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What is non-negative matrix factorization, and why is t...
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See plansNMF decomposes a matrix X (n×m, all entries ≥0) into two lower-rank factors W (n×k) and H (k×m), both also constrained to be non-negative, such that WH approximates X. Plain matrix factorization (via SVD, say) gives a similar low-rank approximation but allows negative entries in the factors, which makes them mathematically optimal but often uninterpretable — a “component” with mixed positive and negative weights doesn’t correspond to anything you can point to in the real world.
Forcing non-negativity changes the character of the solution: because you can only add parts together, never subtract, NMF tends to learn a parts-based, additive representation rather than a holistic one — famously, applied to face images it tends to discover eyes, noses, and other localized parts as components, where PCA finds global eigenfaces that mix light and dark across the whole image. The classic ML application is topic modeling and information retrieval: factor a document-term matrix into a term-topic matrix and a topic-document matrix, and both come out non-negative and interpretable as “how much each term belongs to each topic” and “how much each document draws on each topic.” It also shows up in recommender systems as a way to find latent, additive user and item factors from a non-negative interaction matrix. The tradeoff is that NMF’s optimization is non-convex, unlike SVD, so it only guarantees a local optimum and is sensitive to initialization.