We analyse the matrix factorization problem. Given a noisy measurement of a product of two matrices, the problem is to estimate back the original matrices. It arises in many applications such as dictionary learning, blind matrix calibration, sparse principal component analysis, blind source separation, low rank matrix completion, robust principal component analysis or factor analysis. We use the tools of statistical mechanics - the cavity and replica methods - to analyze the achievability and tractability of the inference problems in the setting of Bayes-optimal inference, which amounts to assuming that the two matrices have random independent elements generated from some known distribution, and this information is available to the inference algorithm. In this setting, we compute the minimal mean-squared-error achievable in principle in any computational time, and the error that can be achieved by an efficient approximate message passing algorithm. The computation is based on the asymptotic state-evolution analysis of the algorithm. The performance that our analysis predicts, both in terms of the achieved mean-squared-error, and in terms of sample complexity, is extremely promising and motivating for a further development of the algorithm.

使用统计力学工具分析了矩阵分解问题的可实现性和计算可处理性，在贝叶斯最优推断设置下计算任意计算时间内可能实现的最小均方误差和有效近似迭代推理算法可以达到的误差。

贝叶斯最优矩阵分解中的相变和样本复杂度