群不变谱嵌入

中文摘要 AI 辅助

谱嵌入方法广泛用于具有内在低维结构的高维数据集的降维和聚类。许多实际感兴趣的数据集在旋转等对称下具有不变性，但标准谱嵌入方法未考虑，将对称相关数据点视为无关。本文方法是将对称直接纳入用于谱嵌入的亲和核。分析了具有紧致李群\(G\)给出对称的黎曼数据流形\(M\)的情况，证明在合适条件下，由三种不变核构造的图拉普拉斯算子在商空间\(M/G\)上逐点收敛到显式二阶微分算子。分析表明随着有效维度根据群的维度下降，收敛速度提高。在具有\(\mathrm{SO}(2)\)或\(\mathrm{SO}(3)\)对称的数据集上验证了该方法，结果表明\(G\)不变谱嵌入能恢复数据的内在几何，而标准谱嵌入即使在无限数据极限下也无法做到。

英文摘要

Spectral embedding methods are widely used for dimensionality reduction and clustering of high-dimensional datasets with intrinsic low-dimensional structures. Although many datasets of practical interest exhibit invariance under symmetries such as rotations, standard spectral embedding methods do not account for this, treating symmetry-related data points as unrelated. Our approach to this problem is to incorporate the symmetries directly into the affinity kernels used for spectral embedding. We analyze the case of a Riemannian data manifold $M$ with symmetries given by a compact Lie group~$G$ and prove that, under suitable conditions, graph Laplacians constructed from three types of invariant kernels converge pointwise to explicit second-order differential operators on the quotient space $M/G$. Our analysis implies improved convergence rates, as the effective dimension drops according to the dimension of the group. We validate our approach on datasets with $\mathrm{SO}(2)$ or $\mathrm{SO}(3)$ symmetry, and show that $G$-invariant spectral embedding recovers the intrinsic geometry of the data, in contrast to standard spectral embedding, which fails to do so even in the limit of infinite data.

发表机构

• Tel Aviv University（特拉维夫大学）
• University of Texas at Austin（德克萨斯大学奥斯汀分校）
• KTH Royal Institute of Technology（皇家理工学院）

机构由 AI 辅助整理，请以论文原文为准。