We consider the problem of the estimation of a high-dimensional probability distribution from i.i.d. samples of the distribution using model classes of functions in tree-based tensor formats, a particular case of tensor networks associated with a dimension partition tree. The distribution is assumed to admit a density with respect to a product measure, possibly discrete for handling the case of discrete random variables. After discussing the representation of classical model classes in tree-based tensor formats, we present learning algorithms based on empirical risk minimization using a $L^2$ contrast. These algorithms exploit the multilinear parametrization of the formats to recast the nonlinear minimization problem into a sequence of empirical risk minimization problems with linear models. A suitable parametrization of the tensor in tree-based tensor format allows to obtain a linear model with orthogonal bases, so that each problem admits an explicit expression of the solution and cross-validation risk estimates. These estimations of the risk enable the model selection, for instance when exploiting sparsity in the coefficients of the representation. A strategy for the adaptation of the tensor format (dimension tree and tree-based ranks) is provided, which allows to discover and exploit some specific structures of high-dimensional probability distributions such as independence or conditional independence. We illustrate the performances of the proposed algorithms for the approximation of classical probabilistic models (such as Gaussian distribution, graphical models, Markov chain).

中文翻译：

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使用树张量网络学习高维概率分布

我们考虑使用基于树的张量格式的函数模型类从分布的 iid 样本估计高维概率分布的问题，这是与维度划分树相关的张量网络的特殊情况。假设分布允许与产品度量相关的密度，可能是离散的，用于处理离散随机变量的情况。在讨论了基于树的张量格式的经典模型类的表示之后，我们提出了基于经验风险最小化的学习算法，使用 $L^2$ 对比。这些算法利用格式的多线性参数化将非线性最小化问题重铸成一系列具有线性模型的经验风险最小化问题。基于树的张量格式的张量的合适参数化允许获得具有正交基的线性模型，以便每个问题都承认解决方案和交叉验证风险估计的明确表达。这些风险估计使模型选择成为可能，例如在利用表示系数中的稀疏性时。提供了一种适应张量格式（维度树和基于树的等级）的策略，它允许发现和利用高维概率分布的某些特定结构，例如独立性或条件独立性。我们说明了所提出的算法在逼近经典概率模型（如高斯分布、图形模型、马尔可夫链）时的性能。