Dimensionality reduction of SDPs through sketching

TL;DR

This paper introduces a sketching method for semidefinite programs using Johnson-Lindenstrauss transforms, enabling dimension reduction while approximately preserving solutions, thus improving computational efficiency and storage needs.

Contribution

It presents a novel approach to reduce SDP dimensions via positive maps, with theoretical analysis of the method's effectiveness and limitations.

Findings

Sketching reduces SDP size while maintaining feasibility or approximate optimality.

The method improves computational complexity and storage requirements.

Limitations of positive linear sketches are characterized.

Abstract

We show how to sketch semidefinite programs (SDPs) using positive maps in order to reduce their dimension. More precisely, we use Johnson\hyp{}Lindenstrauss transforms to produce a smaller SDP whose solution preserves feasibility or approximates the value of the original problem with high probability. These techniques allow to improve both complexity and storage space requirements. They apply to problems in which the Schatten 1-norm of the matrices specifying the SDP and also of a solution to the problem is constant in the problem size. Furthermore, we provide some results which clarify the limitations of positive, linear sketches in this setting.