核范数正则化的贝叶斯矩阵补全
Nuclear Norm-Regularized Bayesian Matrix Completion
- Cornell University（康奈尔大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对核范数正则化贝叶斯矩阵补全，提出首个具有非渐近保证的多项式时间采样器，通过离散化噪声精度并利用热力学积分构建后验，证明了可行性。
AI中文摘要：
矩阵补全问题，即从带有噪声的观测中估计矩阵缺失项的问题，是推荐系统和面板数据中的反事实结果估计等众多问题的基础。许多算法使用正则化最小二乘法来解决该问题，通常以核范数作为正则化项，但这种方法产生点估计，没有内置的不确定性量化。贝叶斯公式是一种自然的替代方案，如果噪声方差已知，基于核范数的先验产生对数凹后验。不幸的是，在实践中，噪声方差先验未知，因此对于完全贝叶斯方法，必须对其施加先验。我们首次为此模型提供了具有显式非渐近保证的采样器：多项式于矩阵维数和目标精度的倒数。我们的技术是将噪声精度的分布离散化到网格上，并通过热力学积分构建分类后验。这种扩展并非矩阵补全所特有，可能在其他非对数凹采样问题中有用，其中非对数凹性仅限于单个变量，其余变量的联合分布是非光滑的。我们的贡献是一个可行性结果：我们证明了该模型存在多项式时间的贝叶斯采样器，并且由此产生的复杂度虽然是多项式的，但在当前问题规模下并不打算作为可部署的算法。
英文摘要：
Matrix completion, the problem of estimating missing entries in a matrix from noisily observed ones, underlies a diverse array of problems such as recommender systems and counterfactual outcome estimation in panel data. Many algorithms address the problem using regularized least squares, often with the nuclear norm as a regularizer, but this method yields a point estimate with no built-in uncertainty quantification. A Bayesian formulation is a natural alternative, and if the noise variance is known, the nuclear norm-based prior yields a log-concave posterior. Unfortunately, in practice, the noise variance will not be known a priori, so for a fully Bayesian approach, a prior must be imposed on it. We give the first sampler for this model with an explicit non-asymptotic guarantee: polynomial in the matrix dimensions and in the reciprocal of the target accuracy. Our technique is to discretize the distribution of the noise precision onto a grid and build a categorical posterior via thermodynamic integration. This extension is not specific to matrix completion and may be useful in other non-log-concave sampling problems where the non-log-concavity is restricted to a single variable and the joint distribution of the remaining variables is nonsmooth. Our contribution is a feasibility result: we show that a polynomial-time Bayesian sampler for this model exists at all, and the resulting complexity, while polynomial, is not intended as a deployable algorithm at current problem scales.
