In this paper we consider the following sparse recovery problem. We have query access to a vector @math such that @math is @math -sparse (or nearly @math -sparse) for some orthogonal transform @math . The goal is to output an approximation (in an @math sense) to @math in sublinear time. This problem has been well-studied in the special case that @math is the Discrete Fourier Transform (DFT), and a long line of work has resulted in sparse Fast Fourier Transforms that run in time @math . However, for transforms @math 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 @math ---for solving the sparse recovery problem for orthogonal transforms @math that arise from orthogonal polynomials. More precisely, our algorithm works for any @math 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 @math -sparse sparse recovery problem to the @math -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.
In this paper, we propose a general strategy for rapidly computing sparse Legendre expansions. The resulting methods yield a new class of fast algorithms capable of approximating a given function f : [ź1, 1] ź ź with a near-optimal linear combination of s Legendre polynomials of degree ≤ N in just (slogN)O(1) @math -time. When s ź N, these algorithms exhibit sublinear runtime complexities in N, as opposed to traditional Ω(NlogN)-time methods for computing all of the first N Legendre coefficients of f. Theoretical as well as numerical results demonstrate the effectiveness of the proposed methods. We consider the problem of recovering polynomials that are sparse with respect to the basis of Legendre polynomials from a small number of random samples. In particular, we show that a Legendre s-sparse polynomial of maximal degree N can be recovered from [email protected]?slog^4(N) random samples that are chosen independently according to the Chebyshev probability measure [email protected](x)[email protected]^-^1(1-x^2)^-^1^ ^2dx. As an efficient recovery method, @?"1-minimization can be used. We establish these results by verifying the restricted isometry property of a preconditioned random Legendre matrix. We then extend these results to a large class of orthogonal polynomial systems, including the Jacobi polynomials, of which the Legendre polynomials are a special case. Finally, we transpose these results into the setting of approximate recovery for functions in certain infinite-dimensional function spaces.