Living on the edge: Phase transitions in convex programs with random data

TL;DR

This paper rigorously explains phase transitions in convex optimization problems with random data, introducing the statistical dimension to predict transition behavior and applying it to various inverse problems.

Contribution

It introduces the statistical dimension as a key tool to analyze phase transitions in convex programs with random data, providing precise predictions of transition thresholds.

Findings

Phase transitions are explained through the concentration of intrinsic volumes.

The statistical dimension accurately predicts the location and width of phase transitions.

Applications include sparse recovery, demixing, and cone programming with random constraints.

Abstract

Recent research indicates that many convex optimization problems with random constraints exhibit a phase transition as the number of constraints increases. For example, this phenomenon emerges in the $\ell_1$ minimization method for identifying a sparse vector from random linear measurements. Indeed, the $\ell_1$ approach succeeds with high probability when the number of measurements exceeds a threshold that depends on the sparsity level; otherwise, it fails with high probability. This paper provides the first rigorous analysis that explains why phase transitions are ubiquitous in random convex optimization problems. It also describes tools for making reliable predictions about the quantitative aspects of the transition, including the location and the width of the transition region. These techniques apply to regularized linear inverse problems with random measurements, to demixing problems under a random incoherence model, and also to cone programs with random affine constraints. The applied results depend on foundational research in conic geometry. This paper introduces a summary parameter, called the statistical dimension, that canonically extends the dimension of a linear subspace to the class of convex cones. The main technical result demonstrates that the sequence of intrinsic volumes of a convex cone concentrates sharply around the statistical dimension. This fact leads to accurate bounds on the probability that a randomly rotated cone shares a ray with a fixed cone.