标题： 非负矩阵分解中求解非单调变分不等式的固定点算法 显示英文标题

标题： Fixed Point Algorithm for Solving Nonmonotone Variational Inequalities in Nonnegative Matrix Factorization

摘要： 非负矩阵分解（NMF），即将数据矩阵近似表示为两个非负矩阵的乘积，是机器学习和数据分析中的关键问题。 解决NMF的一种方法是将问题表述为一个非凸优化问题，即在非负性约束下最小化数据矩阵与两个非负矩阵乘积之间的距离，然后使用迭代算法求解该问题。 常用的算法是乘法更新算法和交替最小二乘算法。 尽管这两种算法收敛速度快，但它们可能无法收敛到与距离函数梯度的非单调变分不等式解相等的驻点。 本文提出了一种基于Krasnosel'ski\u \i -Mann不动点算法的迭代算法来求解该问题。 收敛性分析表明，在某些假设条件下，所提出算法生成的序列的任何聚点都属于变分不等式的解集。 将{\tt '乘'}和{\tt 'als'}算法以及所提出的算法应用于各种NMF问题，结果表明所提出的算法具有快速收敛性和有效性。 显示英文摘要

摘要： Nonnegative matrix factorization (NMF), which is the approximation of a data matrix as the product of two nonnegative matrices, is a key issue in machine learning and data analysis. One approach to NMF is to formulate the problem as a nonconvex optimization problem of minimizing the distance between a data matrix and the product of two nonnegative matrices with nonnegativity constraints and then solve the problem using an iterative algorithm. The algorithms commonly used are the multiplicative update algorithm and the alternating least-squares algorithm. Although both algorithms converge quickly, they may not converge to a stationary point to the problem that is equal to the solution to a nonmonotone variational inequality for the gradient of the distance function. This paper presents an iterative algorithm for solving the problem that is based on the Krasnosel'ski\u\i-Mann fixed point algorithm. Convergence analysis showed that, under certain assumptions, any accumulation point of the sequence generated by the proposed algorithm belongs to the solution set of the variational inequality. Application of the {\tt 'mult'} and {\tt'als'} algorithms in MATLAB and the proposed algorithm to various NMF problems showed that the proposed algorithm had fast convergence and was effective.