Suppose x is an unknown vector in Ropfm (a digital image or signal); we plan to measure n general linear functionals of x and then reconstruct. If x is known to be compressible by transform coding with a known transform, and we reconstruct via the nonlinear procedure defined here, the number of measurements n can be dramatically smaller than the size m. Thus, certain natural classes of images with m pixels need only n=O(m1/4log5/2(m)) nonadaptive nonpixel samples for faithful recovery, as opposed to the usual m pixel samples. More specifically, suppose x has a sparse representation in some orthonormal basis (e.g., wavelet, Fourier) or tight frame (e.g., curvelet, Gabor)-so the coefficients belong to an lscrp ball for 0<ples1. The N most important coefficients in that expansion allow reconstruction with lscr2 error O(N1/2-1p/). It is possible to design n=O(Nlog(m)) nonadaptive measurements allowing reconstruction with accuracy comparable to that attainable with direct knowledge of the N most important coefficients. Moreover, a good approximation to those N important coefficients is extracted from the n measurements by solving a linear program-Basis Pursuit in signal processing. The nonadaptive measurements have the character of "random" linear combinations of basis/frame elements. Our results use the notions of optimal recovery, of n-widths, and information-based complexity. We estimate the Gel'fand n-widths of lscrp balls in high-dimensional Euclidean space in the case 0<ples1, and give a criterion identifying near- optimal subspaces for Gel'fand n-widths. We show that "most" subspaces are near-optimal, and show that convex optimization (Basis Pursuit) is a near-optimal way to extract information derived from these near-optimal subspaces

Compressed sensing

Wireless Communications

/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

This article presents new results on using a greedy algorithm, orthogonal matching pursuit (OMP), to solve the sparse approximation problem over redundant dictionaries. It provides a sufficient condition under which both OMP and Donoho's basis pursuit (BP) paradigm can recover the optimal representation of an exactly sparse signal. It leverages this theory to show that both OMP and BP succeed for every sparse input signal from a wide class of dictionaries. These quasi-incoherent dictionaries offer a natural generalization of incoherent dictionaries, and the cumulative coherence function is introduced to quantify the level of incoherence. This analysis unifies all the recent results on BP and extends them to OMP. Furthermore, the paper develops a sufficient condition under which OMP can identify atoms from an optimal approximation of a nonsparse signal. From there, it argues that OMP is an approximation algorithm for the sparse problem over a quasi-incoherent dictionary. That is, for every input signal, OMP calculates a sparse approximant whose error is only a small factor worse than the minimal error that can be attained with the same number of terms.

/pdf/greed-is-good-algorithmic-results-for-sparse-approximation-3sdp2l8ldg.pdf

Greed is good: algorithmic results for sparse approximation

Low-rank matrix approximations, such as the truncated singular value decomposition and the rank-revealing QR decomposition, play a central role in data analysis and scientific computing. This work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing low-rank matrix approximation. These techniques exploit modern computational architectures more fully than classical methods and open the possibility of dealing with truly massive data sets. This paper presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions. These methods use random sampling to identify a subspace that captures most of the action of a matrix. The input matrix is then compressed—either explicitly or implicitly—to this subspace, and the reduced matrix is manipulated deterministically to obtain the desired low-rank factorization. In many cases, this approach beats its classical competitors in terms of accuracy, robustness, and/or speed. These claims are supported by extensive numerical experiments and a detailed error analysis. The specific benefits of randomized techniques depend on the computational environment. Consider the model problem of finding the $k$ dominant components of the singular value decomposition of an $m \times n$ matrix. (i) For a dense input matrix, randomized algorithms require $\bigO(mn \log(k))$ floating-point operations (flops) in contrast to $ \bigO(mnk)$ for classical algorithms. (ii) For a sparse input matrix, the flop count matches classical Krylov subspace methods, but the randomized approach is more robust and can easily be reorganized to exploit multiprocessor architectures. (iii) For a matrix that is too large to fit in fast memory, the randomized techniques require only a constant number of passes over the data, as opposed to $\bigO(k)$ passes for classical algorithms. In fact, it is sometimes possible to perform matrix approximation with a single pass over the data.

/pdf/finding-structure-with-randomness-probabilistic-algorithms-11r587jy79.pdf

Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions

Transmit beamforming and receive combining are simple methods for exploiting the significant diversity that is available in multiple-input multiple-output (MIMO) wireless systems. Unfortunately, optimal performance requires either complete channel knowledge or knowledge of the optimal beamforming vector; both are hard to realize. In this article, a quantized maximum signal-to-noise ratio (SNR) beamforming technique is proposed where the receiver only sends the label of the best beamforming vector in a predetermined codebook to the transmitter. By using the distribution of the optimal beamforming vector in independent and identically distributed Rayleigh fading matrix channels, the codebook design problem is solved and related to the problem of Grassmannian line packing. The proposed design criterion is flexible enough to allow for side constraints on the codebook vectors. Bounds on the codebook size are derived to guarantee full diversity order. Results on the density of Grassmannian line packings are derived and used to develop bounds on the codebook size given a capacity or SNR loss. Monte Carlo simulations are presented that compare the probability of error for different quantization strategies.

Grassmannian beamforming for multiple-input multiple-output wireless systems

Suppose we wish to recover a signal \input amssym $\font\abc=cmmib10\def\bi#1{\hbox{\abc#1}} {\bi x} \in {\Bbb C}^n$ from m intensity measurements of the form , ; that is, from data in which phase information is missing. We prove that if the vectors are sampled independently and uniformly at random on the unit sphere, then the signal x can be recovered exactly (up to a global phase factor) by solving a convenient semidefinite program–-a trace-norm minimization problem; this holds with large probability provided that m is on the order of , and without any assumption about the signal whatsoever. This novel result demonstrates that in some instances, the combinatorial phase retrieval problem can be solved by convex programming techniques. Finally, we also prove that our methodology is robust vis-a-vis additive noise. © 2012 Wiley Periodicals, Inc.

/pdf/phaselift-exact-and-stable-signal-recovery-from-magnitude-3wnl6whapu.pdf

PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming

A stylized compressed sensing radar is proposed in which the time-frequency plane is discretized into an N times N grid. Assuming the number of targets K is small (i.e., K Lt N2), then we can transmit a sufficiently ldquoincoherentrdquo pulse and employ the techniques of compressed sensing to reconstruct the target scene. A theoretical upper bound on the sparsity K is presented. Numerical simulations verify that even better performance can be achieved in practice. This novel-compressed sensing approach offers great potential for better resolution over classical radar.

High-Resolution Radar via Compressed Sensing

https://www.math.ucdavis.edu/~strohmer/papers/2002/grass.pdf

Grassmannian frames with applications to coding and communication

Suppose we wish to recover a signal x in C^n from m intensity measurements of the form | |^2, i = 1, 2,..., m; that is, from data in which phase information is missing. We prove that if the vectors z_i are sampled independently and uniformly at random on the unit sphere, then the signal x can be recovered exactly (up to a global phase factor) by solving a convenient semidefinite program---a trace-norm minimization problem; this holds with large probability provided that m is on the order of n log n, and without any assumption about the signal whatsoever. This novel result demonstrates that in some instances, the combinatorial phase retrieval problem can be solved by convex programming techniques. Finally, we also prove that our methodology is robust vis a vis additive noise.

Thomas Strohmer

Papers

Grassmannian beamforming for multiple-input multiple-output wireless systems

PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming

High-Resolution Radar via Compressed Sensing

Grassmannian frames with applications to coding and communication

PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming