非負 行列因子分解 (NMF) is a computational technique in the field of 機械学習 and データ分析. It involves decomposing a given non-negative matrix into two non-negative matrices, typically referred to as basis and coefficient matrices. The purpose of this factorization is to identify hidden patterns or structures within the data, making it particularly useful for tasks such as トピックモデリング, 画像処理, and 協調フィルタリング.
数学的には、非負値行列が与えられたとき V (with dimensions m×n）、NMFは次の二つの非負値行列を見つけることを目指します W (basis matrix, of dimensions m×r)と H (coefficient matrix, of dimensions r×n）となるように：
V ≈ W * H
ここで、 r is the rank or number of components to be extracted, and the approximation seeks to minimize the difference between the original matrix and the product of the two factorized matrices. One of the key properties of NMF is that it allows for a parts-based representation of the data, as all components are constrained to be non-negative, leading to more interpretable results. This is in contrast to other matrix factorization techniques, such as Singular Value 分解 (SVD), which can yield negative values and thus may be less intuitive for certain applications.
NMF has applications across various domains, including image compression, document clustering, and レコメンデーションシステム, where it helps in extracting meaningful features from complex datasets. Its simplicity and effectiveness in revealing latent structures make it a popular choice among data scientists and researchers.