论文标题
有限的单纯结构矩阵分解：算法，可识别性和应用程序
Bounded Simplex-Structured Matrix Factorization: Algorithms, Identifiability and Applications
论文作者
论文摘要
在本文中，我们提出了一种新的低级矩阵分解模型，称为有界的单纯形成矩阵分解（BSSMF）。给定输入矩阵$ x $和分解等级$ r $，BSSMF寻找带有$ r $列的矩阵$ w $和带有$ r $行的矩阵$ h $，以至于$ x \ y o $ x \ of y o $ w $中的条目是$ w $的条目，这些条目是有界的，即$ h $属于$ h $属于$ h $属于$ h $属于$ h $属于$ h $属于$ h $属于$ h的属于$ hy的属于$ h is pob las属于$ hys属于$ h的列。 BSSMF概括了非负矩阵分解（NMF）和单纯结构的矩阵分解（SSMF）。当输入矩阵$ x $的条目属于给定间隔时，BSSMF特别适合。例如，当$ x $的行代表图像时，或$ x $是一个评分矩阵，例如在Netflix和Movielens数据集中，其中$ x $的条目属于Interval $ [1,5] $。单纯结构的矩阵$ h $不仅会导致易于理解的分解，从而提供了$ x $的列的软聚类，而且暗示着$ wh $的每个列的条目属于与$ w $的列相同的间隔。在本文中，我们首先提出了BSSMF的快速算法，即使在$ x $中缺少数据的情况下。然后，我们为BSSMF提供可识别性条件，也就是说，我们提供了BSSMF承认独特分解的条件，直到琐碎的歧义。最后，我们说明了BSSMF对两个应用程序的有效性：在一组图像中提取特征，以及推荐系统的矩阵完成问题。
In this paper, we propose a new low-rank matrix factorization model dubbed bounded simplex-structured matrix factorization (BSSMF). Given an input matrix $X$ and a factorization rank $r$, BSSMF looks for a matrix $W$ with $r$ columns and a matrix $H$ with $r$ rows such that $X \approx WH$ where the entries in each column of $W$ are bounded, that is, they belong to given intervals, and the columns of $H$ belong to the probability simplex, that is, $H$ is column stochastic. BSSMF generalizes nonnegative matrix factorization (NMF), and simplex-structured matrix factorization (SSMF). BSSMF is particularly well suited when the entries of the input matrix $X$ belong to a given interval; for example when the rows of $X$ represent images, or $X$ is a rating matrix such as in the Netflix and MovieLens datasets where the entries of $X$ belong to the interval $[1,5]$. The simplex-structured matrix $H$ not only leads to an easily understandable decomposition providing a soft clustering of the columns of $X$, but implies that the entries of each column of $WH$ belong to the same intervals as the columns of $W$. In this paper, we first propose a fast algorithm for BSSMF, even in the presence of missing data in $X$. Then we provide identifiability conditions for BSSMF, that is, we provide conditions under which BSSMF admits a unique decomposition, up to trivial ambiguities. Finally, we illustrate the effectiveness of BSSMF on two applications: extraction of features in a set of images, and the matrix completion problem for recommender systems.
