# CoVariance Filters and Neural Networks over Hilbert Spaces

arXiv: https://arxiv.org/abs/2509.13178 · HTML: https://mlsanity.com/paper/2509.13178/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 31 references that are in the map.
- First posted on arXiv 16 Sep 2025.

## Builds on (most cited first, 19 of 19)

- [The UCR Time Series Archive](https://mlsanity.com/paper/1810.07758/) (2018)
- [Quantum Machine Learning](https://mlsanity.com/paper/1611.09347/) (2016)
- [Stability Properties of Graph Neural Networks](https://mlsanity.com/paper/1905.04497/) (2019)
- [Graph Neural Networks in Network Neuroscience](https://mlsanity.com/paper/2106.03535/) (2021)
- [Graph Filters for Signal Processing and Machine Learning on Graphs](https://mlsanity.com/paper/2211.08854/) (2022)
- [Stationary Graph Processes and Spectral Estimation](https://mlsanity.com/paper/1603.04667/) (2016)
- [Stationary signal processing on graphs](https://mlsanity.com/paper/1601.02522/) (2016)
- [Tangent Bundle Convolutional Learning: from Manifolds to Cellular Sheaves and Back](https://mlsanity.com/paper/2303.11323/) (2023)
- [coVariance Neural Networks](https://mlsanity.com/paper/2205.15856/) (2022)
- [Explainable Brain Age Prediction using coVariance Neural Networks](https://mlsanity.com/paper/2305.18370/) (2023)
- [Algorithms for Learning Graphs in Financial Markets](https://mlsanity.com/paper/2012.15410/) (2020)
- [Fast Robust PCA on Graphs](https://mlsanity.com/paper/1507.08173/) (2015)
- [Learning Undirected Graphs in Financial Markets](https://mlsanity.com/paper/2005.09958/) (2020)
- [Deep Transfer Learning with Graph Neural Network for Sensor-Based Human Activity Recognition](https://mlsanity.com/paper/2203.07910/) (2022)
- [Spatiotemporal Covariance Neural Networks](https://mlsanity.com/paper/2409.10068/) (2024)
- [Fair CoVariance Neural Networks](https://mlsanity.com/paper/2409.08558/) (2024)
- [Transferability of coVariance Neural Networks and Application to Interpretable Brain Age Prediction using Anatomical Features](https://mlsanity.com/paper/2305.01807/) (2023)
- [Covariance Density Neural Networks](https://mlsanity.com/paper/2505.11139/) (2025)
- [Convolutional Filtering with RKHS Algebras](https://mlsanity.com/paper/2411.01341/) (2024)

## Abstract (arXiv)

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.