Optimal Phase Transitions in Compressed Sensing

Compressed sensing deals with efficient recovery of analog signals from linear encodings. This paper presents a statistical study of compressed sensing by modeling the input signal as an i.i.d. process with known distribution. Three classes of encoders are considered, namely optimal nonlinear, optimal linear and random linear encoders. Focusing on optimal decoders, we investigate the fundamental tradeoff between measurement rate and reconstruction fidelity gauged by error probability and noise sensitivity in the absence and presence of measurement noise, respectively. The optimal phase transition threshold is determined as a functional of the input distribution and compared to suboptimal thresholds achieved by popular reconstruction algorithms. In particular, we show that Gaussian sensing matrices incur no penalty on the phase transition threshold with respect to optimal nonlinear encoding. Our results also provide a rigorous justification of previous results based on replica heuristics in the weak-noise regime.Comment: to appear in IEEE Transactions of Information Theor

Phase Diagram and Approximate Message Passing for Blind Calibration and Dictionary Learning

We consider dictionary learning and blind calibration for signals and matrices created from a random ensemble. We study the mean-squared error in the limit of large signal dimension using the replica method and unveil the appearance of phase transitions delimiting impossible, possible-but-hard and possible inference regions. We also introduce an approximate message passing algorithm that asymptotically matches the theoretical performance, and show through numerical tests that it performs very well, for the calibration problem, for tractable system sizes.Comment: 5 page

Lossless Linear Analog Compression

We establish the fundamental limits of lossless linear analog compression by considering the recovery of random vectors ${\boldsymbol{\mathsf{x}}}\in{\mathbb R}^m$ from the noiseless linear measurements ${\boldsymbol{\mathsf{y}}}=\boldsymbol{A}{\boldsymbol{\mathsf{x}}}$ with measurement matrix $\boldsymbol{A}\in{\mathbb R}^{n\times m}$. Specifically, for a random vector ${\boldsymbol{\mathsf{x}}}\in{\mathbb R}^m$ of arbitrary distribution we show that ${\boldsymbol{\mathsf{x}}}$ can be recovered with zero error probability from $n>\inf\underline{\operatorname{dim}}_\mathrm{MB}(U)$ linear measurements, where $\underline{\operatorname{dim}}_\mathrm{MB}(\cdot)$ denotes the lower modified Minkowski dimension and the infimum is over all sets $U\subseteq{\mathbb R}^{m}$ with $\mathbb{P}[{\boldsymbol{\mathsf{x}}}\in U]=1$. This achievability statement holds for Lebesgue almost all measurement matrices $\boldsymbol{A}$. We then show that $s$-rectifiable random vectors---a stochastic generalization of $s$-sparse vectors---can be recovered with zero error probability from $n>s$ linear measurements. From classical compressed sensing theory we would expect $n\geq s$ to be necessary for successful recovery of ${\boldsymbol{\mathsf{x}}}$. Surprisingly, certain classes of $s$-rectifiable random vectors can be recovered from fewer than $s$ measurements. Imposing an additional regularity condition on the distribution of $s$-rectifiable random vectors ${\boldsymbol{\mathsf{x}}}$, we do get the expected converse result of $s$ measurements being necessary. The resulting class of random vectors appears to be new and will be referred to as $s$-analytic random vectors

Replica Symmetry Breaking in Compressive Sensing

For noisy compressive sensing systems, the asymptotic distortion with respect to an arbitrary distortion function is determined when a general class of least-square based reconstruction schemes is employed. The sampling matrix is considered to belong to a large ensemble of random matrices including i.i.d. and projector matrices, and the source vector is assumed to be i.i.d. with a desired distribution. We take a statistical mechanical approach by representing the asymptotic distortion as a macroscopic parameter of a spin glass and employing the replica method for the large-system analysis. In contrast to earlier studies, we evaluate the general replica ansatz which includes the RS ansatz as well as RSB. The generality of the solution enables us to study the impact of symmetry breaking. Our numerical investigations depict that for the reconstruction scheme with the "zero-norm" penalty function, the RS fails to predict the asymptotic distortion for relatively large compression rates; however, the one-step RSB ansatz gives a valid prediction of the performance within a larger regime of compression rates.Comment: 7 pages, 3 figures, presented at ITA 201

Efficient high-dimensional entanglement imaging with a compressive sensing, double-pixel camera

