A robust multi-dimensional sparse Fourier transform in the continuous setting

05/13/2020
by   Yaonan Jin, et al.
0

Sparse Fourier transform (Sparse FT) is the problem of learning an unknown signal, whose frequency spectrum is dominated by a small amount of k individual frequencies, through fast algorithms that use as few samples as possible in the time domain. The last two decades have seen an extensive study on such problems, either in the one-/multi-dimensional discrete setting [Hassanieh, Indyk, Katabi, and Price STOC'12; Kapralov STOC'16] or in the one-dimensional continuous setting [Price and Song FOCS'15]. Despite this rich literature, the most general multi-dimensional continuous case remains mysterious. This paper initiates the study on the Sparse FT problem in the multi-dimensional continuous setting. Our main result is a randomized non-adaptive algorithm that uses sublinear samples and runs in sublinear time. In particular, the sample duration bound required by our algorithm gives a non-trivial improvement over [Price and Song FOCS'15], which studies the same problem in the one-dimensional continuous setting. The dimensionality in the continuous setting, different from both the discrete cases and the one-dimensional continuous case, turns out to incur many new challenges. To overcome these issues, we develop a number of new techniques for constructing the filter functions, designing the permutation-then-hashing schemes, sampling the Fourier measurements, and locating the frequencies. We believe these techniques can find their applications in the future studies on the Sparse FT problem.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
02/22/2021

A High-dimensional Sparse Fourier Transform in the Continuous Setting

In this paper, we theoretically propose a new hashing scheme to establis...
research
07/08/2019

Multiscale High-Dimensional Sparse Fourier Algorithms for Noisy Data

We develop an efficient and robust high-dimensional sparse Fourier algor...
research
12/21/2017

Multi-dimensional Graph Fourier Transform

Many signals on Cartesian product graphs appear in the real world, such ...
research
02/27/2019

Dimension-independent Sparse Fourier Transform

The Discrete Fourier Transform (DFT) is a fundamental computational prim...
research
07/09/2019

Reconstruction under outliers for Fourier-sparse functions

We consider the problem of learning an unknown f with a sparse Fourier s...
research
03/03/2010

Properties of the Discrete Pulse Transform for Multi-Dimensional Arrays

This report presents properties of the Discrete Pulse Transform on multi...
research
04/30/2019

Estimating the Frequency of a Clustered Signal

We consider the problem of locating a signal whose frequencies are "off ...

Please sign up or login with your details

Forgot password? Click here to reset