HALSstabilointia

HALSstabilointia refers to a set of techniques designed to improve the numerical stability and convergence of the Hierarchical Alternating Least Squares (HALS) algorithm when used for nonnegative matrix factorization (NMF). The term is used in Finnish writings to describe methods that constrain or adjust HALS updates to prevent instability, slow convergence, or divergence on difficult data.

HALS is an iterative optimization method that updates factors of a nonnegative matrix factorization by solving a sequence of nonnegative least-squares problems, typically updating one column or row at a time while keeping the other factors fixed. While efficient, HALS can exhibit oscillations or slow convergence on ill-conditioned or noisy datasets. HALSstabilointia aims to mitigate these issues without changing the fundamental HALS framework.

Common stabilization techniques include damping or scaling of updates, adaptive regularization, and normalization of factor matrices. Damping reduces the magnitude of updates to prevent overshooting, while adaptive regularization introduces penalty terms that help control the conditioning of the subproblems. Normalization keeps scale factors balanced, reducing numerical drift. Additional approaches may involve line searches along update directions, selective update rules, or modest momentum terms. Each technique trades off speed, accuracy, and bias, and may introduce hyperparameters that require tuning.

Applications of HALSstabilointia span signal and image processing, hyperspectral data analysis, audio source separation, and bioinformatics, particularly where data are noisy or poorly conditioned. Limitations include added computational cost, the need for parameter tuning, and potential bias from regularization. See also nonnegative matrix factorization, HALS, and stabilization methods for iterative optimization.