Learning mixtures of sparse linear regressions using sparse graph codes
In this paper, we consider the mixture of sparse linear regressions model. Let β<sup>(1)</sup>,…, β<sup>(L)</sup> ∊ C<sup>n</sup> be L unknown sparse parameter vectors with a total of K non-zero coefficients. Noisy linear measurements are obtained in the form y<inf>i</inf> = x<inf>i</inf><sup>H</sup>β<sup>(ℓi)</sup> + w<inf>i</inf>, each of which is generated randomly from one of the sparse vectors with the label ℓ<inf>i</inf> unknown. The goal is to estimate the parameter vectors efficiently with low sample and computational costs. This problem presents significant challenges as one needs to simultaneously solve the demixing problem of recovering the labels ℓ<inf>1</inf> as well as the estimation problem of recovering the sparse vectors β <sup>(ℓ)</sup>. Our solution to the problem leverages the connection between modern coding theory and statistical inference. We introduce a new algorithm, Mixed-Coloring, which samples the mixture strategically using query vectors x<inf>i</inf> constructed based on ideas from sparse graph codes. Our novel code design allows for both efficient demixing and parameter estimation. The algorithm achieves the order-optimal sample and time complexities of Θ (K) in the noiseless setting, and near-optimal Θ (K polylog(n)) complexities in the noisy setting. In one of our experiments, to recover a mixture of two regressions with dimension n = 500 and sparsity K = 50, our algorithm is more than 300 times faster than EM algorithm, with about 1/3 of its sample cost.