In mixture models, anisotropic noise within each cluster is widely present in real-world data. This work investigates both computationally efficient procedures and fundamental statistical limits for clustering in high-dimensional anisotropic mixtures. We propose a new clustering method, Covariance Projected Spectral Clustering (COPO), which adapts to a wide range of dependent noise structures. We first project the data onto a low-dimensional space via eigen-decomposition of a diagonal-deleted Gram matrix. Our central methodological idea is to sharpen clustering in this embedding space by a covariance-aware reassignment step, using quadratic distances induced by estimated projected covariances. Through a novel row-wise analysis of the subspace estimation step in weak-signal regimes, which is of independent interest, we establish tight performance guarantees and algorithmic upper bounds for COPO, covering both Gaussian noise with flexible covariance and general noise with local dependence. To characterize the fundamental difficulty of clustering high-dimensional anisotropic Gaussian mixtures, we further establish two distinct and complementary minimax lower bounds, each highlighting different covariance-driven barriers. Our results show that COPO attains minimax-optimal misclustering rates in Gaussian settings. Extensive simulation studies across diverse noise structures, along with a real data application, demonstrate the superior empirical performance of our method.

翻译：在混合模型中，各簇内的各向异性噪声广泛存在于现实世界数据中。本研究探讨了高维各向异性混合模型聚类的计算高效方法与基本统计极限。我们提出了一种新的聚类方法——协方差投影谱聚类（COPO），该方法能适应广泛的依赖噪声结构。我们首先通过对角线删除格拉姆矩阵的特征分解，将数据投影到低维空间。我们的核心方法思想是，通过一个协方差感知的重分配步骤，利用估计的投影协方差诱导的二次距离，在此嵌入空间中锐化聚类。通过对弱信号区域中子空间估计步骤进行新颖的行方向分析（这本身也具有独立的研究价值），我们为COPO建立了严格的性能保证和算法上界，涵盖了具有灵活协方差的高斯噪声和具有局部依赖的一般噪声。为了刻画高维各向异性高斯混合模型聚类的根本困难，我们进一步建立了两个不同且互补的极小极大下界，每个下界都突出了不同的协方差驱动障碍。我们的结果表明，COPO在高斯设置下达到了极小极大最优的误聚类率。跨多种噪声结构的广泛模拟研究以及一个真实数据应用，证明了我们方法优越的实证性能。