NMF

implementation of Non-negative Matrix Factorisation Algorithms

Non-Negative Matrix Factorization

Non-Negative Matrix Factorization (NMF) is a recent technique for dimensionality reduction and data analysis that yields a parts based, sparse nonnegative representation for nonnegative input data. NMF has found a wide variety of applications, including text analysis, document clustering, face/image recognition, language modeling, speech processing and many others. Despite these numerous applications, the algorithmic development for computing the NMF factors has been relatively deficient.

NMF can be applied To the statistical analysis Of multivariate data In the following manner. Given a Set Of Of multivariate n-dimensional data vectors, the vectors are placed In the columns Of an n x m matrix V where m Is the number Of examples In the data Set. This matrix Is Then approximately factorized into an n x r matrix W (weights matrix) And an r x m matrix H (features matrix), where r Is the number Of features defined by the user. Usually r Is chosen To be smaller than n Or m, so that W And H are smaller than the original matrix V. This results In a compressed version Of the original data matrix.

NMF 算法将矩阵 A 分解为两个矩阵：基矩阵 W 和系数矩阵 H。这两个矩阵的乘积将尽可能地接近原始矩阵 A。

• 基矩阵 W 表示的是 A 的潜在特征或主题，
• 而系数矩阵 H 表示的是每个样本（在矩阵 A 中是每一行）在这些特征或主题上的分布。

在 NMF 中，基矩阵 W 和系数矩阵 H 的乘积近似等于原始矩阵 A。通过这种方式，NMF 能够揭示矩阵 A 中的 潜在结构和特征。例如，在基矩阵 W 的每一列可能代表了矩阵 A 中的主要特征或主题，而系数矩阵 H 的 每一行则表示原始矩阵 A 中对应样本在这些特征或主题上的分布情况。