Spike-and-slab posterior sampling with n ≈ k measurements

Determine whether there exists a polynomial-time algorithm that, for any signal-to-noise ratio σ^{-1} > 0, returns a high-accuracy sample from the spike-and-slab posterior π(θ | X, y) in the linear model y = Xθ* + w with w ∼ N(0, σ^2 I_n), using only n = Θ(k) linear measurements, where k is the expected sparsity under the spike-and-slab prior.

Background

The paper resolves a long-standing challenge by providing the first provable algorithms for sampling from the spike-and-slab posterior in high dimensions that work for arbitrary signal-to-noise ratios and use a sublinear number of measurements n in the ambient dimension d. Their algorithms achieve correctness when n scales as at least k^3 polylog(d) (or k^5 polylog(d) for a faster near-linear-time variant), where k is the expected sparsity.

The authors note that, analogous to sparse recovery theory, a natural limit is to work with n on the order of the sparsity level k. Achieving sampling with n ≈ k would significantly strengthen their results, but they emphasize that new ideas seem necessary beyond those developed in the paper.

This question is posed as a refinement of their motivating inquiry about whether polynomial-time spike-and-slab posterior sampling is possible with n = o(d) measurements for arbitrary signal-to-noise ratios. The open problem asks for the same guarantee at the natural sparsity-driven limit n ≈ k.