Nonnegative Matrix Factorization
Chapter 9: Matrix Decompositions and Beyond — Non-Neg Matrix Fact (NMF)
From the book
Chapter 9: Matrix Decompositions and Beyond. In the chapter mind map this icon labels Non-Neg Matrix Fact (NMF):$V \approx WH$. The discussion below is excerpted and lightly edited from § Theorem: Non-Negative Matrix Factorization in Mathematics for AI and Machine Learning.
For a non-negative matrix $A \in \mathbb{R}_{\ge 0}^{M \times N}$ (i.e., $A_{ij} \ge 0$ for all $i, j$), there exist non-negative matrices $W \in \mathbb{R}_{\ge 0}^{M \times r}$ and $H \in \mathbb{R}_{\ge 0}^{r \times N}$ such that
where the approximation minimizes $||A - WH||_F^2$ subject to $W \ge 0$ and $H \ge 0$.
What this drawing shows
What you see. Represents a nonnegative matrix decomposed into additive nonnegative parts, useful for parts-based representations.
In the mind map. Chapter 9 — Non-Neg Matrix Fact (NMF). See From the book above for definitions, figures, and worked examples.
Where to read next
Read the full definitions, figures, and worked examples in Chapter 9: Matrix Decompositions and Beyond — see the mind-map node Non-Neg Matrix Fact (NMF).