# Cross-Space Adaptive Filter: Integrating Graph Topology and Node Attributes for Alleviating the Over-smoothing Problem

arXiv: https://arxiv.org/abs/2401.14876 · HTML: https://mlsanity.com/paper/2401.14876/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 51 references that are in the map.
- First posted on arXiv 26 Jan 2024.

## Builds on (most cited first, 25 of 25)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [Simplifying Graph Convolutional Networks](https://mlsanity.com/paper/1902.07153/) (2019)
- [Predict then Propagate: Graph Neural Networks meet Personalized PageRank](https://mlsanity.com/paper/1810.05997/) (2018)
- [Representation Learning on Graphs with Jumping Knowledge Networks](https://mlsanity.com/paper/1806.03536/) (2018)
- [Adaptive Universal Generalized PageRank Graph Neural Network](https://mlsanity.com/paper/2006.07988/) (2020)
- [Beyond Low-frequency Information in Graph Convolutional Networks](https://mlsanity.com/paper/2101.00797/) (2021)
- [PairNorm: Tackling Oversmoothing in GNNs](https://mlsanity.com/paper/1909.12223/) (2019)
- [Design Space for Graph Neural Networks](https://mlsanity.com/paper/2011.08843/) (2020)
- [Node Similarity Preserving Graph Convolutional Networks](https://mlsanity.com/paper/2011.09643/) (2020)
- [Be Confident! Towards Trustworthy Graph Neural Networks via Confidence Calibration](https://mlsanity.com/paper/2109.14285/) (2021)
- [Convolutional Neural Networks on Graphs with Chebyshev Approximation, Revisited](https://mlsanity.com/paper/2202.03580/) (2022)
- [A Survey on Graph Structure Learning: Progress and Opportunities](https://mlsanity.com/paper/2103.03036/) (2021)
- [Why Do Deep Residual Networks Generalize Better than Deep Feedforward Networks? -- A Neural Tangent Kernel Perspective](https://mlsanity.com/paper/2002.06262/) (2020)
- [Single Image Defocus Deblurring Using Kernel-Sharing Parallel Atrous Convolutions](https://mlsanity.com/paper/2108.09108/) (2021)
- [Label Efficient Semi-Supervised Learning via Graph Filtering](https://mlsanity.com/paper/1901.09993/) (2019)
- [AdaGNN: Graph Neural Networks with Adaptive Frequency Response Filter](https://mlsanity.com/paper/2104.12840/) (2021)
- [Confidence May Cheat: Self-Training on Graph Neural Networks under Distribution Shift](https://mlsanity.com/paper/2201.11349/) (2022)
- [Random Feature Expansions for Deep Gaussian Processes](https://mlsanity.com/paper/1610.04386/) (2016)
- [JuryGCN: Quantifying Jackknife Uncertainty on Graph Convolutional Networks](https://mlsanity.com/paper/2210.05959/) (2022)
- [Relation Learning on Social Networks with Multi-Modal Graph Edge Variational Autoencoders](https://mlsanity.com/paper/1911.05465/) (2019)
- [Kernel Mean Estimation and Stein's Effect](https://mlsanity.com/paper/1306.0842/) (2013)
- [Message Passing in Graph Convolution Networks via Adaptive Filter Banks](https://mlsanity.com/paper/2106.09910/) (2021)
- [Kernel Mean Estimation via Spectral Filtering](https://mlsanity.com/paper/1411.0900/) (2014)

## Built on by (most cited first, 4 of 4)

- [ReMindRAG: Low-Cost LLM-Guided Knowledge Graph Traversal for Efficient RAG](https://mlsanity.com/paper/2510.13193/) (2025)
- [Grad: Guided Relation Diffusion Generation for Graph Augmentation in Graph Fraud Detection](https://mlsanity.com/paper/2512.18133/) (2025)
- [Adaptive Local Clustering over Attributed Graphs](https://mlsanity.com/paper/2503.20488/) (2025)
- [CONSTRUCTA: Automating Commercial Construction Schedules in Fabrication Facilities with Large Language Models](https://mlsanity.com/paper/2502.12066/) (2025)

## Abstract (arXiv)

The vanilla Graph Convolutional Network (GCN) uses a low-pass filter to extract low-frequency signals from graph topology, which may lead to the over-smoothing problem when GCN goes deep. To this end, various methods have been proposed to create an adaptive filter by incorporating an extra filter (e.g., a high-pass filter) extracted from the graph topology. However, these methods heavily rely on topological information and ignore the node attribute space, which severely sacrifices the expressive power of the deep GCNs, especially when dealing with disassortative graphs. In this paper, we propose a cross-space adaptive filter, called CSF, to produce the adaptive-frequency information extracted from both the topology and attribute spaces. Specifically, we first derive a tailored attribute-based high-pass filter that can be interpreted theoretically as a minimizer for semi-supervised kernel ridge regression. Then, we cast the topology-based low-pass filter as a Mercer's kernel within the context of GCNs. This serves as a foundation for combining it with the attribute-based filter to capture the adaptive-frequency information. Finally, we derive the cross-space filter via an effective multiple-kernel learning strategy, which unifies the attribute-based high-pass filter and the topology-based low-pass filter. This helps to address the over-smoothing problem while maintaining effectiveness. Extensive experiments demonstrate that CSF not only successfully alleviates the over-smoothing problem but also promotes the effectiveness of the node classification task.