SPRIGHT: A Fast and Robust Framework for Sparse Walsh-Hadamard Transform

We consider the problem of stably computing the Walsh-Hadamard Transform (WHT) of some N -length input vector in the presence of noise, where the N -point Walsh spectrum is K-sparse with K = O(N) scaling sub-linearly in the input dimension N for some 0 < δ < 1. Note that K is linear in N (i.e. δ = 1), then similar to the standard Fast Fourier Transform (FFT) algorithm, the classic Fast WHT (FWHT) algorithm offers an O(N) sample cost and O(N logN) computational cost, which are order optimal. Over the past decade, there has been a resurgence in research related to the computation of Discrete Fourier Transform (DFT) for some length-N input signal that has a K-sparse N -point Fourier spectrum. In particular, through a sparse-graph code design, our earlier work on the Fast Fourier Aliasing-based Sparse Transform (FFAST) algorithm [1] computes the K-sparse DFT in time O(K logK) by taking O(K) noiseless samples. Inspired by the coding-theoretic design framework in [1], Scheibler et al. in [2] proposed the Sparse Fast Hadamard Transform (SparseFHT) algorithm that elegantly computes the K-sparse WHT in the absence of noise using O(K logN) samples in time O(K logN). However, the SparseFHT algorithm explicitly exploits the noiseless nature of the problem, and is not equipped to deal with scenarios where the observations are corrupted by noise, as is true in general. Therefore, a question of critical interest is whether this coding-theoretic framework can be made robust to noise. Further, if the answer is yes, what is the extra price that needs to be paid for being robust to noise? In this paper, we show, quite interestingly, that there is no extra price that needs to be paid for being robust to noise other than a constant factor. In other words, we can maintain the same scaling for the sample complexity O(K logN) and the computational complexity O(K logN) as those of the noiseless case, using our proposed SParse Robust Iterative Graph-based Hadamard Transform (SPRIGHT) algorithm. Similar to the FFAST algorithm [1] and the SparseFHT algorithm [2], the proposed SPRIGHT framework succeeds with high probability with respect to a random ensemble of signals with sparse Walsh spectra, where the support of the non-zero WHT coefficients is uniformly random. Experiments further corroborate the robustness of the SPRIGHT framework as well as its scaling performance.