低温下的高磁化采样：伊辛模型与贝叶斯稀疏线性回归
High-Magnetization Sampling at Low Temperatures: Ising Models and Bayesian Sparse Linear Regression
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中文总结 AI 辅助
本文提出利用稀疏性的采样框架，用于高磁化汉明切片上的伊辛模型和贝叶斯稀疏线性回归，分别改进了低温SK模型采样和测量复杂度。
中文摘要 AI 辅助
稀疏性是优化和统计学中一种强大的结构资源。我们开发了在汉明切片 $\mathcal{X}_k^d:=\{\mathbf{x}\in\{\pm 1\}^d:|\{i:\mathbf{x}_i=1\}|=k\}$ 上利用稀疏性的采样框架，适用于 $k\ll d$（即 $\mathcal{X}_k^d$ 为“高磁化”状态）的高维场景。我们利用这些框架为伊辛模型和贝叶斯稀疏线性回归研究中的经典问题设计了改进的采样器。我们的第一个主要结果考虑了限制在固定磁化切片 $\mathcal{X}_k^d$ 上的 Sherrington--Kirkpatrick (SK) 模型。我们给出了一个多项式时间采样器，用于任意逆温度 $\beta>0$、任意外场下、满足 $k\le c_\beta d$（$c_\beta$ 为适当常数）的固定磁化 SK 模型。通过将该结果与用于估计归一化常数的退火策略相结合，我们在足够强的外场强度 $h$ 下，获得了任意低温下 SK 模型的多项式时间采样器。在大 $\beta$ 极限下，我们的框架允许在接近 Almeida--Thouless 线的场强（该线划分了复制对称和复制对称破缺区域，见 [dAT78]）的常数因子范围内进行采样，相对于最近工作 [BAR26] 所需的场强 $h(\beta)$，实现了多项式改进。我们的第二个主要结果涉及多项式时间贝叶斯稀疏线性回归的测量复杂度。最近的工作 [KSTZ25] 展示了如何在任意信噪比下，从期望稀疏度为 $k$ 的典型高斯 spike-and-slab 后验分布中采样，给定 $n\gtrsim k^3\log^3 d$ 个高斯测量。我们将这一要求改进为 $n\gtrsim k^{3/2}\log^2 d+k\log^3 d$，这得益于支撑我们两个结果的通用稀疏感知框架。
英文摘要
Sparsity is a powerful structural resource in optimization and statistics. We develop frameworks for leveraging sparsity in sampling problems over the Hamming slice $\mathcal{X}_k^d:=\{\mathbf{x}\in\{\pm 1\}^d:|\{i:\mathbf{x}_i=1\}|=k\}$, in high-dimensional regimes where $k\ll d$ (i.e., where $\mathcal{X}_k^d$ 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 $\mathcal{X}_k^d$. We give a polynomial-time sampler for fixed-magnetization SK models at any inverse temperature $β>0$, under arbitrary external fields, provided that $k\le c_βd$ 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 $n\gtrsim k^3\log^3 d$ Gaussian measurements. We improve this requirement to $n\gtrsim k^{3/2}\log^2 d+k\log^3 d$, using a common sparsity-aware framework underlying both our results.
发表机构
- University of Texas at Austin（德克萨斯大学奥斯汀分校）
机构由 AI 辅助整理，请以论文原文为准。
