Spectrahedral Lifts of Convex Sets

TL;DR

This paper surveys recent advances in representing convex sets as spectrahedral lifts, highlighting how such representations relate to factorizations through convex cones and their implications across multiple fields.

Contribution

It provides a comprehensive overview of spectrahedral lifts of convex sets and establishes a connection between these lifts and factorizations via convex cones, advancing understanding in convex geometry and optimization.

Findings

Projection representations are governed by factorizations through convex cones.

Spectrahedral lifts enable simpler representations of complex convex sets.

The approach links convex geometry with algebraic and combinatorial methods.

Abstract

Efficient representations of convex sets are of crucial importance for many algorithms that work with them. It is well-known that sometimes, a complicated convex set can be expressed as the projection of a much simpler set in higher dimensions called a lift of the original set. This is a brief survey of recent developments in the topic of lifts of convex sets. Our focus will be on lifts that arise from affine slices of real positive semidefinite cones known as psd or spectrahedral lifts. The main result is that projection representations of a convex set are controlled by factorizations, through closed convex cones, of an operator that comes from the convex set. This leads to several research directions and results that lie at the intersection of convex geometry, combinatorics, real algebraic geometry, optimization, computer science and more.