Matrix factorization techniques decompose a matrix into simpler, structured components to solve computational problems efficiently. Common methods include LU decomposition, QR decomposition, Singular Value Decomposition (SVD), Cholesky decomposition, and Non-Negative Matrix Factorization (NMF). Each technique serves distinct purposes, from solving linear systems to dimensionality reduction, depending on the matrix properties and application needs.
LU decomposition splits a square matrix into a lower triangular matrix (L) and an upper triangular matrix (U). This is useful for solving systems of linear equations, as triangular matrices simplify forward and backward substitution. For example, in circuit analysis or structural engineering simulations, LU decomposition helps efficiently compute solutions for large systems. QR decomposition factors a matrix into an orthogonal matrix (Q) and an upper triangular matrix ®. Orthogonal matrices preserve vector lengths and angles, making QR ideal for least-squares regression, a common task in machine learning. Libraries like NumPy use QR to solve overdetermined systems where data points outnumber variables.
Singular Value Decomposition (SVD) breaks any rectangular matrix into three components: U (left singular vectors), Σ (diagonal matrix of singular values), and V (right singular vectors). SVD is foundational in recommendation systems, where it identifies latent user-item preferences, and in Principal Component Analysis (PCA) for reducing data dimensionality. Cholesky decomposition is specific to symmetric positive-definite matrices, decomposing them into a lower triangular matrix (L) and its transpose. This method is computationally efficient for problems like portfolio optimization in finance or solving partial differential equations in physics.
Non-Negative Matrix Factorization (NMF) constrains factors to non-negative values, making it suitable for datasets where negative values lack meaning. For instance, in text mining, NMF identifies topics by decomposing a term-document matrix into non-negative topic-word and document-topic matrices. Similarly, in image processing, NMF can separate an image into additive components (e.g., facial features). Each technique’s utility depends on context: LU and Cholesky excel in numerical stability, QR in orthogonal transformations, SVD in uncovering latent patterns, and NMF in interpretability for non-negative data.