# Living on the edge: A geometric theory of phase transitions in convex optimization

Amelunxen, Dennis and Lotz, Martin and Mccoy, Michael B. and Tropp, Joel A. (2013) Living on the edge: A geometric theory of phase transitions in convex optimization. [MIMS Preprint]

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## Abstract

Recent empirical 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 samples. Indeed, this approach succeeds with high probability when the number of samples 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. These applications depend on foundational research in conic geometry. This paper introduces a new 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 conic intrinsic volumes of a convex cone concentrates sharply near the statistical dimension. This fact leads to an approximate version of the conic kinematic formula that gives bounds on the probability that a randomly oriented cone shares a ray with a fixed cone.

Item Type: MIMS Preprint Compressed sensing, convex optimization, convex geometry, concentration of measure, phase transition MSC 2010, the AMS's Mathematics Subject Classification > 51 Geometry (See also algebraic geometry)MSC 2010, the AMS's Mathematics Subject Classification > 52 Convex and discrete geometryMSC 2010, the AMS's Mathematics Subject Classification > 60 Probability theory and stochastic processesMSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysisMSC 2010, the AMS's Mathematics Subject Classification > 90 Operations research, mathematical programming Dr. Martin Lotz 28 Mar 2013 08 Nov 2017 18:18 http://eprints.maths.manchester.ac.uk/id/eprint/1963

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