Transcendental Regularization of Finite Mixtures:Theoretical Guarantees and Practical Limitations
This work addresses a known bottleneck in unsupervised learning for researchers and practitioners using mixture models, though it is incremental in its practical impact.
The paper tackles the degeneracy problem in finite mixture models by introducing transcendental regularization, which prevents component collapse while maintaining asymptotic efficiency, resulting in stable estimation but only modest improvements in classification accuracy in high dimensions.
Finite mixture models are widely used for unsupervised learning, but maximum likelihood estimation via EM suffers from degeneracy as components collapse. We introduce transcendental regularization, a penalized likelihood framework with analytic barrier functions that prevent degeneracy while maintaining asymptotic efficiency. The resulting Transcendental Algorithm for Mixtures of Distributions (TAMD) offers strong theoretical guarantees: identifiability, consistency, and robustness. Empirically, TAMD successfully stabilizes estimation and prevents collapse, yet achieves only modest improvements in classification accuracy-highlighting fundamental limits of mixture models for unsupervised learning in high dimensions. Our work provides both a novel theoretical framework and an honest assessment of practical limitations, implemented in an open-source R package.