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 1 minimization method for identifying a sparse vector from random linear measurements. Indeed, the 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. MOTIVATIONA phase transition is a sharp change in the character of a computational problem as its parameters vary. Recent research suggests that phase transitions emerge in many random convex optimization problems from mathematical signal processing and computational statistics; for example, see [DT09b, Sto09, OH10, CSPW11, DGM13, MT14b]. This paper proves that the locations of these phase transitions are determined by geometric invariants associated with the mathematical programs. Our analysis provides the first complete account of transition phenomena in random linear inverse problems, random demixing problems, and random cone programs.1.1. Vignette: Compressed sensing. To illustrate our goals, we discuss the compressed sensing problem, a familiar example where a phase transition is plainly visible in numerical experiments [DT09b]. Let x 0 ∈ R d be an unknown vector with s nonzero entries. Let A be an m × d random matrix whose entries are independent standard normal variables, and suppose we have access to the vector(1.1)This serves as a model for data acquisition: we interpret z 0 as a collection of m independent linear measurements of the unknown x 0 . The compressed sensing problem requires us to identify x 0 given only the measurement vector z 0 and the realization of the measurement matrix A. When the number m of measurements is smaller than the ambient dimension d , we cannot solve this inverse problem unless we take advantage of the prior knowledge that x 0 is sparse. |x i |. This approach is sensible because the 1 norm of a vector can serv...

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The theory of intrinsic volumes of convex cones has recently found striking applications in areas such as convex optimization and compressive sensing. This article provides a self-contained account of the combinatorial theory of intrinsic volumes for polyhedral cones. Direct derivations of the general Steiner formula, the conic analogues of the Brianchon-Gram-Euler and the Gauss-Bonnet relations, and the principal kinematic formula are given. In addition, a connection between the characteristic polynomial of a hyperplane arrangement and the intrinsic volumes of the regions of the arrangement, due to Klivans and Swartz, is generalized and some applications are presented.

Abstract. We prove a general theorem providing smoothed analysis estimates for conic condition numbers of problems of numerical analysis. Our probability estimates depend only on geometric invariants of the corresponding sets of ill-posed inputs. Several applications to linear and polynomial equation solving show that the estimates obtained in this way are easy to derive and quite accurate. The main theorem is based on a volume estimate of ε-tubular neighborhoods around a real algebraic subvariety of a sphere, intersected with a spherical disk of radius σ. Besides ε and σ, this bound depends only on the dimension of the sphere and on the degree of the defining equations.

Smoothed analysis of complexity bounds and condition numbers has been done, so far, on a case by case basis. In this paper we consider a reasonably large class of condition numbers for problems over the complex numbers and we obtain smoothed analysis estimates for elements in this class depending only on geometric invariants of the corresponding sets of ill-posed inputs. These estimates are for a version of smoothed analysis proposed in this paper which, to the best of our knowledge, appears to be new. Several applications to linear and polynomial equation solving show that estimates obtained in this way are easy to derive and quite accurate.