Compressed sensing theory explains why Lasso programs recover structured high-dimensional signals with minimax order-optimal error. Yet, the optimal choice of the program's governing parameter is often unknown in practice. It is still unclear how variation of the governing parameter impacts recovery error in compressed sensing, which is otherwise provably stable and robust. We establish a novel notion of instability in Lasso programs when the measurement matrix is identity. This is the proximal denoising setup. We prove asymptotic cusp-like behaviour of the risk as a function of the parameter choice, and illustrate the theory with numerical simulations. For example, a 0.1 underestimate of a Lasso parameter can increase the error significantly; and a 50 underestimate can cause the error to increase by a factor of 1e9. We hope that revealing parameter instability regimes of Lasso programs helps to inform a practitioner's choice.
PD is a simple model that elucidates crucial properties of models in general @cite_16 . As a central model for denoising, it lays the groundwork for CS, deconvolution and inpainting problems @cite_30 . A fundamental signal recovery phase transition in CS is predicted by geometric properties of PD @cite_32 , because the minimax risk for PD is equal to the statistical dimension of the signal class @cite_8 . This quantity is a generalized version of @math introduced above.
@cite_30 @cite_16 @cite_32 @cite_8
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Recovering an unknown complex signal from the magnitude of linear combinations of the signal is referred to as phase retrieval. We present an exact performance analysis of a recently proposed convex-optimization-formulation for this problem, known as PhaseMax. Standard convex-relaxation-based methods in phase retrieval resort to the idea of “lifting” which makes them computationally inefficient, since the number of unknowns is effectively squared. In contrast, PhaseMax is a novel convex relaxation that does not increase the number of unknowns. Instead it relies on an initial estimate of the true signal which must be externally provided. In this paper, we investigate the required number of measurements for exact recovery of the signal in the large system limit and when the linear measurement matrix is random with iid standard normal entries. If @math denotes the dimension of the unknown complex signal and @math the number of phaseless measurements, then in the large system limit, @math measurements is necessary and sufficient to recover the signal with high probability, where @math is the angle between the initial estimate and the true signal. Our result indicates a sharp phase transition in the asymptotic regime which matches the empirical result in numerical simulations. Compressive sensing (CS) is an alternative to Shannon Nyquist sampling for the acquisition of sparse or compressible signals that can be well approximated by just K ? N elements from an N -dimensional basis. Instead of taking periodic samples, CS measures inner products with M < N random vectors and then recovers the signal via a sparsity-seeking optimization or greedy algorithm. Standard CS dictates that robust signal recovery is possible from M = O(K log(N K)) measurements. It is possible to substantially decrease M without sacrificing robustness by leveraging more realistic signal models that go beyond simple sparsity and compressibility by including structural dependencies between the values and locations of the signal coefficients. This paper introduces a model-based CS theory that parallels the conventional theory and provides concrete guidelines on how to create model-based recovery algorithms with provable performance guarantees. A highlight is the introduction of a new class of structured compressible signals along with a new sufficient condition for robust structured compressible signal recovery that we dub the restricted amplification property, which is the natural counterpart to the restricted isometry property of conventional CS. Two examples integrate two relevant signal models-wavelet trees and block sparsity-into two state-of-the-art CS recovery algorithms and prove that they offer robust recovery from just M = O(K) measurements. Extensive numerical simulations demonstrate the validity and applicability of our new theory and algorithms. In this paper we develop a novel computational sensing framework for sensing and recovering structured signals. When trained on a set of representative signals, our framework learns to take undersampled measurements and recover signals from them using a deep convolutional neural network. In other words, it learns a transformation from the original signals to a near-optimal number of undersampled measurements and the inverse transformation from measurements to signals. This is in contrast to traditional compressive sensing (CS) systems that use random linear measurements and convex optimization or iterative algorithms for signal recovery. We compare our new framework with @math -minimization from the phase transition point of view and demonstrate that it outperforms @math -minimization in the regions of phase transition plot where @math -minimization cannot recover the exact solution. In addition, we experimentally demonstrate how learning measurements enhances the overall recovery performance, speeds up training of recovery framework, and leads to having fewer parameters to learn. The two-parameter Poisson-Dirichlet distribution, denoted PD(α,θ), is a probability distribution on the set of decreasing positive sequences with sum 1. The usual Poisson-Dirichlet distribution with a single parameter θ, introduced by Kingman, is PD(0, θ). Known properties of PD(0, θ), including the Markov chain description due to Vershik, Shmidt and Ignatov, are generalized to the two-parameter case. The size-biased random permutation of PD(α, θ) is a simple residual allocation model proposed by Engen in the context of species diversity, and rediscovered by Perman and the authors in the study of excursions of Brownian motion and Bessel processes. For 0 < α < 1, PD(α, 0) is the asymptotic distribution of ranked lengths of excursions of a Markov chain away from a state whose recurrence time distribution is in the domain of attraction of a stable law of index α. Formulae in this case trace back to work of Darling, Lamperti and Wendel in the 1950s and 1960s. The distribution of ranked lengths of excursions of a one-dimensional Brownian motion is PD(1 2, 0), and the corresponding distribution for a Brownian bridge is PD(1 2, 1 2). The PD(α, 0) and PD(α, α) distributions admit a similar interpretation in terms of the ranked lengths of excursions of a semistable Markov process whose zero set is the range of a stable subordinator of index α.