The SPRIGHT Algorithm for Robust Sparse Hadamard Transforms

In this paper, we consider the problem of computing a K-sparse N-point Hadamard Transforms (HT) from noisy time domain samples, where K = O(Nα) scales sub-linearly in N for some α ∈ (0; 1). The SParse Robust Iterative Graph-based Hadamard Transform (SPRIGHT) algorithm is proposed to recover the sparse HT coefficients in a stable manner that is robust to additive Gaussian noise. In particular, it is shown that the K-sparse HT of the signal can be reconstructed from noisy time domain samples with a vanishing error probability using the same sample complexity O(K logN) as in the noiseless case of [1] and computational complexity1 O(N logN). Last but not least, given the complexity orders of the SPRIGHT algorithm, our numerical experiments further validate that the big-Oh constants in the complexity are small.