Nearly Optimal Deterministic Algorithm for Sparse Walsh-Hadamard Transform

Abstract

For every fixed constant α > 0, we design an algorithm for computing the k -sparse Walsh-Hadamard transform (i.e., Discrete Fourier Transform over the Boolean cube) of an N -dimensional vector x ∈ R N in time k 1 + α (log N ) O (1) . Specifically, the algorithm is given query access to x and computes a k -sparse x˜ ∈ R N satisfying ‖ x˜ − xˆ ‖ 1 ≤ c ‖ xˆ − H k ( xˆ )‖‖‖‖‖‖‖‖ 1 for an absolute constant c > 0, where xˆ is the transform of x and H k ( xˆ ) is its best k -sparse approximation. Our algorithm is fully deterministic and only uses nonadaptive queries to x (i.e., all queries are determined and performed in parallel when the algorithm starts). An important technical tool that we use is a construction of nearly optimal and linear lossless condensers, which is a careful instantiation of the GUV condenser (Guruswami et al. [2009]). Moreover, we design a deterministic and nonadaptive ℓ 1 /ℓ 1 compressed sensing scheme based on general lossless condensers that is equipped with a fast reconstruction algorithm running in time k 1 + α (log N ) O (1) (for the GUV-based condenser) and is of independent interest. Our scheme significantly simplifies and improves an earlier expander-based construction due to Berinde, Gilbert, Indyk, Karloff, and Strauss [Berinde et al. 2008]. Our methods use linear lossless condensers in a black box fashion; therefore, any future improvement on explicit constructions of such condensers would immediately translate to improved parameters in our framework (potentially leading to k (log N ) O (1) reconstruction time with a reduced exponent in the poly-logarithmic factor, and eliminating the extra parameter α). By allowing the algorithm to use randomness while still using nonadaptive queries, the runtime of the algorithm can be improved to õ ( k log 3 N ).