Sergei Chubanov | Searcharxiv

arXiv2025

A polynomial projective algorithm for convex feasibility problems with positive-definite constraints

We study a class of projective transformations of spectraplexes associated with self-dual cones and, on this basis, propose a polynomial-time algorithm for convex feasibility problems with positive definite constraints. At each iteration of the algorithm, either a feasible solution is found or a suitable valid inequality inducing a projective transformation allowing to bring the solution set closer to the center of an associated spectraplex. The closeness to the center is measured in terms of a potential function. The running time of our algorithm makes the existing complexity bounds more precise for the case when the number of equations linking the positive definite variable matrices is not less than the sum of the ranks of the respective positive-semidefinite cones.

math.OC↗

arXiv2023

On the complexity of PAC learning in Hilbert spaces

We study the problem of binary classification from the point of view of learning convex polyhedra in Hilbert spaces, to which one can reduce any binary classification problem. The problem of learning convex polyhedra in finite-dimensional spaces is sufficiently well studied in the literature. We generalize this problem to that in a Hilbert space and propose an algorithm for learning a polyhedron which correctly classifies at least $1- \varepsilon$ of the distribution, with a probability of at least $1 - δ,$ where $\varepsilon$ and $δ$ are given parameters. Also, as a corollary, we improve some previous bounds for polyhedral classification in finite-dimensional spaces.

cs.LG↗