The Discrete Fourier Transform (DFT) is a fundamental computational primitive, and the fastest known algorithm for computing the DFT is the FFT (Fast Fourier Transform) algorithm. One remarkable feature of FFT is the fact that its runtime depends only on the size N of the input vector, but not on the dimensionality of the input domain:The state of the art for Sparse FFT, i.e. the problem of computing the DFT of a signal that has at most k nonzeros in Fourier domain, is very different: all current techniques for sublinear time computation of Sparse FFT incur an exponential dependence on the dimension d in the runtime. In this paper we give the first algorithm that computes the DFT of a ksparse signal in time poly(k, log N ) in any dimension d, avoiding the curse of dimensionality inherent in all previously known techniques. Our main tool is a new class of filters that we refer to as adaptive aliasing filters: these filters allow isolating frequencies of a k-Fourier sparse signal using O(k) samples in time domain and O(k log N ) runtime per frequency, in any dimension d.We also investigate natural average case models of the input signal:(1) worst case support in Fourier domain with randomized coefficients and (2) random locations in Fourier domain with worst case coefficients. Our techniques lead to an O(k 2 ) time algorithm for the former and an O(k) time algorithm for the latter.
The problem of approximately computing the k dominant Fourier coefficients of a vector X quickly, and using few samples in time domain, is known as the Sparse Fourier Transform (sparse FFT) problem. A long line of work on the sparse FFT has resulted in algorithms with O(k log n log(n/k)) runtime [Hassanieh et al., STOC'12] and O(k log n) sample complexity [Indyk et al., FOCS'14]. These results are proved using non-adaptive algorithms, and the latter O(k log n) sample complexity result is essentially the best possible under the sparsity assumption alone: It is known that even adaptive algorithms must use Ω((k log(n/k))/ log log n) samples [Hassanieh et al., STOC'12]. By adaptive, we mean being able to exploit previous samples in guiding the selection of further samples. This paper revisits the sparse FFT problem with the added twist that the sparse coefficients approximately obey a (k 0 , k 1 )-block sparse model. In this model, signal frequencies are clustered in k 0 intervals with width k 1 in Fourier space, and k = k 0 k 1 is the total sparsity. Signals arising in applications are often well approximated by this model with k 0 k.Our main result is the first sparse FFT algorithm for (k 0 , k 1 )-block sparse signals with a sample complexity of O * (k 0 k 1 + k 0 log(1 + k 0 ) log n) at constant signal-to-noise ratios, and sublinear runtime. A similar sample complexity was previously achieved in the works on model-based compressive sensing using random Gaussian measurements, but used Ω(n) runtime. To the best of our knowledge, our result is the first sublinear-time algorithm for model based compressed sensing, and the first sparse FFT result that goes below the O(k log n) sample complexity bound. Interestingly, the aforementioned model-based compressive sensing result that relies on Gaussian measurements is non-adaptive, whereas our algorithm crucially uses adaptivity to achieve the improved sample complexity bound. We prove that adaptivity is in fact necessary in the Fourier setting: Any non-adaptive algorithm must use Ω(k 0 k 1 log n k0k1 ) samples for the (k 0 , k 1 )-block sparse model, ruling out improvements over the vanilla sparsity assumption. Our main technical innovation for adaptivity is a new randomized energy-based importance sampling technique that may be of independent interest.
Reconstructing continuous signals based on a small number of discrete samples is a fundamental problem across science and engineering. In practice, we are often interested in signals with "simple" Fourier structure -e.g., those involving frequencies within a bounded range, a small number of frequencies, or a few blocks of frequencies. 1 More broadly, any prior knowledge about a signal's Fourier power spectrum can constrain its complexity. Intuitively, signals with more highly constrained Fourier structure require fewer samples to reconstruct.We formalize this intuition by showing that, roughly, a continuous signal from a given class can be approximately reconstructed using a number of samples proportional to the statistical dimension of the allowed power spectrum of that class. We prove that, in nearly all settings, this natural measure tightly characterizes the sample complexity of signal reconstruction.Surprisingly, we also show that, up to logarithmic factors, a universal non-uniform sampling strategy can achieve this optimal complexity for any class of signals. We present a simple, efficient, and general algorithm for recovering a signal from the samples taken. For bandlimited and sparse signals, our method matches the state-of-the-art. At the same time, it gives the first computationally and sample efficient solution to a broad range of problems, including multiband signal reconstruction and kriging and Gaussian process regression tasks in one dimension.Our work is based on a novel connection between randomized linear algebra and the problem of reconstructing signals with constrained Fourier structure. We extend tools based on statistical leverage score sampling and column-based matrix reconstruction to the approximation of continuous linear operators that arise in the signal reconstruction problem. We believe that these extensions are of independent interest and serve as a foundation for tackling a broad range of continuous time problems using randomized methods.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.