previously proposed method.

Self-Scaled Barrier Functions on Symmetric Cones and Their Classification

Abstract Self-scaled barrier functions on self-scaled cones were axiomatically introduced by Nesterov and Todd in 1994 as a tool for the construction of primal—dual long-step interior point algorithms. This paper provides firm foundations for these objects by exhibiting their symmetry properties, their close ties with the symmetry groups of their domains of definition, and subsequently their decomposition into irreducible parts and their algebraic classification theory. In the first part we recall the characterization of the family of self-scaled cones as the set of symmetric cones and develop a primal—dual symmetric viewpoint on self-scaled barriers, results that were first discovered by the second author. We then show in a short, simple proof that any pointed, convex cone decomposes into a direct sum of irreducible components in a unique way, a result which can also be of independent interest. We then proceed to showing that any self-scaled barrier function decomposes, in an essentially unique way, into a direct sum of self-scaled barriers defined on the irreducible components of the underlying symmetric cone. Finally, we present a complete algebraic classification of self-scaled barrier functions using the correspondence between symmetric cones and Euclidean—Jordan algebras.

Randomized relaxation methods for the maximum feasible subsystem problem

In the Max FS problem, given an infeasible linear system Ax ≥ b, one wishes to find a feasible subsystem containing a maximum number of inequalities. This NP-hard problem has interesting applications in a variety of fields. In some challenging applications in telecommunications and computational biology one faces very large Max FS instances with up to millions of inequalities in thousands of variables. We propose to tackle large-scale instances of Max FS using randomized and thermal variants of the classical relaxation method for solving systems of linear inequalities. We present a theoretical analysis of one particular version of such a method in which we derive a lower bound on the probability that it identifies an optimal solution within a given number of iterations. This bound, which is expressed as a function of a condition number of the input data, implies that with probability 1 the randomized method identifies an optimal solution after finitely many iterations. We also present computational results obtained for medium- to large-scale instances arising in the planning of digital video broadcasts and in the modelling of the energy functions driving protein folding. Our experiments indicate that these methods perform very well in practice.

The Continuous Newton--Raphson Method Can Look Ahead

This paper is about an intriguing property of the continuous Newton--Raphson method for the minimization of a continuous objective function f: if x is a point in the domain of attraction of a strict local minimizer x*, then the flux line of the Newton--Raphson flow that starts in x approaches x* from a direction that depends only on the behavior of f in arbitrarily small neighborhoods around x and x*. In fact, if

Self-Scaled Barriers for Irreducible Symmetric Cones

\hat{f}

Tail Decay and Moment Estimates of a Condition Number for Random Linear Conic Systems

is a sufficiently benign perturbation of f on an open region D not containing x, then the two flux lines through x defined by the Newton--Raphson vector fields that correspond to f and

Boundedness Theorems for the Relaxation Method

\hat{f}

Large deviations-based upper bounds on the expected relative length of longest common subsequences

differ from one another only within D.

On the Relationship between the Convergence Rates of Iterative and Continuous Processes

Self-scaled barrier functions are fundamental objects in the theory of interior-point methods for linear optimization over symmetric cones, of which linear and semidefinite programming are special cases. In this article we classify the special class of self-scaled barriers which are defined on irreducible symmetric cones. Together with a decomposition theorem for general self-scaled barriers this concludes the algebraic classification theory of these functions.

3D Image Reconstruction from X-Ray Measurements with Overlap

In this paper we study the distribution of