# Filter Learning for Subgraphs: Algebras and Performance Risk Bounds

arXiv: https://arxiv.org/abs/2607.21263 · HTML: https://mlsanity.com/paper/2607.21263/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 48 references that are in the map.
- First posted on arXiv 23 Jul 2026.

## Builds on (most cited first, 21 of 21)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Signal Processing: Overview, Challenges and Applications](https://mlsanity.com/paper/1712.00468/) (2017)
- [PyTorch Geometric Temporal: Spatiotemporal Signal Processing with Neural Machine Learning Models](https://mlsanity.com/paper/2104.07788/) (2021)
- [Connecting the Dots: Identifying Network Structure via Graph Signal Processing](https://mlsanity.com/paper/1810.13066/) (2018)
- [Stationary Graph Processes and Spectral Estimation](https://mlsanity.com/paper/1603.04667/) (2016)
- [EdgeNets:Edge Varying Graph Neural Networks](https://mlsanity.com/paper/2001.07620/) (2020)
- [Stationary signal processing on graphs](https://mlsanity.com/paper/1601.02522/) (2016)
- [Signal Recovery on Graphs: Variation Minimization](https://mlsanity.com/paper/1411.7414/) (2014)
- [An elementary analysis of ridge regression with random design](https://mlsanity.com/paper/2203.08564/) (2022)
- [Bayesian Estimation of Graph Signals](https://mlsanity.com/paper/2103.15520/) (2021)
- [Subgraph-based filterbanks for graph signals](https://mlsanity.com/paper/1509.05642/) (2015)
- [A Hilbert Space Theory of Generalized Graph Signal Processing](https://mlsanity.com/paper/1904.11655/) (2019)
- [Graph Signal Processing over a Probability Space of Shift Operators](https://mlsanity.com/paper/2108.09192/) (2021)
- [Reconstruction of Graph Signals through Percolation from Seeding Nodes](https://mlsanity.com/paper/1507.08364/) (2015)
- [Kernel Based Reconstruction for Generalized Graph Signal Processing](https://mlsanity.com/paper/2308.06949/) (2023)
- [A Graph Signal Processing Perspective of Network Multiple Hypothesis Testing with False Discovery Rate Control](https://mlsanity.com/paper/2408.03142/) (2024)
- [Polynomial graph filter of multiple shifts and distributed implementation of inverse filtering](https://mlsanity.com/paper/2003.11152/) (2020)
- [Graph Signal Sampling Under Stochastic Priors](https://mlsanity.com/paper/2206.00382/) (2022)
- [Partition-wise Graph Filtering: A Unified Perspective Through the Lens of Graph Coarsening](https://mlsanity.com/paper/2505.14033/) (2025)
- [Wide-Sense Stationarity in Generalized Graph Signal Processing](https://mlsanity.com/paper/2112.01127/) (2021)
- [Spatio-Temporal Multi-Subgraph GCN for 3D Human Motion Prediction](https://mlsanity.com/paper/2501.00317/) (2024)

## Abstract (arXiv)

Graph signal processing tasks that leverage spectral information typically assume access to the complete graph topology, which is often unavailable in practice. We propose a systematic framework for subgraph filter learning (SFL), where subgraph-supported operators approximate ambient graph filters under partial observations. We formulate SFL as a statistical learning problem in which optimal subgraph operators are inherently data-dependent. To address the difficulty of directly estimating such operators, we develop a subgraph filter algebra based on distance-aware Laplacian constructions, defining a structured and controllable class of filters for effective approximation. We further establish performance risk bounds under the least squares loss, quantifying how well the learned operator approximates the restricted ambient mapping. Experiments real-world datasets show that, for SFL tasks, the proposed algebraic models consistently outperform polynomial filters, distribution-agnostic operators, and direct numerical filter learning baselines that attempt to recover the underlying structure from data.