
FFT and orthogonal discrete transform on weight lattices of semisimple Lie groups
We give two algebrogeometric inspired approaches to fast algorithms for...
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Fast associated classical orthogonal polynomial transforms
We discuss a fast approximate solution to the associated classical – cla...
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The discrete cosine transform on triangles
The discrete cosine transform is a valuable tool in analysis of data on ...
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Sketching with Kerdock's crayons: Fast sparsifying transforms for arbitrary linear maps
Given an arbitrary matrix A∈ℝ^n× n, we consider the fundamental problem ...
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Sparse Fourier Transform by traversing CooleyTukey FFT computation graphs
Computing the dominant Fourier coefficients of a vector is a common task...
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Predicting sparse circle maps from their dynamics
The problem of identifying a dynamical system from its dynamics is of gr...
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Batch Sparse Recovery, or How to Leverage the Average Sparsity
We introduce a batch version of sparse recovery, where the goal is to re...
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Sparse Recovery for Orthogonal Polynomial Transforms
In this paper we consider the following sparse recovery problem. We have query access to a vector ∈^N such that = is ksparse (or nearly ksparse) for some orthogonal transform . The goal is to output an approximation (in an ℓ_2 sense) to in sublinear time. This problem has been wellstudied in the special case that 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 ·polylog N). However, for transforms other than the DFT (or closely related transforms like the Discrete Cosine Transform), the question is much less settled. In this paper we give sublineartime algorithmsrunning in time (k (N))for solving the sparse recovery problem for orthogonal transforms that arise from orthogonal polynomials. More precisely, our algorithm works for any 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 ksparse sparse recovery problem to the 1sparse sparse recovery problem that holds for any flat orthogonal polynomial transform; then we solve this onesparse recovery problem for transforms derived from Jacobi polynomials.
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