Matrix factorization is an inference problem that has acquired importance due to its vast range of applications that go from dictionary learning to recommendation systems and machine learning with deep networks. The study of its fundamental statistical limits represents a true challenge, and despite a decade-long history of efforts in the community, there is still no closed formula able to describe its optimal performances in the case where the rank of the matrix scales linearly with its size. In the present paper, we study this extensive rank problem, extending the alternative 'decimation' procedure that we recently introduced, and carry out a thorough study of its performance. Decimation aims at recovering one column/line of the factors at a time, by mapping the problem into a sequence of neural network models of associative memory at a tunable temperature. Though being sub-optimal, decimation has the advantage of being theoretically analyzable. We extend its scope and analysis to two families of matrices. For a large class of compactly supported priors, we show that the replica symmetric free entropy of the neural network models takes a universal form in the low temperature limit. For sparse Ising prior, we show that the storage capacity of the neural network models diverges as sparsity in the patterns increases, and we introduce a simple algorithm based on a ground state search that implements decimation and performs matrix factorization, with no need of an informative initialization.

翻译：矩阵分解是一个推演问题，因其在从字典学习、推荐系统到深度网络机器学习等领域的广泛应用而日益重要。其基本统计极限的研究构成真正挑战，尽管学界经过十余年的努力，至今仍未形成能够描述矩阵秩与其尺寸线性扩展时最优表现的封闭公式。本文研究这种大尺度秩问题，拓展了我们近期引入的替代性"抽取"程序，并对其性能进行了彻底研究。抽取旨在通过将问题映射为一系列可调温度下的联想记忆神经网络模型，每次恢复因子矩阵的一列/一行。尽管非最优，抽取的优势在于具有理论可分析性。我们将其实用范围和分析扩展至两类矩阵。对于一大类紧支撑先验，我们证明神经网络模型的副本对称自由熵在低温极限下呈现普适形式。对于稀疏伊辛先验，我们发现神经网络模型的存储容量随模式稀疏性增加而发散，并引入基于基态搜索的简单算法来实现抽取与矩阵分解，无需信息性初始化。