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 k3 polylog(d) (or k5 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.

References

It is plausible that eq:main_q could even be answered affirmatively given the natural limit of $n \approx k$ observations. However, this would likely require fundamentally new techniques, which we leave as an exciting open question.

eq:main_q:

$\begin{gathered} Is there a polynomial-time spike-and-slab posterior sampler \\ that uses $n = o(d)measurementsandworksforarbitrarySNRs measurements and works for arbitrary SNRs ^{-1 > 0$?} \end{gathered} $

— Spike-and-Slab Posterior Sampling in High Dimensions  (2503.02798 - Kumar et al., 4 Mar 2025) in Section 1.2 (Our results)

This motivates the tantalizing open question of whether spike-and-slab posterior sampling is tractable at n = Ω(k\log d) measurements, which would close the gap between frequentist and Bayesian SLR.

— High-Magnetization Sampling at Low Temperatures: Ising Models and Bayesian Sparse Linear Regression  (2609.08873 - Kumar et al., 8 Sep 2026) in Section 1, subsection “Our results,” paragraph following Theorem 1.2

Removing this √k factor beyond the measurement complexity of (frequentist) sparse recovery is an exciting problem, that we leave open as a testbed of ``average-case'' trickle down theorems avoiding the union bound over worst-case dependencies suffered by our approach.

— High-Magnetization Sampling at Low Temperatures: Ising Models and Bayesian Sparse Linear Regression  (2609.08873 - Kumar et al., 8 Sep 2026) in Section 1, subsection “Our techniques,” paragraph “Spike-and-slab posterior sampling”