Evaluating approximations of the semidefinite cone with trace normalized distance

TL;DR

This paper compares different approximations of the semidefinite cone using trace normalized distance, revealing limitations of previous measures and providing exact distances for new measures.

Contribution

It introduces the trace normalized distance as a new measure and computes exact distances between dual cones and the semidefinite cone.

Findings

Normalized distance is not sufficient to evaluate approximations.

Trace normalized distance differs for ${ m DD}_n^*$ and ${ m SDD}_n^*$.

Exact distances are provided for these approximations.

Abstract

We evaluate the dual cone of the set of diagonally dominant matrices (resp., scaled diagonally dominant matrices), namely ${\cal DD}_n^*$ (resp., ${\cal SDD}_n^*$), as an approximation of the semidefinite cone. We prove that the norm normalized distance, proposed by Blekherman et al. (2022), between a set ${\cal S}$ and the semidefinite cone has the same value whenever ${\cal SDD}_n^* \subseteq {\cal S} \subseteq {\cal DD}_n^*$. This implies that the norm normalized distance is not a sufficient measure to evaluate these approximations. As a new measure to compensate for the weakness of that distance, we propose a new distance, called the trace normalized distance. We prove that the trace normalized distance between ${\cal DD}_n^*$ and ${\cal S}^n_+$ has a different value from the one between ${\cal SDD}_n^*$ and ${\cal S}^n_+$ and give the exact values of these distances.