  In this paper, we consider the mixture of sparse linear regressions model.
Let ${\beta}^{(1)},\ldots,{\beta}^{(L)}\in\mathbb{C}^n$ be $ L $ unknown sparse
parameter vectors with a total of $ K $ non-zero coefficients. Noisy linear
measurements are obtained in the form $y_i={x}_i^H {\beta}^{(\ell_i)} + w_i$,
each of which is generated randomly from one of the sparse vectors with the
label $ \ell_i $ 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 $ \ell_i $ as well as the estimation problem
of recovering the sparse vectors $ {\beta}^{(\ell)} $.
  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}_i $
constructed based on ideas from sparse graph codes. Our novel code design
allows for both efficient demixing and parameter estimation. In the noiseless
setting, for a constant number of sparse parameter vectors, our algorithm
achieves the order-optimal sample and time complexities of $\Theta(K)$. In the
presence of Gaussian noise, for the problem with two parameter vectors (i.e.,
$L=2$), we show that the Robust Mixed-Coloring algorithm achieves near-optimal
$\Theta(K polylog(n))$ sample and time complexities. When $K=O(n^{\alpha})$ for
some constant $\alpha\in(0,1)$ (i.e., $K$ is sublinear in $n$), we can achieve
sample and time complexities both sublinear in the ambient dimension. 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 one third of its sample cost.