We implement a double-pixel, compressive sensing camera to efficiently characterize, at high resolution, the spatially entangled fields produced by spontaneous parametric downconversion. This technique leverages sparsity in spatial correlations between entangled photons to improve acquisition times over raster-scanning by a scaling factor up to n^2/log(n) for n-dimensional images. We image at resolutions up to 1024 dimensions per detector and demonstrate a channel capacity of 8.4 bits per photon. By comparing the classical mutual information in conjugate bases, we violate an entropic Einstein-Podolsky-Rosen separability criterion for all measured resolutions. More broadly, our result indicates compressive sensing can be especially effective for higher-order measurements on correlated systems.Comment: 10 pages, 7 figure

Measure What Should be Measured: Progress and Challenges in Compressive Sensing

Is compressive sensing overrated? Or can it live up to our expectations? What will come after compressive sensing and sparsity? And what has Galileo Galilei got to do with it? Compressive sensing has taken the signal processing community by storm. A large corpus of research devoted to the theory and numerics of compressive sensing has been published in the last few years. Moreover, compressive sensing has inspired and initiated intriguing new research directions, such as matrix completion. Potential new applications emerge at a dazzling rate. Yet some important theoretical questions remain open, and seemingly obvious applications keep escaping the grip of compressive sensing. In this paper I discuss some of the recent progress in compressive sensing and point out key challenges and opportunities as the area of compressive sensing and sparse representations keeps evolving. I also attempt to assess the long-term impact of compressive sensing

Compressed Sensing of Approximately-Sparse Signals: Phase Transitions and Optimal Reconstruction

Compressed sensing is designed to measure sparse signals directly in a compressed form. However, most signals of interest are only "approximately sparse", i.e. even though the signal contains only a small fraction of relevant (large) components the other components are not strictly equal to zero, but are only close to zero. In this paper we model the approximately sparse signal with a Gaussian distribution of small components, and we study its compressed sensing with dense random matrices. We use replica calculations to determine the mean-squared error of the Bayes-optimal reconstruction for such signals, as a function of the variance of the small components, the density of large components and the measurement rate. We then use the G-AMP algorithm and we quantify the region of parameters for which this algorithm achieves optimality (for large systems). Finally, we show that in the region where the GAMP for the homogeneous measurement matrices is not optimal, a special "seeding" design of a spatially-coupled measurement matrix allows to restore optimality.Comment: 8 pages, 10 figure

Multi Terminal Probabilistic Compressed Sensing

In this paper, the `Approximate Message Passing' (AMP) algorithm, initially developed for compressed sensing of signals under i.i.d. Gaussian measurement matrices, has been extended to a multi-terminal setting (MAMP algorithm). It has been shown that similar to its single terminal counterpart, the behavior of MAMP algorithm is fully characterized by a `State Evolution' (SE) equation for large block-lengths. This equation has been used to obtain the rate-distortion curve of a multi-terminal memoryless source. It is observed that by spatially coupling the measurement matrices, the rate-distortion curve of MAMP algorithm undergoes a phase transition, where the measurement rate region corresponding to a low distortion (approximately zero distortion) regime is fully characterized by the joint and conditional Renyi information dimension (RID) of the multi-terminal source. This measurement rate region is very similar to the rate region of the Slepian-Wolf distributed source coding problem where the RID plays a role similar to the discrete entropy. Simulations have been done to investigate the empirical behavior of MAMP algorithm. It is observed that simulation results match very well with predictions of SE equation for reasonably large block-lengths.Comment: 11 pages, 13 figures. arXiv admin note: text overlap with arXiv:1112.0708 by other author

Isotropically Random Orthogonal Matrices: Performance of LASSO and Minimum Conic Singular Values

Recently, the precise performance of the Generalized LASSO algorithm for recovering structured signals from compressed noisy measurements, obtained via i.i.d. Gaussian matrices, has been characterized. The analysis is based on a framework introduced by Stojnic and heavily relies on the use of Gordon's Gaussian min-max theorem (GMT), a comparison principle on Gaussian processes. As a result, corresponding characterizations for other ensembles of measurement matrices have not been developed. In this work, we analyze the corresponding performance of the ensemble of isotropically random orthogonal (i.r.o.) measurements. We consider the constrained version of the Generalized LASSO and derive a sharp characterization of its normalized squared error in the large-system limit. When compared to its Gaussian counterpart, our result analytically confirms the superiority in performance of the i.r.o. ensemble. Our second result, derives an asymptotic lower bound on the minimum conic singular values of i.r.o. matrices. This bound is larger than the corresponding bound on Gaussian matrices. To prove our results we express i.r.o. matrices in terms of Gaussians and show that, with some modifications, the GMT framework is still applicable