High-Magnetization Sampling at Low Temperatures: Ising Models and Bayesian Sparse Linear Regression

Abstract: Sparsity is a powerful structural resource in optimization and statistics. We develop frameworks for leveraging sparsity in sampling problems over the Hamming slice $$\mathcal{X}<em>k^{d:={\mathbf{x}\in{\pm} 1}<sup>d:|{i:\mathbf{x}_i=1}|=k}$$, in high-dimensional regimes where $k\ll d$ (i.e., where $\mathcal{X}_k<sup>d$ is \emph{highly magnetized}). We use our frameworks to design improved samplers for canonical problems in the study of \emph{Ising models} and \emph{Bayesian sparse linear regression}. Our first main result considers the \emph{Sherrington--Kirkpatrick} (SK) model restricted to fixed-magnetization slices $\mathcal{X}_k<sup>d$. We give a polynomial-time sampler for fixed-magnetization SK models at any inverse temperature $β&gt;0$, under arbitrary external fields, provided that $k\le c</em>βd$ for an appropriate constant $c_β$. By combining this result with an annealing strategy for estimating normalizing constants, we obtain polynomial-time samplers for the SK model at arbitrarily low temperatures under a sufficiently strong external field of strength $h$. In the large-$β$ limit, our framework permits sampling at field strengths within constant factors of the \emph{Almeida--Thouless line} delineating the replica-symmetric and replica-symmetry-breaking regions ([dAT78]), improving polynomially over the field strength $h(β)$ required by the recent work of [BAR26]. Our second main result concerns the measurement complexity of polynomial-time Bayesian sparse linear regression. Recent work by [KSTZ25] shows how to sample from the canonical \emph{Gaussian spike-and-slab posterior} with expected sparsity $k$, at any signal-to-noise ratio, given $n\gtrsim k^3\log³ d$ Gaussian measurements. We improve this requirement to $$n\gtrsim k^{3/2}\log² d+k\log³ d$$, using a common sparsity-aware framework underlying both our results.