Abstract
A strong measure to handle the high-dimensional models has been principally centered on the compact representation of the input data sets. In this framework, since a wide range of the data sets are given in the matrix forms, and meanwhile, the real-world data are often inherently nonnegative, classic tools of the linear algebra such as nonnegative matrix factorization (NMF) can be significantly helpful. Nowadays, NMF is frequently used in practical disciplines such as pattern recognition, recommendation systems, document clustering and face detection.
In a common plan, various NMF algorithms take a matrix with nonnegative entries as the input, and give two lower dimension matrices with nonnegative entries as the output, such that multiplying the output matrices yields an approximation for the input matrix. Among such schemes, there exists the alternating nonnegative least squares (ANLS) algorithm, in essence characterized by alternatingly solving two least squares subproblems of the NMF model, often by the classic large-scale unconstrained optimization methods.
Here, we discuss augmented NMF models that are computationally capable to make the factorization elements well-conditioned. The augmented model benefits a term to penalize the ill-conditioning in the trajectory toward the solution. To assess the condition number, the measure functions that suggested in the literature to analyze the scaling and sizing of the quasi-Newton updating formulas are employed. Meanwhile, some matrix approximations put forward to relax and simplify the given model. To see at what level the model revisions are effective, some numerical results are presented as well.
References:
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[2] S. Babaie-Kafaki. A survey on the Dai-Liao family of nonlinear conjugate gradient methods, RAIRO-Oper. Res., 57:43-58, 2023.