Structured random measurements in signal processing

Compressed sensing and its extensions have recently triggered interest in randomized signal acquisition. A key finding is that random measurements provide sparse signal reconstruction guarantees for efficient and stable algorithms with a minimal number of samples. While this was first shown for (unstructured) Gaussian random measurement matrices, applications require certain structure of the measurements leading to structured random measurement matrices. Near optimal recovery guarantees for such structured measurements have been developed over the past years in a variety of contexts. This article surveys the theory in three scenarios: compressed sensing (sparse recovery), low rank matrix recovery, and phaseless estimation. The random measurement matrices to be considered include random partial Fourier matrices, partial random circulant matrices (subsampled convolutions), matrix completion, and phase estimation from magnitudes of Fourier type measurements. The article concludes with a brief discussion of the mathematical techniques for the analysis of such structured random measurements.Comment: 22 pages, 2 figure

Improving compressed sensing with the diamond norm

In low-rank matrix recovery, one aims to reconstruct a low-rank matrix from a minimal number of linear measurements. Within the paradigm of compressed sensing, this is made computationally efficient by minimizing the nuclear norm as a convex surrogate for rank. In this work, we identify an improved regularizer based on the so-called diamond norm, a concept imported from quantum information theory. We show that -for a class of matrices saturating a certain norm inequality- the descent cone of the diamond norm is contained in that of the nuclear norm. This suggests superior reconstruction properties for these matrices. We explicitly characterize this set of matrices. Moreover, we demonstrate numerically that the diamond norm indeed outperforms the nuclear norm in a number of relevant applications: These include signal analysis tasks such as blind matrix deconvolution or the retrieval of certain unitary basis changes, as well as the quantum information problem of process tomography with random measurements. The diamond norm is defined for matrices that can be interpreted as order-4 tensors and it turns out that the above condition depends crucially on that tensorial structure. In this sense, this work touches on an aspect of the notoriously difficult tensor completion problem.Comment: 25 pages + Appendix, 7 Figures, published versio

A Tensor Approach to Learning Mixed Membership Community Models

Community detection is the task of detecting hidden communities from observed interactions. Guaranteed community detection has so far been mostly limited to models with non-overlapping communities such as the stochastic block model. In this paper, we remove this restriction, and provide guaranteed community detection for a family of probabilistic network models with overlapping communities, termed as the mixed membership Dirichlet model, first introduced by Airoldi et al. This model allows for nodes to have fractional memberships in multiple communities and assumes that the community memberships are drawn from a Dirichlet distribution. Moreover, it contains the stochastic block model as a special case. We propose a unified approach to learning these models via a tensor spectral decomposition method. Our estimator is based on low-order moment tensor of the observed network, consisting of 3-star counts. Our learning method is fast and is based on simple linear algebraic operations, e.g. singular value decomposition and tensor power iterations. We provide guaranteed recovery of community memberships and model parameters and present a careful finite sample analysis of our learning method. As an important special case, our results match the best known scaling requirements for the (homogeneous) stochastic block model

Beating Randomized Response on Incoherent Matrices

Computing accurate low rank approximations of large matrices is a fundamental data mining task. In many applications however the matrix contains sensitive information about individuals. In such case we would like to release a low rank approximation that satisfies a strong privacy guarantee such as differential privacy. Unfortunately, to date the best known algorithm for this task that satisfies differential privacy is based on naive input perturbation or randomized response: Each entry of the matrix is perturbed independently by a sufficiently large random noise variable, a low rank approximation is then computed on the resulting matrix. We give (the first) significant improvements in accuracy over randomized response under the natural and necessary assumption that the matrix has low coherence. Our algorithm is also very efficient and finds a constant rank approximation of an m x n matrix in time O(mn). Note that even generating the noise matrix required for randomized response already requires time O(mn)