Sampling from Constrained Gibbs Measures: with Applications to High-Dimensional Bayesian Inference
受约束 Gibbs 测度采样及其高维贝叶斯推断应用AI 解读 ↓
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This paper considers a non-standard problem of generating samples from a low-temperature Gibbs distribution with \emph{constrained} support, when some of the coordinates of the mode lie on the boundary. These coordinates are referred to as the non-regular part of the model. We show that in a ``pre-asymptotic'' regime in which the limiting Laplace approximation is not yet valid, the low-temperature Gibbs distribution concentrates on a neighborhood of its mode. Within this region, the distribution is a bounded perturbation of a product measure: a strongly log-concave distribution in the regular part and a one-dimensional exponential-type distribution in each coordinate of the non-regular part. Leveraging this structure, we provide a non-asymptotic sampling guarantee by analyzing the spectral gap of Langevin dynamics. Key examples of low-temperature Gibbs distributions include Bayesian posteriors, and we demonstrate our results on three canonical examples: a high-dimensional logistic regression model, a Poisson linear model, and a Gaussian mixture model.
RBF 核高斯过程最大似然估计的精确渐近理论
Gaussian processes (GPs) are widely used across machine learning, spatial statistics, time-series analysis, optimization, Bayesian statistics, and scientific applications. A central component of a GP model is its kernel, which is typically specified through a parametric family. Among the most widely used choices is the radial basis function (RBF), also known as the squared exponential or Gaussian kernel, owing to its simple form, smoothness, and flexibility. In practice, the…
凸正则化的谱依赖:右旋转不变设计下的基本极限
We study the fundamental limits of convex-regularized estimation in high-dimensional linear regression with right-rotationally invariant design matrices. We show that the asymptotic risks of convex-penalized least squares estimators are lower bounded by the risk of an approximate message passing algorithm known as Bayes VAMP, and we further characterize when the lower bound is attainable. As a technical ingredient in the proof of our lower bound theorem, we characterize the…
基于相容性的贝叶斯推断:渐近极限、变分原理与 Wasserstein 几何
Approximate Bayesian Computation (ABC) replaces intractable likelihood evaluations with comparisons between observed and simulated data. We develop a measure-theoretic formulation of discrepancy-based Bayesian inference in which compatibility is encoded through nonnegative weights. For vanishing thresholds, we establish convergence of the corresponding posterior distributions in total variation. In the large-sample regime, we identify conditions under which the posterior…
大规模需求估计的贝叶斯机器学习方法
This work studies how Bayesian machine learning methods can be used for large-scale demand estimation with many product categories. I compare two model classes, a latent factorization model and a mixed logit model and two Bayesian estimation approaches, Markov Chain Monte Carlo (MCMC) and Variational Inference (VI). The analysis combines a simulation study with an application to supermarket scanner data. The results show that the latent factorization model benefits from…
逐元素独立同分布重尾数据下经验风险最小化的渐近分析
Many real-world datasets exhibit unusually large values far more frequently than predicted by Gaussian models. Heavy-tailed distributions capture this behavior, yet evaluating learning performance under them remains challenging because rare, large feature entries retain non-vanishing effects even in high dimensions. Even in the canonical setting of empirical risk minimization for linear regression with entry-wise i.i.d. symmetric $α$-stable data, a precise asymptotic…
分数岭回归
Regularization methods such as the Lasso combine shrinkage and variable selection, but strong penalization drives their solutions toward the null model. We introduce Fractional Ridge (Fridge) regression, a penalized regression method that instead shrinks toward models of a prespecified target size. Its penalty vanishes whenever the number of nonzero coefficients does not exceed that target. This behavior connects continuous shrinkage with best subset selection. We develop an…