Non-Negative Matrix Factorization

Last Updated : 2 May, 2026

Non-Negative Matrix Factorization (NMF) is a technique used to break down large dataset into smaller meaningful parts while ensuring that all values remain non-negative. This helps in extracting useful features from data and making it easier to analyze and process it.

Matrix Decomposition and Representation in NMF

For a matrix A of dimensions m \times n where each element is \geq 0 NMF factorizes it into two matrices W and H with dimensions m \times k and k \times n respectively where both matrices contain only non-negative elements:

A_{m \times n} \approx W_{m \times k} \cdot H_{k \times n}

where:

  • A : Original input matrix (a linear combination of W and H)
  • W: Feature matrix (basis components)
  • H : Coefficient matrix (weights associated with W)
  • k : Rank (dimensionality of the reduced representation where k \le \min(m, n)

NMF helps to identify hidden patterns in data by assuming that each data point can be represented as a combination of fundamental features found in W.