提出新正则化方法，防止混合模型参数坍塌
Transcendental Regularization of Finite Mixtures:Theoretical Guarantees and Practical Limitations
- 用解析屏障函数惩罚似然，阻止成分坍缩
- 理论保证模型可识别、一致且鲁棒，实验稳定收敛
- 适合关注混合模型理论边界的研究者
有限混合模型广泛用于无监督学习，但通过EM算法进行最大似然估计时会出现成分坍缩的退化问题。本文提出超越正则化（transcendental regularization），一种具有解析屏障函数的惩罚似然框架，可在保持渐近效率的同时防止退化。由此产生的分布混合的超越算法（TAMD）具备强理论保证：可识别性、一致性与鲁棒性。实证上，TAMD有效稳定估计并防止坍缩，但在分类准确率上仅取得适度提升，揭示了高维无监督学习中混合模型的根本局限。本工作既提供新理论框架，也给出对实际局限的诚实评估，并开源实现于R包。
原文摘要 · Abstract (English)
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.
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