# Accuracy of approximate projection to the semidefinite cone

## TL;DR

This paper derives gap-independent error bounds for approximate projections onto the semidefinite cone, showing that small spectral gaps do not necessarily impair accuracy, unlike traditional eigenvalue perturbation bounds.

## Contribution

It introduces new error bounds for semidefinite cone projections that do not depend on spectral gaps, improving understanding of approximation accuracy.

## Key findings

- Error bounds are gap-independent
- Small eigenvalues do not significantly affect projection accuracy
- Traditional bounds can be overly pessimistic in this context

## Abstract

When a projection of a symmetric or Hermitian matrix to the positive semidefinite cone is computed approximately (or to working precision on a computer), a natural question is to quantify its accuracy. A straightforward bound invoking standard eigenvalue perturbation theory (e.g. Davis-Kahan and Weyl bounds) suggests that the accuracy would be inversely proportional to the spectral gap, implying it can be poor in the presence of small eigenvalues. This work shows that a small gap is not a concern for projection onto the semidefinite cone, by deriving error bounds that are gap-independent.