Deterministic Sparse Sublinear FFT with Improved Numerical Stability

Iterative Sparse FFT for M‐sparse Vectors: Deterministic versus Random Sampling

“…Our method is based on a divide-and-conquer technique and requires to solve an equation system of Vandermonde type at each iteration step. To ensure numerical stability of the algorithm, the number of rows of the employed Vandermonde matrices at each step has been chosen adaptively in [3]. In this paper, we compare the results of [3] with a random sampling approach for determining the rows of the coefficient matrix.…”

“…Unfortunately, even an optimal choice of σ j does not always ensure a sufficiently small condition number of A (j+1) . Therefore, in [3] we have extended the algorithm by allowing rectangular Vandermonde matrices with M j ≥ M j rows. It had been observed already in [2] that prime numbers are good candidates for σ j .…”

“…is stable for larger τ but the unstructured coefficient matrices obtained by random sampling possess essentially smaller condition numbers. Worst condition for τmax = 1 (left), τmax = 2 (center) and τmax = 5 (right) after 20 tests for the algorithm in[3].…”

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In a recent paper [3] we have proposed a deterministic stable sparse FFT algorithm for M -sparse vectors of length N = 2 J with runtime O(M 2 log 2 N ) which generalizes the approach in [2]. Our method is based on a divide-and-conquer technique and requires to solve an equation system of Vandermonde type at each iteration step.

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“…The following theorem from [2] serves as the basis for our iterative algorithm in [3] and for our modification using random sampling that we consider here.…”