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Also known as: NMF
A factorization method that decomposes a non-negative matrix into non-negative factors, yielding additive, parts-based representations.
DESCRIPTION
Non-negative Matrix Factorization (NMF) approximates a data matrix as a product of two lower-rank non-negative matrices. The non-negativity constraint encourages sparse, additive components that often align with interpretable parts or topics. NMF is widely used in text mining, recommender systems, and image decomposition. Optimization commonly minimizes Frobenius norm or KL divergence with multiplicative updates or coordinate descent. Model selection involves choosing rank, regularization, and initialization to balance fit and interpretability.
ALTERNATIVE DEFINITIONS
Wikipedia
Non-negative matrix factorization (NMF) is a group of algorithms in multivariate analysis and linear algebra where a matrix V is factorized into (usually) two matrices W and H, with the property that all three matrices have no negative elements.
Canonical description emphasizing non-negativity of all factors.
Nature (Lee & Seung, 1999)
By forcing non-negativity, NMF yields a parts-based representation of the data, because it allows only additive combinations of the basis vectors.
Highlights the hallmark parts-based interpretability.
scikit-learn
NMF finds two non-negative matrices whose product approximates the input matrix. The method is suited to learn parts-based representations.
Concise practitioner-focused definition and use case.
RELATED TERMS
Matrix Factorization Parts-based Representation Topic Modeling KL Divergence Frobenius Norm Sparsity Dictionary Learning Feature Reduction
COMPETING TERMS
Latent Semantic Analysis Latent Dirichlet Allocation
RELATIONS
- Broader: Matrix Factorization
KNOWLEDGE AREAS
TARGET AUDIENCE
Data Scientist AI Engineer Data Analyst
EXTERNAL SOURCES
- Wikipedia - Non-negative matrix factorization
- scikit-learn - API and usage for NMF
- Nature - Seminal parts-based representation paper
- arXiv - Survey of NMF and extensions
TOP QUESTIONS
What makes Non-negative Matrix Factorization interpretable?
Non-negativity enforces additive combinations, leading to parts-based factors often aligned with topics or components.
Category: Definition
How do you choose the rank for NMF?
Use validation curves, reconstruction error, sparsity objectives, or task performance to select rank.
Category: Requirements
What loss functions are common in NMF?
Squared Frobenius norm and generalized KL divergence are standard choices, each favoring different residual structures.
Category: Example
What initialization strategies help NMF?
NNDSVD and random non-negative inits are common; NNDSVD often accelerates convergence.
Category: Requirements
What are common mistakes with NMF?
Using negative inputs without preprocessing or ignoring scaling, which violates the method’s assumptions.
Category: Mistakes
How does NMF compare to LDA for topics?
NMF is deterministic (given init) and linear, while LDA is probabilistic. Interpretability can differ by corpus.
Category: Alternatives
When is NMF preferred over LSA?
When non-negativity aligns with the data (e.g., counts, pixel intensities) and parts-based factors are desired.
Category: Opinion
How do you regularize NMF?
Add L1/L2 penalties to promote sparsity or smoothness, depending on the application.
Category: Requirements
What risks exist with NMF?
Local minima and sensitivity to initialization can produce unstable factors; multiple runs and consensus help.
Category: Risk