论文标题
单纯结构矩阵分解：基于稀疏性的可识别性和可证明的正确算法
Simplex-Structured Matrix Factorization: Sparsity-based Identifiability and Provably Correct Algorithms
论文作者
论文摘要
在本文中，我们提供了新颖的算法，并具有可识别性保证的单纯矩阵分解（SSMF），这是非负矩阵分解的概括。为SSMF提供可识别性结果的当前最新算法依赖于足够分散的条件（SSC），该条件要求数据点在基础向量的凸壳内散布。在大多数情况下，我们提出的算法恢复独特的分解的条件比SSC弱得多。我们只需要在尺寸为$ d-1 $的基础向量的凸面的每个方面都有$ d $点。关键思想是基于提取包含最大点的方面。我们说明了方法对合成数据集和高光谱图像的有效性，这表明它的表现优于最先进的SSMF算法，因为它能够处理更高的噪声水平，排名不足的矩阵，异常值和极大侵犯SSC的输入数据。
In this paper, we provide novel algorithms with identifiability guarantees for simplex-structured matrix factorization (SSMF), a generalization of nonnegative matrix factorization. Current state-of-the-art algorithms that provide identifiability results for SSMF rely on the sufficiently scattered condition (SSC) which requires the data points to be well spread within the convex hull of the basis vectors. The conditions under which our proposed algorithms recover the unique decomposition is in most cases much weaker than the SSC. We only require to have $d$ points on each facet of the convex hull of the basis vectors whose dimension is $d-1$. The key idea is based on extracting facets containing the largest number of points. We illustrate the effectiveness of our approach on synthetic data sets and hyperspectral images, showing that it outperforms state-of-the-art SSMF algorithms as it is able to handle higher noise levels, rank deficient matrices, outliers, and input data that highly violates the SSC.
