Fast Algorithms for the Multi-dimensional Jacobi Polynomial Transform

Bremer, James; Pang, Qiyuan; Yang, Haizhao

doi:10.48550/arxiv.1901.07275

Cited by 1 publication

(2 citation statements)

References 31 publications

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“…By suitably scaling and translating X, we can ensure that the orthogonality is on [−1, 1]. 6 In particular deg(p i ) = i and for any i, j ≥ 0,…”

Section: Orthogonal Polynomialsmentioning

confidence: 99%

“…We note that in this work we consider a setting slightly different than this example, where D = [−1, 1] rather than S 1 .3 We note that even though the work of[11] has in some sense solved the problem of computing any OP transform in nearlinear time, many practical issues still remain to be resolved and the problem of computing OP transforms in near-linear time has seen a lot of research activity recently. We just mention two recent works[6,7] that present near-linear time algorithms for the Jacobi polynomial transforms (and indeed their notion of uniform Jacobi transform corresponds exactly to the Jacobi polynomial transform that we study in this paper). However, these algorithms inherently seem to require at least linear-time and it is not clear how to convert them into sub-linear algorithms, which is the focus of our work.…”

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confidence: 94%

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Sparse Recovery for Orthogonal Polynomial Transforms

Gilbert,

Gu,

et al. 2019

Preprint

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In this paper we consider the following sparse recovery problem. We have query access to a vector x ∈ R N such that x = Fx is k-sparse (or nearly k-sparse) for some orthogonal transform F. The goal is to output an approximation (in an ℓ2 sense) to x in sublinear time. This problem has been well-studied in the special case that F is the Discrete Fourier Transform (DFT), and a long line of work has resulted in sparse Fast Fourier Transforms that run in time O(k • polylogN ). However, for transforms F other than the DFT (or closely related transforms like the Discrete Cosine Transform), the question is much less settled.In this paper we give sublinear-time algorithms-running in time poly(k log(N ))-for solving the sparse recovery problem for orthogonal transforms F that arise from orthogonal polynomials. More precisely, our algorithm works for any F that is an orthogonal polynomial transform derived from Jacobi polynomials. The Jacobi polynomials are a large class of classical orthogonal polynomials (and include Chebyshev and Legendre polynomials as special cases), and show up extensively in applications like numerical analysis and signal processing. One caveat of our work is that we require an assumption on the sparsity structure of the sparse vector, although we note that vectors with random support have this property with high probability.Our approach is to give a very general reduction from the k-sparse sparse recovery problem to the 1-sparse sparse recovery problem that holds for any flat orthogonal polynomial transform; then we solve this one-sparse recovery problem for transforms derived from Jacobi polynomials. Frequently, sparse FFT algorithms are described as implementing such a reduction; however, the technical details of such works are quite specific to the Fourier transform and moreover the actual implementations of these algorithms do not use the 1-sparse algorithm as a black box. In this work we give a reduction that works for a broad class of orthogonal polynomial families, and which uses any 1-sparse recovery algorithm as a black box.1 To be more precise the radial component of a Zernike polynomial is a Gegenbauer and hence, a Jacobi polynomial.

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“…By suitably scaling and translating X, we can ensure that the orthogonality is on [−1, 1]. 6 In particular deg(p i ) = i and for any i, j ≥ 0,…”

Section: Orthogonal Polynomialsmentioning

confidence: 99%

mentioning

confidence: 94%