高维聚类的相变及相关问题
Phase Transitions for High Dimensional Clustering and Related Problems
摘要：
考虑一个两类聚类问题，其中我们观察到$X_i = \ell_i μ+ Z_i$、$Z_i \stackrel{iid}{\sim} N(0, I_p)$、$1 \leq i \leq n$。特征向量$μ\in R^p$是未知的，但可能是稀疏的。类标签$\ell_i\in\{-1, 1\}$也是未知的，主要的兴趣是估计它们。我们对统计极限感兴趣。在二维相空间中标定有用特征的稀有度和强度，我们找到了不可能区域和可能区域的精确划分。在前一种情况下，有用的特征对于成功的聚类来说太罕见/太弱。在后者中，有用的特征足够强大，可以成功聚类。利用Le Cam的实验比较思想，将结果推广到色噪声的情况。我们还将聚类的统计极限研究扩展到信号恢复和假设检验。我们比较了三个问题的统计极限，并揭示了一些有趣的见解。我们提出了经典的PCA和重要特征PCA（IF-PCA）的聚类。对于阈值$t > 0$，IF-PCA通过将经典PCA应用于具有大于$t$的$L^2$ -范数的$X$的所有列来聚类。我们还提出了两种聚合方法。对于“可能性区域”中的任何参数，这些方法中的某些方法都能生成成功的聚类。我们发现一个有趣的相变IF-PCA。我们的结果需要细致的分析，特别是在后选择随机矩阵理论和下界参数。
英文摘要：
Consider a two-class clustering problem where we observe $X_i = \ell_i μ+ Z_i$, $Z_i \stackrel{iid}{\sim} N(0, I_p)$, $1 \leq i \leq n$. The feature vector $μ\in R^p$ is unknown but is presumably sparse. The class labels $\ell_i\in\{-1, 1\}$ are also unknown and the main interest is to estimate them. We are interested in the statistical limits. In the two-dimensional phase space calibrating the rarity and strengths of useful features, we find the precise demarcation for the Region of Impossibility and Region of Possibility. In the former, useful features are too rare/weak for successful clustering. In the latter, useful features are strong enough to allow successful clustering. The results are extended to the case of colored noise using Le Cam's idea on comparison of experiments. We also extend the study on statistical limits for clustering to that for signal recovery and that for hypothesis testing. We compare the statistical limits for three problems and expose some interesting insight. We propose classical PCA and Important Features PCA (IF-PCA) for clustering. For a threshold $t > 0$, IF-PCA clusters by applying classical PCA to all columns of $X$ with an $L^2$-norm larger than $t$. We also propose two aggregation methods. For any parameter in the Region of Possibility, some of these methods yield successful clustering. We find an interesting phase transition for IF-PCA. Our results require delicate analysis, especially on post-selection Random Matrix Theory and on lower bound arguments.
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