We study the problem of estimating the best B term Fourier representation for a given frequency-sparse signal (i.e., vector) $\textbf{A}$ of length $N \gg B$. More explicitly, we investigate how to deterministically identify B of the largest magnitude frequencies of $\hat{\textbf{A}}$, and estimate their coefficients, in polynomial$(B,\log N)$ time. Randomized sub-linear time algorithms which have a small (controllable) probability of failure for each processed signal exist for solving this problem. However, for failure intolerant applications such as those involving mission-critical hardware designed to process many signals over a long lifetime, deterministic algorithms with no probability of failure are highly desirable. In this paper we build on the deterministic Compressed Sensing results of Cormode and Muthukrishnan (CM) \cite{CMDetCS3,CMDetCS1,CMDetCS2} in order to develop the first known deterministic sub-linear time sparse Fourier Transform algorithm suitable for failure intolerant applications. Furthermore, in the process of developing our new Fourier algorithm, we present a simplified deterministic Compressed Sensing algorithm which improves on CM's algebraic compressibility results while simultaneously maintaining their results concerning exponential decay.
Deep Dive into A Deterministic Sub-linear Time Sparse Fourier Algorithm via Non-adaptive Compressed Sensing Methods.
We study the problem of estimating the best B term Fourier representation for a given frequency-sparse signal (i.e., vector) $\textbf{A}$ of length $N \gg B$. More explicitly, we investigate how to deterministically identify B of the largest magnitude frequencies of $\hat{\textbf{A}}$, and estimate their coefficients, in polynomial$(B,\log N)$ time. Randomized sub-linear time algorithms which have a small (controllable) probability of failure for each processed signal exist for solving this problem. However, for failure intolerant applications such as those involving mission-critical hardware designed to process many signals over a long lifetime, deterministic algorithms with no probability of failure are highly desirable. In this paper we build on the deterministic Compressed Sensing results of Cormode and Muthukrishnan (CM) \cite{CMDetCS3,CMDetCS1,CMDetCS2} in order to develop the first known deterministic sub-linear time sparse Fourier Transform algorithm suitable for failure intole
In many applications only the top few most energetic terms of a signal's Fourier Transform (FT) are of interest. In such applications the Fast Fourier Transform (FFT), which computes all FT terms, is computationally wasteful. To make our point, we next consider a simple application-based example in which the FFT can be replaced by faster approximate Fourier methods:
Imagine a signal/function f : [0, 2π] → C of the form
consisting of a single unknown frequency ω ∈ (-N, N] (e.g., consider a windowed sinusoidal portion of a wideband frequency-hopping signal [21]). Sampling at the Nyquist-rate would dictate the need for at least 2N equally spaced samples from f in order to discover ω via the FFT without aliasing [3]. Thus, we would have to compute the FFT of the 2N-length vector
However, if we use aliasing to our advantage we can correctly determine ω with significantly fewer f -samples taken in parallel.
Compressed Sensing (CS) methods [4,28,26,6,7] provide a robust framework for reducing the number of measurements required to estimate a sparse signal. For this reason CS methods are useful in areas such as MR imaging [23,24] and analog-to-digital conversion [21,20] where measurement costs are high. The general CS setup is as follows: Let A be an N-length signal/vector with complex valued entries and Ψ be a full rank N × N change of basis matrix. Furthermore, suppose that Ψ • A is sparse (i.e., only k ≪ N entries of Ψ • A are significant/large in magnitude). CS methods deal with generating a K × N measurement matrix, M, with the smallest number of rows possible (i.e., K minimized) so that the k significant entries of Ψ • A can be well approximated by the K-element vector result of
Note that CS is inherently algorithmic since a procedure for recovering Ψ • A’s largest k-entries from the result of Equation 1 must be specified.
For the remainder of this paper we will consider the special CS case where Ψ is the N × N Discrete Fourier Transform matrix. Hence, we have
Our problem of interest is to find, and estimate the coefficients of, the k significant entries (i.e., frequencies) of  given a frequency-sparse (i.e., smooth) signal A. In this case the deterministic Fourier CS measurement matrixes, M • Ψ, produced by [28,26,6,7] require super-linear O(KN)-time to multiply by A in Equation 1. Similarly, the energetic frequency recovery procedure of [4,9] requires super-linear time in N. Hence, none of [4,28,9,26,6,7] have both sub-linear measurement and reconstruction time.
Existing randomized sub-linear time Fourier algorithms [15,19,16] not only show great promise for decreasing measurement costs, but also for speeding up the numerical solution of computationally challenging multi-scale problems [8,18]. However, these algorithms are not deterministic and so can produce incorrect results with some small probability on each input signal. Thus, they aren’t appropriate for long-lived failure intolerant applications.
In this paper we build on the deterministic Compressed Sensing methods of Cormode and Muthukrishnan (CM) [26,6,7] in order to construct the first known deterministic sub-linear time sparse Fourier algorithm. In order to produce our new Fourier algorithm we must modify CM’s work in two ways. First, we alter CM’s measurement construction in order to allow sub-linear time computation of Fourier measurements via aliasing. Thus, our algorithm can deterministically approximate the result of Equation 1 in time K 2 •polylog(N). Second, CM use a k-strongly selective collection of sets [17] to construct their measurements for algebraically compressible signals. We introduce the generalized notion of a K-majority k-strongly selective collection of sets which leads us to a new reconstruction algorithm with better algebraic compressibility results than CM’s algorithm. As a result, our deterministic sub-linear time Fourier algorithm has better then previously possible algebraic compressibility behavior.
The main contributions of this paper are:
We present a new deterministic compressed sensing algorithm that both (i) improves on CM’s algebraically compressible signal results, and (ii) has comparable measurement/run time requirements to CM’s algorithm for exponentially decaying signals.
We present the first known deterministic sub-linear time sparse DFT. In the process, we explicitly demonstrate the connection between compressed sensing and sub-linear time Fourier transform methods.
We introduce K-majority k-strongly selective collections of sets which have potential applications to streaming algorithms along the lines of [25,13].
The remainder of this paper is organized as follows: In section 2 we introduce relevant definitions and terminology. Then, in section 3 we define K-majority k-strongly selective collections of sets and use them to construct our compressed sensing measurements. Section 4 contains our new deterministic compressed sensing algorithm along with analysis of it’s accuracy and run
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