Matrix Factorization
Last Updated: July 29, 2026 | By Mihail Sebastian | AI Dictionary
Splits one large table into smaller ones whose product rebuilds it, exposing hidden factors and filling in the values the original table never recorded.
What is Matrix Factorization?
Matrix factorization splits a large matrix into two or three smaller matrices whose product reconstructs the original. The smaller matrices describe rows and columns in terms of a limited set of hidden factors, so the method doubles as dimensionality reduction.
Its usefulness comes from what happens to the gaps. A table of which users rated which films is mostly empty, and multiplying the two factor matrices back together produces a value in every cell, including the ones nobody filled in. Those filled-in cells are predictions.
Types of Matrix Factorization
- Singular value decomposition (SVD) splits a matrix into three parts and ranks the factors by how much of the original variation each explains. Principal component analysis is usually computed this way.
- Non-negative matrix factorization (NMF) forbids negative numbers in the factors, so each one adds rather than cancels. On text, the resulting factors read like topics: a list of words with positive weights.
- Alternating least squares (ALS) fixes one factor matrix, solves for the other, then swaps and repeats. It handles matrices with missing entries and parallelizes across machines, which is why large recommender systems use it.
Applications of Matrix Factorization
- Recommendation: Predicting a rating or click for user-item pairs with no history, which is the standard collaborative filtering setup.
- Topic modeling: NMF on a document-term matrix produces interpretable groups of co-occurring words without labeled data.
- Missing data: Reconstructing sensor readings or survey responses that were never recorded, when the columns are correlated enough to support the guess.
Example of Matrix Factorization
A streaming service has 500,000 subscribers and 20,000 titles, and each subscriber has rated a few dozen of them. The ratings table is more than 99% empty.
ALS factors it into a 500,000 by 40 user matrix and a 20,000 by 40 title matrix. Each user becomes 40 numbers, each film becomes 40 numbers, and a predicted rating is the dot product of the two. Nobody chose what the 40 factors mean; they emerge from the ratings, and some turn out to track recognizable things like a taste for subtitled films.
This approach was the workhorse of the leading entries in the Netflix Prize competition, won in 2009. It also carries a known governance problem: the factors are unnamed, so a factor that happens to align with a user’s neighborhood, age, or language is indistinguishable from one that tracks genre preference, and the recommendations it drives are hard to audit for bias.
Related AI terms: Dimensionality Reduction · Principal Component Analysis · Latent Space · Sparsity · Tensors
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