Bayesian nonparametrics are a class of probabilistic models in which the model size is inferred from data. A recently developed methodology in this field is small-variance asymptotic analysis, a mathematical technique for deriving learning algorithms that capture much of the flexibility of Bayesian nonparametric inference algorithms, but are simpler to implement and less computationally expensive. Past work on small-variance analysis of Bayesian nonparametric inference algorithms has exclusively considered batch models trained on a single, static dataset, which are incapable of capturing time evolution in the latent structure of the data. This work presents a small-variance analysis of the maximum a posteriori filtering problem for a temporally varying mixture model with a Markov dependence structure, which captures temporally evolving clusters within a dataset. Two clustering algorithms result from the analysis: D-Means, an iterative clustering algorithm for linearly separable, spherical clusters; and SD-Means, a spectral clustering algorithm derived from a kernelized, relaxed version of the clustering problem. Empirical results from experiments demonstrate the advantages of using D-Means and SD-Means over contemporary clustering algorithms, in terms of both computational cost and clustering accuracy.

中文翻译：

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基于马尔可夫链混合模型小方差分析的动态聚类算法

贝叶斯非参数是一类概率模型，其中模型大小是从数据中推断出来的。该领域中最近开发的一种方法是小方差渐近分析，这是一种用于派生学习算法的数学技术，该算法捕获了贝叶斯非参数推理算法的许多灵活性，但更易于实现且计算成本较低。过去有关贝叶斯非参数推理算法的小方差分析的工作专门考虑了在单个静态数据集上训练的批处理模型，这些模型无法捕获数据潜在结构中的时间演变。这项工作针对具有马尔可夫依赖结构的时变混合模型，提出了最大后验滤波问题的小方差分析，捕获数据集中随时间变化的聚类。分析得出两种聚类算法：D-Means，一种线性可分离的球形聚类的迭代聚类算法；SD-Means是一种光谱聚类算法，它是从聚类问题的核化，宽松版本中得出的。实验的经验结果证明，就计算成本和聚类准确性而言，使用D-Means和SD-Means优于现代聚类算法。