SMTf

SMTf, or Sparse Matrix–Tensor Factorization, is a mathematical framework for decomposing high-dimensional data into a small set of sparse latent factors. It extends traditional tensor factorization by enforcing sparsity on the factor matrices, promoting interpretable components that are active only on subsets of the data modes. This sparsity can improve interpretability and reduce storage and computation for large datasets.

In its typical form, SMTf targets a tensor X of order three, X ∈ R^{I×J×K}. The goal is to find factor matrices A ∈ R^{I×R}, B ∈ R^{J×R}, C ∈ R^{K×R} and a weight vector λ ∈ R^R such that X is approximated by a sum of R rank-one components: X ≈ ∑_{r=1}^R λ_r a_r ∘ b_r ∘ c_r, where a_r, b_r, and c_r denote the r-th columns of A, B, and C, respectively. Sparsity constraints are applied to these columns, typically via L1 regularization or hard-thresholding, to produce concise, feature-selective factors.

Algorithms for SMTf commonly use alternating optimization (ALS) or gradient-based methods, iteratively updating one factor while keeping the others fixed. Because the problem is nonconvex, convergence depends on initialization and regularization settings. Numerous variants exist, including nonnegative SMTf, probabilistic SMTf with sparsity priors, and models incorporating group sparsity or temporal dynamics.

Applications span data mining and analytics, including recommender systems, social network analysis, neuroscience, remote sensing, and image or video data. By producing compact, interpretable representations, SMTf supports pattern discovery, data compression, and improved downstream learning tasks. See also tensor decomposition and sparse coding.