We study the problem of estimating the best B term Fourier representation for a given frequency-sparse signal (i.e., vector) @math of length @math . More explicitly, we investigate how to deterministically identify B of the largest magnitude frequencies of @math , and estimate their coefficients, in polynomial @math time. Randomized sub-linear time algorithms which have a small (controllable) probability of failure for each processed signal exist for solving this problem. However, for failure intolerant applications such as those involving mission-critical hardware designed to process many signals over a long lifetime, deterministic algorithms with no probability of failure are highly desirable. In this paper we build on the deterministic Compressed Sensing results of Cormode and Muthukrishnan (CM) CMDetCS3,CMDetCS1,CMDetCS2 in order to develop the first known deterministic sub-linear time sparse Fourier Transform algorithm suitable for failure intolerant applications. Furthermore, in the process of developing our new Fourier algorithm, we present a simplified deterministic Compressed Sensing algorithm which improves on CM's algebraic compressibility results while simultaneously maintaining their results concerning exponential decay.
Compressed Sensing (CS) methods @cite_2 @cite_20 @cite_22 @cite_15 @cite_27 provide a robust framework for reducing the number of measurements required to estimate a sparse signal. For this reason CS methods are useful in areas such as MR imaging @cite_8 @cite_6 and analog-to-digital conversion @cite_10 @cite_12 where measurement costs are high. The general CS setup is as follows: Let be an @math -length signal vector with complex valued entries and @math be a full rank @math change of basis matrix. Furthermore, suppose that @math is sparse (i.e., only @math entries of @math are significant large in magnitude). CS methods deal with generating a @math measurement matrix, @math , with the smallest number of rows possible (i.e., @math minimized) so that the @math significant entries of @math can be well approximated by the @math -element vector result of Note that CS is inherently algorithmic since a procedure for recovering @math 's largest @math -entries from the result of Equation must be specified.
@cite_22 @cite_8 @cite_6 @cite_27 @cite_2 @cite_15 @cite_10 @cite_12 @cite_20
2032618720 1676074242 2147361252 2949903712
Compressed sensing (CS) decoding algorithms can efficiently recover an N -dimensional real-valued vector x to within a factor of its best k-term approximation by taking m = O(klogN k) measurements y = Phix. If the sparsity or approximate sparsity level of x were known, then this theoretical guarantee would imply quality assurance of the resulting CS estimate. However, because the underlying sparsity of the signal x is unknown, the quality of a CS estimate x using m measurements is not assured. It is nevertheless shown in this paper that sharp bounds on the error ||x - x ||lN2 can be achieved with almost no effort. More precisely, suppose that a maximum number of measurements m is preimposed. One can reserve 10 log p of these m measurements and compute a sequence of possible estimates ( xj)j=1p to x from the m -10logp remaining measurements; the errors ||x - xj ||lN2 for j = 1, ..., p can then be bounded with high probability. As a consequence, numerical upper and lower bounds on the error between x and the best k-term approximation to x can be estimated for p values of k with almost no cost. This observation has applications outside CS as well. Compressed sensing (CS) demonstrates that sparse signals can be recovered from underdetermined linear measurements. We focus on the joint sparse recovery problem where multiple signals share the same common sparse support sets, and they are measured through the same sensing matrix. Leveraging a recent information theoretic characterization of single signal CS, we formulate the optimal minimum mean square error (MMSE) estimation problem, and derive a belief propagation algorithm, its relaxed version, for the joint sparse recovery problem and an approximate message passing algorithm. In addition, using density evolution, we provide a sufficient condition for exact recovery. We relate compressed sensing (CS) with Bayesian experimental design and provide a novel efficient approximate method for the latter, based on expectation propagation. In a large comparative study about linearly measuring natural images, we show that the simple standard heuristic of measuring wavelet coefficients top-down systematically outperforms CS methods using random measurements; the sequential projection optimisation approach of (Ji & Carin, 2007) performs even worse. We also show that our own approximate Bayesian method is able to learn measurement filters on full images efficiently which outperform the wavelet heuristic. To our knowledge, ours is the first successful attempt at "learning compressed sensing" for images of realistic size. In contrast to common CS methods, our framework is not restricted to sparse signals, but can readily be applied to other notions of signal complexity or noise models. We give concrete ideas how our method can be scaled up to large signal representations. The Compressive Sensing (CS) framework aims to ease the burden on analog-to-digital converters (ADCs) by reducing the sampling rate required to acquire and stably recover sparse signals. Practical ADCs not only sample but also quantize each measurement to a finite number of bits; moreover, there is an inverse relationship between the achievable sampling rate and the bit depth. In this paper, we investigate an alternative CS approach that shifts the emphasis from the sampling rate to the number of bits per measurement. In particular, we explore the extreme case of 1-bit CS measurements, which capture just their sign. Our results come in two flavors. First, we consider ideal reconstruction from noiseless 1-bit measurements and provide a lower bound on the best achievable reconstruction error. We also demonstrate that i.i.d. random Gaussian matrices describe measurement mappings achieving, with overwhelming probability, nearly optimal error decay. Next, we consider reconstruction robustness to measurement errors and noise and introduce the Binary @math -Stable Embedding (B @math SE) property, which characterizes the robustness measurement process to sign changes. We show the same class of matrices that provide almost optimal noiseless performance also enable such a robust mapping. On the practical side, we introduce the Binary Iterative Hard Thresholding (BIHT) algorithm for signal reconstruction from 1-bit measurements that offers state-of-the-art performance.