On the structure of low-rank matrices that approximate the identity matrix

TL;DR

This paper establishes lower bounds on the number of significant elements in low-rank matrices approximating the identity, revealing structural limitations and answering a question posed by Kashin.

Contribution

It provides new lower bounds on the sparsity and element magnitude of low-rank approximations to the identity matrix, advancing understanding of their structure.

Findings

At least c(K)N^2 elements satisfy |A_{i,j}| > c(K)n^{-1/2} for n ≤ K log N.

Number of nonzero elements in A is at least c log(N)/(n log(2 + n/ log N)).

Answers a question of B.S. Kashin regarding matrix element bounds.

Abstract

Consider a matrix $A$ of rank $n$ that approximates the $N\times N$ identity matrix with elementwise error at most $1/3$. We give a lower bound on the number of elements s.t. $|A_{i,j}|>\gamma$, for a certain threshold. Two corollaries are obtained. 1. If $n \le K\log N$ with some $K$, then at least $c(K)N^2$ elements satisfy $|A_{i,j}|>c(K)n^{-1/2}$. This answers a question of B.S. Kashin. 2. The number of nonzero elements in $A$ is at least $c\log(N)/(n\log(2+n/\log N))$.