{"title":"An adaptive sublinear-time block sparse fourier transform","authors":"V. Cevher, M. Kapralov, J. Scarlett, A. Zandieh","doi":"10.1145/3055399.3055462","DOIUrl":null,"url":null,"abstract":"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(klognlog(n/k)) runtime [Hassanieh et al., STOC'12] and O(klogn) sample complexity [Indyk et al., FOCS'14]. This paper revisits the sparse FFT problem with the added twist that the sparse coefficients approximately obey a (k0,k1)-block sparse model. In this model, signal frequencies are clustered in k0 intervals with width k1 in Fourier space, and k= k0k1 is the total sparsity. Our main result is the first sparse FFT algorithm for (k0, k1)-block sparse signals with a sample complexity of O*(k0k1 + k0log(1+ k0)logn) at constant signal-to-noise ratios, and sublinear runtime. Our algorithm crucially uses adaptivity to achieve the improved sample complexity bound, and we provide a lower bound showing that this is essential in the Fourier setting: Any non-adaptive algorithm must use Ω(k0k1logn/k0k1) samples for the (k0,k1)-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.","PeriodicalId":20615,"journal":{"name":"Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing","volume":"22 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2017-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"15","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1145/3055399.3055462","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 15
Abstract
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(klognlog(n/k)) runtime [Hassanieh et al., STOC'12] and O(klogn) sample complexity [Indyk et al., FOCS'14]. This paper revisits the sparse FFT problem with the added twist that the sparse coefficients approximately obey a (k0,k1)-block sparse model. In this model, signal frequencies are clustered in k0 intervals with width k1 in Fourier space, and k= k0k1 is the total sparsity. Our main result is the first sparse FFT algorithm for (k0, k1)-block sparse signals with a sample complexity of O*(k0k1 + k0log(1+ k0)logn) at constant signal-to-noise ratios, and sublinear runtime. Our algorithm crucially uses adaptivity to achieve the improved sample complexity bound, and we provide a lower bound showing that this is essential in the Fourier setting: Any non-adaptive algorithm must use Ω(k0k1logn/k0k1) samples for the (k0,k1)-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.