A lower bound for the Balan--Jiang matrix problem

TL;DR

This paper establishes a probabilistic lower bound on the sum of -norms of factors in rank-1 decompositions of positive semidefinite matrices, advancing understanding of the Balan--Jiang matrix problem.

Contribution

It provides the first probabilistic construction demonstrating a fundamental lower bound for the Balan--Jiang matrix problem.

Findings

Any rank-1 decomposition of certain positive semidefinite matrices requires large -norm factors.

The lower bound scales with -norm and matrix size n.

The result is independent of the matrix dimension n.

Abstract

We prove the existence of a positive semidefinite matrix $A \in \mathbb{R}^{n \times n}$ such that any decomposition into rank-1 matrices has to have factors with a large $\ell^1-$norm, more precisely $$ \sum_{k} x_k x_k^*=A \quad \implies \quad \sum_k \|x_k\|^2_{1} \geq c \sqrt{n} \|A\|_{1},$$ where $c$ is independent of $n$. This provides a lower bound for the Balan--Jiang matrix problem. The construction is probabilistic.