论文检索

基于最优传输的贝叶斯后验采样生成模型AI 解读

Abstract We investigate the problem of sampling from posterior distributions with intractable normalizing constants in Bayesian inference. Building on transport map-based generative models for posterior sampling, we propose an optimal transport (OT)-constrained transport map class that learns a deterministic map from a reference distribution to the target posterior through constrained optimization. The proposed class exploits structural properties of OT maps and allows…

通过变分近似估计有限混合模型的成分数AI 解读

This work introduces a new method for selecting the number of components in finite mixture models (FMMs) using variational Bayes, inspired by the large-sample properties of the Evidence Lower Bound (ELBO) derived from mean-field (MF) variational approximation. Specifically, we establish matching upper and lower bounds for the ELBO without assuming conjugate priors, suggesting the consistency of model selection for FMMs based on maximizing the ELBO. As a by-product, we show…

再生核希尔伯特空间中非参数随机逼近的可扩展推断AI 解读

Stochastic approximation (SA) is a powerful and scalable computational method for iteratively estimating the solution of optimization problems in the presence of randomness, particularly well suited for large-scale and streaming data settings. In this work we propose a theoretical framework for stochastic approximation (SA) applied to nonparametric least squares in reproducing kernel Hilbert spaces (RKHS), enabling online statistical inference in nonparametric regression…

利用生成式神经网络进行双重稳健条件独立检验AI 解读

Abstract This article addresses the problem of testing the conditional independence of two generic random vectors X and Y given a third random vector Z, which plays an important role in statistical and machine learning applications. We propose a new non-parametric testing procedure that avoids explicitly estimating any conditional distributions but instead requires sampling from the two marginal conditional distributions of X given Z and Y given Z. We further propose using a…

坐标上升变分推断的收敛性AI 解读

As a computational alternative to Markov chain Monte Carlo approaches, variational inference (VI) is becoming more and more popular for approximating intractable posterior distributions in large-scale Bayesian models due to its comparable efficacy and superior efficiency. Several recent works provide theoretical justifications of VI by proving its statistical optimality for parameter estimation under various settings; meanwhile, formal analysis on the algorithmic convergence…

通过平均场变分近似进行贝叶斯模型选择AI 解读

Abstract This article considers Bayesian model selection via mean-field (MF) variational approximation. Towards this goal, we study the non-asymptotic properties of MF inference that allows latent variables and model misspecification. Concretely, we show a Bernstein–von Mises (BvM) theorem for the variational distribution from MF under possible model misspecification, which implies the distributional convergence of MF variational approximation to a normal distribution…

主题模型学习：可识别性与有限样本分析AI 解读

Topic models provide a useful text-mining tool for learning, extracting, and discovering latent structures in large text corpora. Although a plethora of methods have been proposed for topic modeling, lacking in the literature is a formal theoretical investigation of the statistical identifiability and accuracy of latent topic estimation. In this article, we propose a maximum likelihood estimator (MLE) of latent topics based on a specific integrated likelihood that is…

对抗损失下未知子流形上的分布估计极小极大速率AI 解读

Statistical inference from high-dimensional data with low-dimensional structures has recently attracted a lot of attention. In machine learning, deep generative modelling approaches implicitly estimate distributions of complex objects by creating new samples from the underlying distribution, and have achieved great success in generating synthetic realistic-looking images and texts. A key step in these approaches is the extraction of latent features or representations…

通过隐式正则化进行高维线性回归AI 解读

Summary Many statistical estimators for high-dimensional linear regression are $M$-estimators, formed through minimizing a data-dependent square loss function plus a regularizer. This work considers a new class of estimators implicitly defined through a discretized gradient dynamic system under overparameterization. We show that, under suitable restricted isometry conditions, overparameterization leads to implicit regularization: if we directly apply gradient descent to the…

主成分分析的分布式估计：扩展特征子空间分析AI 解读

The growing size of modern datasets brings many challenges to the existing statistical estimation approaches, which calls for new distributed methodologies. This article studies distributed estimation for a fundamental statistical machine learning problem, principal component analysis (PCA). Despite the massive literature on top eigenvector estimation, much less is presented for the top-L-dim (L > 1) eigenspace estimation, especially in a distributed manner. We propose a…

基于指数倾斜经验似然的风险最小化贝叶斯推断AI 解读

Abstract The celebrated Bernstein von-Mises theorem ensures credible regions from a Bayesian posterior to be well-calibrated when the model is correctly-specified, in the frequentist sense that their coverage probabilities tend to the nominal values as data accrue. However, this conventional Bayesian framework is known to lack robustness when the model is misspecified or partly specified, for example, in quantile regression, risk minimization based supervised/unsupervised…

