Convex regularization has become a popular approach to solve large scale inverse or data separation problems. A prominent example is the problem of identifying a sparse signal from linear samples my minimizing the l_1 norm under linear constraints. Recent empirical research indicates that many convex regularization problems on random data exhibit a phase transition phenomenon: the probability of successfully recovering a signal changes abruptly from zero to one as the number of constraints increases past a certain threshold. We present a rigorous analysis that explains why phase transitions are ubiquitous in convex optimization. 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, to demixing problems, and to cone programs with random affine constraints. These applications depend on a new summary parameter, the statistical dimension of cones, that canonically extends the dimension of a linear subspace to the class of convex cones.
Joint work with Dennis Amelunxen, Mike McCoy and Joel Tropp.