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Explicit constructions for compressed sensing of sparse signals
 In Proceedings of the 19th Annual ACMSIAM Symposium on Discrete Algorithms
, 2008
"... Over the recent years, a new approach for obtaining a succinct approximate representation of ndimensional ..."
Abstract

Cited by 33 (3 self)
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Over the recent years, a new approach for obtaining a succinct approximate representation of ndimensional
Simple and Practical Algorithm for Sparse Fourier Transform
"... We consider the sparse Fourier transform problem: given a complex vector x of length n, and a parameter k, estimate the k largest (in magnitude) coefficients of the Fourier transform of x. The problem is of key interest in several areas, including signal processing, audio/image/video compression, an ..."
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Cited by 13 (5 self)
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We consider the sparse Fourier transform problem: given a complex vector x of length n, and a parameter k, estimate the k largest (in magnitude) coefficients of the Fourier transform of x. The problem is of key interest in several areas, including signal processing, audio/image/video compression, and learning theory. We propose a new algorithm for this problem. The algorithm leverages techniques from digital signal processing, notably Gaussian and DolphChebyshev filters. Unlike the typical approach to this problem, our algorithm is not iterative. That is, instead of estimating “large ” coefficients, subtracting them and recursing on the reminder, it identifies and estimates the k largest coefficients in “one shot”, in a manner akin to sketching/streaming algorithms. The resulting algorithm is structurally simpler than its predecessors. As a consequence, we are able to extend considerably the range of sparsity, k, for which the algorithm is faster than FFT, both in theory and practice. 1
Nearly Optimal Sparse Fourier Transform
"... We consider the problem of computing the ksparse approximation to the discrete Fourier transform of an ndimensional signal. We show: • An O(k log n)time algorithm for the case where the input signal has at most k nonzero Fourier coefficients, and • An O(k log n log(n/k))time algorithm for gener ..."
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Cited by 10 (4 self)
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We consider the problem of computing the ksparse approximation to the discrete Fourier transform of an ndimensional signal. We show: • An O(k log n)time algorithm for the case where the input signal has at most k nonzero Fourier coefficients, and • An O(k log n log(n/k))time algorithm for general input signals. Both algorithms achieve o(n log n) time, and thus improve over the Fast Fourier Transform, for any k = o(n). Further, they are the first known algorithms that satisfy this property. Also, if one assumes that the Fast Fourier Transform is optimal, the algorithm for the exactly ksparse case is optimal for any k = n Ω(1). We complement our algorithmic results by showing that any algorithm for computing the sparse Fourier transform of a general signal must use at least Ω(k log(n/k) / log log n) signal samples, even if it is allowed The discrete Fourier transform (DFT) is one of the most important and widely used computational tasks. Its applications are broad and include signal processing, communications, and audio/image/video compression. Hence, fast algorithms for DFT are highly valuable. Currently, the fastest such algorithm is the Fast Fourier
Sampleoptimal averagecase sparse fourier transform in two dimensions. Unpublished manuscript
, 2012
"... We present the first sampleoptimal sublinear time algorithms for the sparse Discrete Fourier Transform over a twodimensional √ n × √ n grid. Our algorithms are analyzed for average case signals. For signals whose spectrum is exactly sparse, our algorithms use O(k) samples and run in O(klogk) time ..."
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Cited by 2 (1 self)
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We present the first sampleoptimal sublinear time algorithms for the sparse Discrete Fourier Transform over a twodimensional √ n × √ n grid. Our algorithms are analyzed for average case signals. For signals whose spectrum is exactly sparse, our algorithms use O(k) samples and run in O(klogk) time, wherek is the expected sparsity of the signal. For signals whose spectrum is approximately sparse, our algorithm usesO(klogn) samples and runs in O(klog 2 n) time; the latter algorithm works for k = Θ ( √ n). The number of samples used by our algorithms matches the known lower bounds for the respective signal models. By a known reduction, our algorithms give similar results for the onedimensional sparse Discrete Fourier Transform whennis a power of a small composite number (e.g.,n = 6 t). 1
Statement of Research
, 2012
"... My goal in research is to discover theoretical insights that can guide practitioners in the creation of useful systems. To this end, I try to focus on relatively simple algorithms that are feasible to implement and have small bigOh constants; when finding lower bounds, I look for ones that give gui ..."
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My goal in research is to discover theoretical insights that can guide practitioners in the creation of useful systems. To this end, I try to focus on relatively simple algorithms that are feasible to implement and have small bigOh constants; when finding lower bounds, I look for ones that give guidance in the creation of efficient algorithms. To calibrate my understanding of the relation between theory and practice, I implement about half my algorithms and analyze their empirical performance. With this in mind, I have decided to focus my research on the interdisciplinary area of sparse recovery, which includes aspects of compressive sensing and streaming algorithms. The goal of sparse recovery is to acquire and process “sparse ” data from a small number of samples. The topic offers the opportunity to develop theoretical and mathematical techniques that apply to big data problems in diverse areas such as data stream analysis and signal processing. On the theory side, sparse recovery involves fascinating techniques from algorithms, statistics, probability theory, and information theory. On the application side, sparse recovery lets me interact with researchers that understand the practical constraints involved in creating real systems. It is a great area to develop theoretical results that can make an impact in practice. My first paper in the area shows fundamental limitations on the “standard ” compressive sensing framework, by showing that the number of samples required by the seminal work of Candès, Romberg, and Tao is in fact optimal. This led me to investigate two methods for circumventing these lower bounds: making the sampling process adaptive and incorporating additional structural assumptions on the signals. I have shown that adaptivity enables a significant—and in some cases exponential—reduction in the number of samples required for sparse recovery. I have also shown the first linear time algorithm to exploit one of the most common additional structural assumptions. My research on algorithms that are highly efficient in both number of samples and processing time has culminated in algorithms to compute the Fourier transform efficiently when its output is sparse. These algorithms are faster than the ubiquitous Fast Fourier Transform for moderately sparse data, both in theory and in practice.
Sparse Recovery and Fourier Sampling
, 2013
"... the last decade a broad literature has arisen studying sparse recovery, the estimation of sparse vectors from low dimensional linear projections. Sparse recovery has a wide variety of applications such as streaming algorithms, image acquisition, and disease testing. A particularly important subclass ..."
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the last decade a broad literature has arisen studying sparse recovery, the estimation of sparse vectors from low dimensional linear projections. Sparse recovery has a wide variety of applications such as streaming algorithms, image acquisition, and disease testing. A particularly important subclass of sparse recovery is the sparse Fourier transform, which considers the computation of a discrete Fourier transform when the output is sparse. Applications of the sparse Fourier transform include medical imaging, spectrum sensing, and purely computation tasks involving convolution. This thesis describes a coherent set of techniques that achieve optimal or nearoptimal upper and lower bounds for a variety of sparse recovery problems. We give the following stateoftheart algorithms for recovery of an approximately ksparse vector in n dimensions: • Two sparse Fourier transform algorithms, respectively taking O(k log n log(n/k)) time and O(k log n log c log n) samples. The latter is within log c log n of the optimal sample complexity when k < n 1−ɛ. • An algorithm for adaptive sparse recovery using O(k log log(n/k)) measurements, showing