具有统计保证的 α 变分推断AI 解读

We provide statistical guarantees for a family of variational approximations to Bayesian posterior distributions, called $\alpha $-VB, which has close connections with variational approximations of tempered posteriors in the literature. The standard variational approximation is a special case of $\alpha $-VB with $\alpha =1$. When $\alpha \in (0,1]$, a novel class of variational inequalities are developed for linking the Bayes risk under the variational approximation to the…

通信高效的分布式统计推断AI 解读

We present a communication-efficient surrogate likelihood (CSL) framework for solving distributed statistical inference problems. CSL provides a communication-efficient surrogate to the global likelihood that can be used for low-dimensional estimation, high-dimensional regularized estimation, and Bayesian inference. For low-dimensional estimation, CSL provably improves upon naive averaging schemes and facilitates the construction of confidence intervals. For high-dimensional…

贝叶斯分数后验AI 解读

We consider the fractional posterior distribution that is obtained by updating a prior distribution via Bayes theorem with a fractional likelihood function, a usual likelihood function raised to a fractional power. First, we analyze the contraction property of the fractional posterior in a general misspecified framework. Our contraction results only require a prior mass condition on certain Kullback–Leibler (KL) neighborhood of the true parameter (or the KL divergence…

适应平滑度与稀疏性的贝叶斯回归树集成AI 解读

Summary Ensembles of decision trees are a useful tool for obtaining flexible estimates of regression functions. Examples of these methods include gradient-boosted decision trees, random forests and Bayesian classification and regression trees. Two potential shortcomings of tree ensembles are their lack of smoothness and their vulnerability to the curse of dimensionality. We show that these issues can be overcome by instead considering sparsity inducing soft decision trees in…

任意预测变量空间上的分位数平面联合估计AI 解读

In spite of the recent surge of interest in quantile regression, joint estimation of linear quantile planes remains a great challenge in statistics and econometrics. We propose a novel parameterization that characterizes any collection of noncrossing quantile planes over arbitrarily shaped convex predictor domains in any dimension by means of unconstrained scalar, vector and function valued parameters. Statistical models based on this parameterization inherit a fast…

核方法的随机压缩：快速且最优的非参数回归AI 解读

Kernel ridge regression (KRR) is a standard method for performing nonparametric regression over reproducing kernel Hilbert spaces. Given $n$ samples, the time and space complexity of computing the KRR estimate scale as $\mathcal{O}(n^{3})$ and $\mathcal{O}(n^{2})$, respectively, and so is prohibitive in many cases. We propose approximations of KRR based on $m$-dimensional randomized sketches of the kernel matrix, and study how small the projection dimension $m$ can be chosen…

高维贝叶斯变量选择的计算复杂度AI 解读

We study the computational complexity of Markov chain Monte Carlo (MCMC) methods for high-dimensional Bayesian linear regression under sparsity constraints. We first show that a Bayesian approach can achieve variable-selection consistency under relatively mild conditions on the design matrix. We then demonstrate that the statistical criterion of posterior concentration need not imply the computational desideratum of rapid mixing of the MCMC algorithm. By introducing a…

用于高维分类的贝叶斯条件张量分解AI 解读

In many application areas, data are collected on a categorical response and high-dimensional categorical predictors, with the goals being to build a parsimonious model for classification while doing inferences on the important predictors. In settings such as genomics, there can be complex interactions among the predictors. By using a carefully-structured Tucker factorization, we define a model that can characterize any conditional probability, while facilitating variable…

贝叶斯流形回归AI 解读

There is increasing interest in the problem of nonparametric regression with high-dimensional predictors. When the number of predictors $D$ is large, one encounters a daunting problem in attempting to estimate a $D$-dimensional surface based on limited data. Fortunately, in many applications, the support of the data is concentrated on a $d$-dimensional subspace with $d\ll D$. Manifold learning attempts to estimate this subspace. Our focus is on developing computationally…

高维非参数回归的极小极大最优估计AI 解读

Minimax $L_{2}$ risks for high-dimensional nonparametric regression are derived under two sparsity assumptions: (1) the true regression surface is a sparse function that depends only on $d=O(\log n)$ important predictors among a list of $p$ predictors, with $\log p=o(n)$; (2) the true regression surface depends on $O(n)$ predictors but is an additive function where each additive component is sparse but may contain two or more interacting predictors and may have a smoothness…