Valuedecompositions

Valuedecompositions refer to a class of matrix factorization techniques designed to decompose a data matrix into a sum of latent components, each endowed with a value weight that measures its contribution to the data. The approach emphasizes interpreting these weights as the importance or significance of individual components in explaining the observed data.

For a data matrix X in R^{m×n}, a valuedecomposition of rank r seeks factor matrices U in R^{m×r}, V in R^{n×r}, and a diagonal matrix Λ in R^{r×r} with nonnegative entries such that X is approximated by the sum of rank-1 terms: X ≈ ∑_{k=1}^r λ_k u_k v_k^T. Here the columns u_k and v_k are the left and right value patterns, and λ_k is the associated value of the k-th component. Nonnegativity and sometimes normalization constraints are commonly imposed to promote interpretability and parts-based representations.

The decomposition is typically found by alternating optimization, with projections onto nonnegative orthants when required. Variants often incorporate sparsity, orthogonality, or unit-norm constraints to address scale ambiguity and to improve identifiability and interpretability of the components.

Valuedecompositions relate to several established techniques. They generalize nonnegative matrix factorization by introducing explicit value weights, and they reduce to standard SVD under more relaxed constraints that permit negative weights and orthogonality. They share with PCA the aim of dimensionality reduction while emphasizing component-wise valuation.

Applications include pattern discovery in images, gene expression data, recommender systems, and market research, where ranking components by their values aids interpretation. Limitations include potential non-uniqueness and sensitivity to initialization, requiring careful model selection and validation.