发表机构
Federal University of São Carlos（圣卡洛斯联邦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出SHOPCA，一种融入微分几何信息的无监督降维方法，通过混合系数α正则化协方差矩阵，基于谱特征间隙无监督选α，在50余个数据集上较PCA提升聚类质量，小样本场景优于UMAP。
AI 中文摘要
本文提出了SHOPCA（基于形状算子的主成分分析），一种将微分几何信息融入经典主成分分析（PCA）协方差结构的新型无监督度量学习与降维方法。SHOPCA使用平均形状算子对全局协方差矩阵进行正则化，该算子定义为从数据流形估计得到的局部形状算子绝对值的平均值，引导主成分向同时具有最大方差和有效曲率的方向移动。单个经迹归一化的混合系数α控制正则化过程，当α=0时恢复标准PCA，当α→∞时得到曲率驱动的嵌入。我们进一步引入一种完全无监督的准则，基于正则化协方差矩阵的谱特征间隙选择α，在不使用类标签的情况下最大化前d个特征值与其余特征值之间的相对分离度。我们在50多个真实基准数据集上对SHOPCA进行评估，与PCA、ISOMAP和UMAP使用调整兰德指数（ARI）、标准化互信息（NMI）、Fowlkes-Mallows指数（FM）和V测度进行比较。结果表明，在广泛的数据集范围内，SHOPCA相较于PCA始终提升聚类质量，且在小样本场景下优于UMAP——此时基于迭代邻域的流形估计可能会出现性能下降。SHOPCA计算上易于处理、参数高效，适用于需要完全无监督、几何感知降维的领域。
英文摘要
In this paper, we propose SHOPCA (Shape Operator-based Principal Component Analysis), a novel method for unsupervised metric learning and dimensionality reduction that incorporates differential geometric information into the covariance structure of classical PCA. SHOPCA regularizes the global covariance matrix using the mean shape operator, defined as the average of the absolute local shape operators estimated from the data manifold, steering principal components toward directions of both maximum variance and informative curvature. A single trace-normalized mixing coefficient $α$ controls the regularization, recovering standard PCA at $α= 0$ and a curvature-driven embedding as $α\to \infty$. We further introduce a fully unsupervised criterion for selecting $α$ based on the spectral eigengap of the regularized covariance matrix, maximizing the relative separation between the top-$d$ and remaining eigenvalues without using class labels. We evaluate SHOPCA on more than 50 real-world benchmark datasets, comparing it with PCA, ISOMAP, and UMAP using Adjusted Rand Index (ARI), Normalized Mutual Information (NMI), Fowlkes-Mallows index (FM), and V-measure. Results show that SHOPCA consistently improves clustering quality over PCA across a broad range of datasets and surpasses UMAP on small-sample settings, where iterative neighborhood-based manifold estimation can degrade. SHOPCA is computationally tractable, parameter-efficient, and applicable to domains requiring fully unsupervised, geometry-aware dimensionality reduction.
Comments23 pages, 4 figures, 4 tables
