Model assisted variable clustering: Minimax-optimal recovery and algorithms
模型辅助变量聚类：极小极大最优恢复与算法AI 解读 ↓
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模型辅助变量聚类：极小极大最优恢复与算法
- 研究问题
- 如何以总体协方差模型定义变量簇，使聚类结果具有统计解释，并确定不同分离度量下精确恢复的难度界限？
- 方法与思路
- 作者建立 G 块协方差模型，以两种簇分离度量分析恢复难度，提出 COD 与 PECOK 算法，并比较谱聚类。
- 材料陈述的贡献
- 摘要称推导两种度量下不同的极小极大分离阈值，研究算法最优性，并对修正凸松弛的变量聚类方法作统计分析。
- 阅读关注点
- 应核对总体簇的可识别定义、两种分离度量、算法各自的最优性条件，以及模拟和数据分析支持的具体范围。
- 阅读边界
- 索引摘要未注明稿件版本，未列出阈值表达式、完整条件或比较数值；不能认定两种算法在所有设定中均最优。
原文摘要
The problem of variable clustering is that of estimating groups of similar components of a $p$-dimensional vector $X=(X_{1},\ldots ,X_{p})$ from $n$ independent copies of $X$. There exists a large number of algorithms that return data-dependent groups of variables, but their interpretation is limited to the algorithm that produced them. An alternative is model-based clustering, in which one begins by defining population level clusters relative to a model that embeds notions of similarity. Algorithms tailored to such models yield estimated clusters with a clear statistical interpretation. We take this view here and introduce the class of $G$-block covariance models as a background model for variable clustering. In such models, two variables in a cluster are deemed similar if they have similar associations will all other variables. This can arise, for instance, when groups of variables are noise corrupted versions of the same latent factor. We quantify the difficulty of clustering data generated from a $G$-block covariance model in terms of cluster proximity, measured with respect to two related, but different, cluster separation metrics. We derive minimax cluster separation thresholds, which are the metric values below which no algorithm can recover the model-defined clusters exactly, and show that they are different for the two metrics. We therefore develop two algorithms, COD and PECOK, tailored to $G$-block covariance models, and study their minimax-optimality with respect to each metric. Of independent interest is the fact that the analysis of the PECOK algorithm, which is based on a corrected convex relaxation of the popular $K$-means algorithm, provides the first statistical analysis of such algorithms for variable clustering. Additionally, we compare our methods with another popular clustering method, spectral clustering. Extensive simulation studies, as well as our data analyses, confirm the applicability of our approach.