将卷积网络扩展到无限维希尔伯特空间，提升信号处理的鲁棒性与泛化能力。
CoVariance Filters and Neural Networks over Hilbert Spaces
- 基于协方差算子构建希尔伯特空间上的滤波器，实现对高维信号的卷积学习。
- 实验证明其能恢复滤波后信号的函数主成分分析（FPCA）结构，性能优于MLP和传统方法。
- 适用于时间序列等复杂信号，特别适合需要强泛化能力的场景。
协方差神经网络（VNNs）在有限维希尔伯特空间上对信号的样本协方差矩阵进行图卷积，因其鲁棒性和可迁移性而受到关注。然而，该理论在无限维希尔伯特空间中的扩展仍不明确。本文首次提出一种面向无限维希尔伯特空间信号的新卷积学习框架，核心为（样本）协方差算子。我们构造性地定义了希尔伯特协方差滤波器（HVFs），并设计了由非线性激活连接的多滤波器堆叠架构——希尔伯特协方差网络（HVNs）。提出了一种合理的离散化方法，并证明了经验型HVFs可恢复滤波后信号的函数主成分分析（FPCA）结构。通过从多元实值函数到再生核希尔伯特空间（RKHS）的多种示例展示了框架的通用性。最后，在合成与真实时间序列分类任务上验证了HVNs，结果表明其性能优于全连接网络（MLP）和基于FPCA的分类器。
原文摘要 · Abstract (English)
CoVariance Neural Networks (VNNs) perform graph convolutions on the empirical covariance matrix of signals defined over finite-dimensional Hilbert spaces, motivated by robustness and transferability properties. Yet, little is known about how these arguments extend to infinite-dimensional Hilbert spaces. In this work, we take a first step by introducing a novel convolutional learning framework for signals defined over infinite-dimensional Hilbert spaces, centered on the (empirical) covariance operator. We constructively define Hilbert coVariance Filters (HVFs) and design Hilbert coVariance Networks (HVNs) as stacks of HVF filterbanks with nonlinear activations. We propose a principled discretization procedure, and we prove that empirical HVFs can recover the Functional PCA (FPCA) of the filtered signals. We then describe the versatility of our framework with examples ranging from multivariate real-valued functions to reproducing kernel Hilbert spaces. Finally, we validate HVNs on both synthetic and real-world time-series classification tasks, showing robust performance compared to MLP and FPCA-based classifiers.
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