317 research outputs found
Nearly Optimal Sparse Fourier Transform
We consider the problem of computing the k-sparse approximation to the
discrete Fourier transform of an n-dimensional signal. We show:
* An O(k log n)-time randomized algorithm for the case where the input signal
has at most k non-zero Fourier coefficients, and
* An O(k log n log(n/k))-time randomized 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). 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 k-sparse case is optimal for any k =
n^{\Omega(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
\Omega(k log(n/k)/ log log n) signal samples, even if it is allowed to perform
adaptive sampling.Comment: 28 pages, appearing at STOC 201
Deterministic Sparse Fourier Transform with an ?_{?} Guarantee
In this paper we revisit the deterministic version of the Sparse Fourier
Transform problem, which asks to read only a few entries of and design a recovery algorithm such that the output of the
algorithm approximates , the Discrete Fourier Transform (DFT) of .
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 samples and a similar runtime
with the guarantee. We focus on the stronger
guarantee and the closely related problem of incoherent
matrices. We list our contributions as follows.
1. We find a deterministic collection of samples for the
recovery in time , and a deterministic
collection of samples for the sparse
recovery in time .
2. We give new deterministic constructions of incoherent matrices that are
row-sampled submatrices of the DFT matrix, via a derandomization of Bernstein's
inequality and bounds on exponential sums considered in analytic number theory.
Our first construction matches a previous randomized construction of Nelson,
Nguyen and Woodruff (RANDOM'12), where there was no constraint on the form of
the incoherent matrix.
Our algorithms are nearly sample-optimal, since a lower bound of is known, even for the case where the sensing matrix can be
arbitrarily designed. A similar lower bound of is
known for incoherent matrices.Comment: ICALP 2020--presentation improved according to reviewers' comment
Theoretical and Experimental Analysis of a Randomized Algorithm for Sparse Fourier Transform Analysis
We analyze a sublinear RAlSFA (Randomized Algorithm for Sparse Fourier
Analysis) that finds a near-optimal B-term Sparse Representation R for a given
discrete signal S of length N, in time and space poly(B,log(N)), following the
approach given in \cite{GGIMS}. Its time cost poly(log(N)) should be compared
with the superlinear O(N log N) time requirement of the Fast Fourier Transform
(FFT). A straightforward implementation of the RAlSFA, as presented in the
theoretical paper \cite{GGIMS}, turns out to be very slow in practice. Our main
result is a greatly improved and practical RAlSFA. We introduce several new
ideas and techniques that speed up the algorithm. Both rigorous and heuristic
arguments for parameter choices are presented. Our RAlSFA constructs, with
probability at least 1-delta, a near-optimal B-term representation R in time
poly(B)log(N)log(1/delta)/ epsilon^{2} log(M) such that
||S-R||^{2}<=(1+epsilon)||S-R_{opt}||^{2}. Furthermore, this RAlSFA
implementation already beats the FFTW for not unreasonably large N. We extend
the algorithm to higher dimensional cases both theoretically and numerically.
The crossover point lies at N=70000 in one dimension, and at N=900 for data on
a N*N grid in two dimensions for small B signals where there is noise.Comment: 21 pages, 8 figures, submitted to Journal of Computational Physic
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