李群代数卷积滤波器
Lie Group Algebra Convolutional Filters
AI总结：
提出基于代数信号处理的李群代数卷积滤波器框架，通过指数映射与滤波器/信号分离采样实现任意空间信号处理，并给出唯一性、误差界、稳定性及纽结分类等实验验证。
摘要：
本文提出一个利用代数信号处理（ASP）在任意空间上利用李群对称性的框架。我们表明，传统群卷积是更一般的李群代数同态的一个特定实例，该同态与根植于给定李群 G 的李群代数 L¹(G) 中的代数信号模型相关联。利用这一事实，我们解耦了李群卷积的离散化，阐明了两个独立的采样实例：滤波器与信号。为离散化滤波器，我们利用连接李群及其相关李代数的指数映射。我们表明，从数据中学习到的离散李群滤波器在 L¹(G) 中确定了唯一滤波器，并说明在给定谱表示的情况下，这种表示唯一性如何由滤波器带宽定义。我们还推导了 L¹(G) 中滤波器相对于其学习到的离散表示进行近似的误差界。所提出框架允许处理任意维度空间上的信号，并且在这些空间中群的某些元素的作用不一定定义良好。最后，我们表明多图卷积信号模型是李群信号处理模型的自然离散实现，并利用这一联系建立李群代数滤波器的稳定性结果。为数值评估我们的结果，我们用这些滤波器构建神经网络，并将其应用于多个数据集，包括一个纽结分类问题。
英文摘要：
In this paper we propose a framework to leverage Lie group symmetries on arbitrary spaces exploiting \textit{algebraic signal processing} (ASP). We show that traditional group convolutions are one particular instantiation of a more general Lie group algebra homomorphism associated to an algebraic signal model rooted in the Lie group algebra $L^{1}(G)$ for given Lie group $G$. Exploiting this fact, we decouple the discretization of the Lie group convolution elucidating two separate sampling instances: the filter and the signal. To discretize the filters, we exploit the exponential map that links a Lie group with its associated Lie algebra. We show that the discrete Lie group filter learned from the data determines a unique filter in $L^{1}(G)$, and we show how this uniqueness of representation is defined by the bandwidth of the filter given a spectral representation. We also derive error bounds for the approximations of the filters in $L^{1}(G)$ with respect to its learned discrete representations. The proposed framework allows the processing of signals on spaces of arbitrary dimension and where the actions of some elements of the group are not necessarily well defined. Finally, we show that multigraph convolutional signal models come as the natural discrete realization of Lie group signal processing models, and we use this connection to establish stability results for Lie group algebra filters. To evaluate numerically our results, we build neural networks with these filters and we apply them in multiple datasets, including a knot classification problem.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
