Yanxi Hou

Sixiao Zheng, Ke Fan, Yanxi Hou et al.

Clustering is an essential task to unsupervised learning. It tries to automatically separate instances into coherent subsets. As one of the most well-known clustering algorithms, k-means assigns sample points at the boundary to a unique cluster, while it does not utilize the information of sample distribution or density. Comparably, it would potentially be more beneficial to consider the probability of each sample in a possible cluster. To this end, this paper generalizes k-means to model the distribution of clusters. Our novel clustering algorithm thus models the distributions of distances to centroids over a threshold by Generalized Pareto Distribution (GPD) in Extreme Value Theory (EVT). Notably, we propose the concept of centroid margin distance, use GPD to establish a probability model for each cluster, and perform a clustering algorithm based on the covering probability function derived from GPD. Such a GPD k-means thus enables the clustering algorithm from the probabilistic perspective. Correspondingly, we also introduce a naive baseline, dubbed as Generalized Extreme Value (GEV) k-means. GEV fits the distribution of the block maxima. In contrast, the GPD fits the distribution of distance to the centroid exceeding a sufficiently large threshold, leading to a more stable performance of GPD k-means. Notably, GEV k-means can also estimate cluster structure and thus perform reasonably well over classical k-means. Thus, extensive experiments on synthetic datasets and real datasets demonstrate that GPD k-means outperforms competitors. The github codes are released in https://github.com/sixiaozheng/EVT-K-means.

L. Jeff Hong, Yanxi Hou, Qingkai Zhang et al.

Stochastic simulation models effectively capture complex system dynamics but are often too slow for real-time decision-making. Traditional metamodeling techniques learn relationships between simulator inputs and a single output summary statistic, such as the mean or median. These techniques enable real-time predictions without additional simulations. However, they require prior selection of one appropriate output summary statistic, limiting their flexibility in practical applications. We propose a new concept: generative metamodeling. It aims to construct a "fast simulator of the simulator," generating random outputs significantly faster than the original simulator while preserving approximately equal conditional distributions. Generative metamodels enable rapid generation of numerous random outputs upon input specification, facilitating immediate computation of any summary statistic for real-time decision-making. We introduce a new algorithm, quantile-regression-based generative metamodeling (QRGMM), and establish its distributional convergence and convergence rate. Extensive numerical experiments demonstrate QRGMM's efficacy compared to other state-of-the-art generative algorithms in practical real-time decision-making scenarios.