The main technical result demonstrates that the sequence of intrinsic volumes of aĬonvex cone concentrates sharply around the statistical dimension. This paper introduces a summary parameter, called the statistical dimension, that canonically extends the dimension of a linear The applied results depend on foundational research in conic geometry. These techniquesĪpply to regularized linear inverse problems with random measurements, to demixing problems under a random incoherence model,Īnd also to cone programs with random affine constraints. It also describes tools for making reliable predictionsĪbout the quantitative aspects of the transition, including the location and the width of the transition region. Transitions are ubiquitous in random convex optimization problems. This paper provides the first rigorous analysis that explains why phase Level otherwise, it fails with high probability. Indeed, the ℓ1 approach succeeds with high probability when the number of measurements exceeds a threshold that depends on the sparsity For example, this phenomenon emerges in the ℓ1 minimization method for identifying a sparse vector from random linear measurements. Recent research indicates that many convex optimization problems with random constraints exhibit a phase transition as the In each case, the theoretical analysis of the convex demixing method closely matches its empirical behavior. Some applications include (1) demixing two signals that are sparse in mutually incoherent bases, (2) decoding spread-spectrum transmissions in the presence of impulsive errors, and (3) removing sparse corruptions from a low-rank matrix. The difficulty of separating two structured, incoherent signals depends only on the total complexity of the two structures. ![]() For an observation from this model, this approach identifies a summary statistic that reflects the complexity of a particular signal. This work introduces a randomized signal model that ensures that the two structures are incoherent, i.e., generically oriented. ![]() ![]() This paper describes and analyzes a framework, based on convex optimization, for solving these demixing problems, and many others. Examples include the problem of separating a signal that is sparse with respect to one basis from a signal that is sparse with respect to a second basis, and the problem of decomposing an observed matrix into a low-rank matrix plus a sparse matrix. The main ingredient in the proofs is a connection between the number of faces of the tessellation and the number of faces of the Weyl chambers of the corresponding type that are intersected non-trivially by a certain linear subspace in general position.ĭemixing refers to the challenge of identifying two structured signals given only the sum of the two signals and prior information about their structures. We compute the number of cones in this tessellation and the expectations of various geometric functionals for random cones drawn from this random tessellation. The hyperplanes \begin This tessellation is closely related to the Weyl chambers of type B n B_n. We consider $d$-dimensional random vectors $Y_1,\dots,Y_n$ that satisfy a mild general position assumption a.s.
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