Abstract-This paper considers the model problem of reconstructing an object from incomplete frequency samples. Consider a discrete-time signal and a randomly chosen set of frequencies . Is it possible to reconstruct from the partial knowledge of its Fourier coefficients on the set ?A typical result of this paper is as follows. Suppose that is a superposition offor some constant 0. We do not know the locations of the spikes nor their amplitudes. Then with probability at least 1 ( ), can be reconstructed exactly as the solution to the 1 minimization problemIn short, exact recovery may be obtained by solving a convex optimization problem. We give numerical values for which depend on the desired probability of success. Our result may be interpreted as a novel kind of nonlinear sampling theorem. In effect, it says that any signal made out of spikes may be recovered by convex programming from almost every set of frequencies of size ( log ). Moreover, this is nearly optimal in the sense that any method succeeding with probability 1 ( ) would in general require a number of frequency samples at least proportional to log .The methodology extends to a variety of other situations and higher dimensions. For example, we show how one can reconstruct a piecewise constant (one-or two-dimensional) object from incomplete frequency samples-provided that the number of jumps (discontinuities) obeys the condition above-by minimizing other convex functionals such as the total variation of .

This article is about a curious phenomenon. Suppose we have a data matrix, which is the superposition of a low-rank component and a sparse component. Can we recover each component individually? We prove that under some suitable assumptions, it is possible to recover both the low-rank and the sparse components exactly by solving a very convenient convex program called Principal Component Pursuit ; among all feasible decompositions, simply minimize a weighted combination of the nuclear norm and of the ℓ 1 norm. This suggests the possibility of a principled approach to robust principal component analysis since our methodology and results assert that one can recover the principal components of a data matrix even though a positive fraction of its entries are arbitrarily corrupted. This extends to the situation where a fraction of the entries are missing as well. We discuss an algorithm for solving this optimization problem, and present applications in the area of video surveillance, where our methodology allows for the detection of objects in a cluttered background, and in the area of face recognition, where it offers a principled way of removing shadows and specularities in images of faces.

onventional approaches to sampling signals or images follow Shannon's celebrated theorem: the sampling rate must be at least twice the maximum frequency present in the signal (the so-called Nyquist rate). In fact, this principle underlies nearly all signal acquisition protocols used in consumer audio and visual electronics, medical imaging devices, radio receivers, and so on. (For some signals, such as images that are not naturally bandlimited, the sampling rate is dictated not by the Shannon theorem but by the desired temporal or spatial resolution. However, it is common in such systems to use an antialiasing low-pass filter to bandlimit the signal before sampling, and so the Shannon theorem plays an implicit role.) In the field of data conversion, for example, standard analog-to-digital converter (ADC) technology implements the usual quantized Shannon representation: the signal is uniformly sampled at or above the Nyquist rate.

Abstract-This paper considers a natural error correcting problem with real valued input/output. We wish to recover an input vector from corrupted measurements = + . Here,is an by (coding) matrix and is an arbitrary and unknown vector of errors. Is it possible to recover exactly from the data ?We prove that under suitable conditions on the coding matrix , the input is the unique solution to the 1 -minimization problem ( This work is related to the problem of finding sparse solutions to vastly underdetermined systems of linear equations. There are also significant connections with the problem of recovering signals from highly incomplete measurements. In fact, the results introduced in this paper improve on our earlier work. Finally, underlying the success of 1 is a crucial property we call the uniform uncertainty principle that we shall describe in detail.Index Terms-Basis pursuit, decoding of (random) linear codes, duality in optimization, Gaussian random matrices, 1 minimization, linear codes, linear programming, principal angles, restricted orthonormality, singular values of random matrices, sparse solutions to underdetermined systems.

Suppose we wish to recover a vector x 0 ∈ R m (e.g., a digital signal or image) from incomplete and contaminated observations y = Ax 0 + e; A is an n × m matrix with far fewer rows than columns (n m) and e is an error term. Is it possible to recover x 0 accurately based on the data y?To recover x 0 , we consider the solution x to the 1 -regularization problemwhere is the size of the error term e. We show that if A obeys a uniform uncertainty principle (with unit-normed columns) and if the vector x 0 is sufficiently sparse, then the solution is within the noise levelAs a first example, suppose that A is a Gaussian random matrix; then stable recovery occurs for almost all such A's provided that the number of nonzeros of x 0 is of about the same order as the number of observations. As a second instance, suppose one observes few Fourier samples of x 0 ; then stable recovery occurs for almost any set of n coefficients provided that the number of nonzeros is of the order of n/(log m) 6 .In the case where the error term vanishes, the recovery is of course exact, and this work actually provides novel insights into the exact recovery phenomenon discussed in earlier papers. The methodology also explains why one can also very nearly recover approximately sparse signals.