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

arXiv: https://arxiv.org/abs/1305.0243 · HTML: https://mlsanity.com/paper/1305.0243/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 May 2013.

## Built on by (most cited first, 1 of 1)

- [The Nonconvex Geometry of Linear Inverse Problems](https://mlsanity.com/paper/2101.02776/) (2021)

## Abstract (arXiv)

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}.