Convexity of the image of a quadratic map via the relative entropy distance

TL;DR

This paper establishes an upper bound on the relative entropy distance between the convex hull of the image of a quadratic map and the image itself, revealing insights into the convexity properties of quadratic maps.

Contribution

It proves a universal bound on the relative entropy distance for quadratic maps defined by positive definite forms, connecting convex hulls and image sets.

Findings

Bound of 4.8 on the relative entropy for points in the convex hull.

Existence of convex combinations with entropy less than 15/√m.

Quantitative analysis of convexity properties of quadratic maps.

Abstract

Let psi: R^n --> R^k be a map defined by k positive definite quadratic forms on R^n. We prove that the relative entropy (Kullback-Leibler) distance from the convex hull of the image of psi to the image of psi is bounded above by an absolute constant. More precisely, we prove that for every point a=(a_1, ..., a_k) in the convex hull of the image of psi such that a_1 + ... +a_k =1 there is a point b=(b_1, ..., b_k) in the image of psi such that b_1 + ... + b_k =1 and such that a_1 ln(a_1/b_1) + ... + a_k ln(a_k/b_k) < 4.8. Similarly, we prove that for any integer m one can choose a convex combination b of at most m points from the image of psi such that a_1 ln(a_1/b_1) + ... + a_k ln(a_k/b_k) < 15/sqrt{m}.