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.
Let @math denote a set of polynomials with complex coefficients. Let @math denote any set of sample points . For any @math , the discrete polynomial transform of @math (with respect to @math and @math ) is defined as the collection of sums, @math , where @math for some associated weight function @math . These sorts of transforms find important applications in areas such as medical imaging and signal processing. In this paper, we present fast algorithms for computing discrete orthogonal polynomial transforms. For a system of @math orthogonal polynomials of degree at most @math , we give an @math algorithm for computing a discrete polynomial transform at an arbitrary set of points instead of the @math operations required by direct evaluation. Our algorithm depends only on the fact that orthogonal polynomial sets satisfy a three-term recurrence and thus it may be applied to any such set of discretely sampled functions. In particular, sampled orthogonal polynomials generate the vector space of functions on a distance transitive graph. As a direct application of our work, we are able to give a fast algorithm for computing subspace decompositions of this vector space which respect the action of the symmetry group of such a graph. This has direct applications to treating computational bottlenecks in the spectral analysis of data on distance transitive graphs, and we discuss this in some detail.