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Register a new userSparse Fourier Transform by traversing Cooley-Tukey FFT computation graphs
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Added by
Amir Zandieh
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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Computing the dominant Fourier coefficients of a vector is a common task in many fields, such as signal processing, learning theory, and computational complexity. In the Sparse Fast Fourier Transform (Sparse FFT) problem, one is given oracle access to a $d$-dimensional vector $x$ of size $N$, and is asked to compute the best $k$-term approximation of its Discrete Fourier Transform, quickly and using few samples of the input vector $x$. While the sample complexity of this problem is quite well understood, all previous approaches either suffer from an exponential dependence of runtime on the dimension $d$ or can only tolerate a trivial amount of noise. This is in sharp contrast with the classical FFT algorithm of Cooley and Tukey, which is stable and completely insensitive to the dimension of the input vector: its runtime is $O(Nlog N)$ in any dimension $d$. In this work, we introduce a new high-dimensional Sparse FFT toolkit and use it to obtain new algorithms, both on the exact, as well as in the case of bounded $ell_2$ noise. This toolkit includes i) a new strategy for exploring a pruned FFT computation tree that reduces the cost of filtering, ii) new structural properties of adaptive aliasing filters recently introduced by Kapralov, Velingker and ZandiehSODA19, and iii) a novel lazy estimation argument, suited to reducing the cost of estimation in FFT tree-traversal approaches. Our robust algorithm can be viewed as a highly optimized sparse, stable extension of the Cooley-Tukey FFT algorithm. Finally, we explain the barriers we have faced by proving a conditional quadratic lower bound on the running time of the well-studied non-equispaced Fourier transform problem. This resolves a natural and frequently asked question in computational Fourier transforms. Lastly, we provide a preliminary experimental evaluation comparing the runtime of our algorithm to FFTW and SFFT 2.0.
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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: FFT runs in time $O(Nlog N)$ irrespective of whether the DFT in question is on $mathbb{Z}_N$ or $mathbb{Z}_n^d$ for some $d>1$, where $N=n^d$. 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 $k$-sparse signal in time $text{poly}(k, log N)$ 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 $O(klog N)$ 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 $widetilde O(k^2)$ time algorithm for the former and an $widetilde O(k)$ time algorithm for the latter.
In this paper we revisit the deterministic version of the Sparse Fourier Transform problem, which asks to read only a few entries of $x in mathbb{C}^n$ and design a recovery algorithm such that the output of the algorithm approximates $hat x$, the Discrete Fourier Transform (DFT) of $x$. The randomized case has been well-understood, while the main work in the deterministic case is that of Merhi et al.@ (J Fourier Anal Appl 2018), which obtains $O(k^2 log^{-1}k cdot log^{5.5}n)$ samples and a similar runtime with the $ell_2/ell_1$ guarantee. We focus on the stronger $ell_{infty}/ell_1$ guarantee and the closely related problem of incoherent matrices. We list our contributions as follows. 1. We find a deterministic collection of $O(k^2 log n)$ samples for the $ell_infty/ell_1$ recovery in time $O(nk log^2 n)$, and a deterministic collection of $O(k^2 log^2 n)$ samples for the $ell_infty/ell_1$ sparse recovery in time $O(k^2 log^3n)$. 2. We give new deterministic constructions of incoherent matrices that are row-sampled submatrices of the DFT matrix, via a derandomization of Bernsteins inequality and bounds on exponential sums considered in analytic number theory. Our first construction matches a previous randomized construction of Nelson, Nguyen and Woodruff (RANDOM12), where there was no constraint on the form of the incoherent matrix. Our algorithms are nearly sample-optimal, since a lower bound of $Omega(k^2 + k log n)$ is known, even for the case where the sensing matrix can be arbitrarily designed. A similar lower bound of $Omega(k^2 log n/ log k)$ is known for incoherent matrices.
In this paper, we theoretically propose a new hashing scheme to establish the sparse Fourier transform in high-dimensional space. The estimation of the algorithm complexity shows that this sparse Fourier transform can overcome the curse of dimensionality. To the best of our knowledge, this is the first polynomial-time algorithm to recover the high-dimensional continuous frequencies.
In this paper, we consider the extensively studied problem of computing a $k$-sparse approximation to the $d$-dimensional Fourier transform of a length $n$ signal. Our algorithm uses $O(k log k log n)$ samples, is dimension-free, operates for any universe size, and achieves the strongest $ell_infty/ell_2$ guarantee, while running in a time comparable to the Fast Fourier Transform. In contrast to previous algorithms which proceed either via the Restricted Isometry Property or via filter functions, our approach offers a fresh perspective to the sparse Fourier Transform problem.
In this paper a deterministic sparse Fourier transform algorithm is presented which breaks the quadratic-in-sparsity runtime bottleneck for a large class of periodic functions exhibiting structured frequency support. These functions include, e.g., the oft-considered set of block frequency sparse functions of the form $$f(x) = sum^{n}_{j=1} sum^{B-1}_{k=0} c_{omega_j + k} e^{i(omega_j + k)x},~~{ omega_1, dots, omega_n } subset left(-leftlceil frac{N}{2}rightrceil, leftlfloor frac{N}{2}rightrfloorright]capmathbb{Z}$$ as a simple subclass. Theoretical error bounds in combination with numerical experiments demonstrate that the newly proposed algorithms are both fast and robust to noise. In particular, they outperform standard sparse Fourier transforms in the rapid recovery of block frequency sparse functions of the type above.
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