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Graph Spectral Filtering with Chebyshev Interpolation for Recommendation

Chanwoo Kim, Jinkyu Sung, Yebonn Han, Joonseok Lee

TL;DR

This work targets two fundamental bottlenecks in graph-convolutional collaborative filtering: the limited expressiveness of embeddings and the overly simple neighborhood aggregation. It introduces ChebyCF, a graph spectral filtering framework that operates directly on raw interaction signals and uses Chebyshev interpolation to approximate a flexible non-linear transfer on an item-item graph, complemented by an ideal pass filter and degree-based normalization. The method achieves state-of-the-art results on multiple benchmarks (e.g., LastFM, Gowalla, Amazon-Book) with faster inference than prior top models, highlighting the value of full-spectrum, learnable spectral filtering in CF. By shifting from embedding-centered to spectral-filter-centered reasoning, ChebyCF demonstrates practical advantages in both accuracy and efficiency, suggesting a promising direction for future spectral-method–driven CF research.

Abstract

Graph convolutional networks have recently gained prominence in collaborative filtering (CF) for recommendations. However, we identify potential bottlenecks in two foundational components. First, the embedding layer leads to a latent space with limited capacity, overlooking locally observed but potentially valuable preference patterns. Also, the widely-used neighborhood aggregation is limited in its ability to leverage diverse preference patterns in a fine-grained manner. Building on spectral graph theory, we reveal that these limitations stem from graph filtering with a cut-off in the frequency spectrum and a restricted linear form. To address these issues, we introduce ChebyCF, a CF framework based on graph spectral filtering. Instead of a learned embedding, it takes a user's raw interaction history to utilize the full spectrum of signals contained in it. Also, it adopts Chebyshev interpolation to effectively approximate a flexible non-linear graph filter, and further enhances it by using an additional ideal pass filter and degree-based normalization. Through extensive experiments, we verify that ChebyCF overcomes the aforementioned bottlenecks and achieves state-of-the-art performance across multiple benchmarks and reasonably fast inference. Our code is available at https://github.com/chanwoo0806/ChebyCF.

Graph Spectral Filtering with Chebyshev Interpolation for Recommendation

TL;DR

This work targets two fundamental bottlenecks in graph-convolutional collaborative filtering: the limited expressiveness of embeddings and the overly simple neighborhood aggregation. It introduces ChebyCF, a graph spectral filtering framework that operates directly on raw interaction signals and uses Chebyshev interpolation to approximate a flexible non-linear transfer on an item-item graph, complemented by an ideal pass filter and degree-based normalization. The method achieves state-of-the-art results on multiple benchmarks (e.g., LastFM, Gowalla, Amazon-Book) with faster inference than prior top models, highlighting the value of full-spectrum, learnable spectral filtering in CF. By shifting from embedding-centered to spectral-filter-centered reasoning, ChebyCF demonstrates practical advantages in both accuracy and efficiency, suggesting a promising direction for future spectral-method–driven CF research.

Abstract

Graph convolutional networks have recently gained prominence in collaborative filtering (CF) for recommendations. However, we identify potential bottlenecks in two foundational components. First, the embedding layer leads to a latent space with limited capacity, overlooking locally observed but potentially valuable preference patterns. Also, the widely-used neighborhood aggregation is limited in its ability to leverage diverse preference patterns in a fine-grained manner. Building on spectral graph theory, we reveal that these limitations stem from graph filtering with a cut-off in the frequency spectrum and a restricted linear form. To address these issues, we introduce ChebyCF, a CF framework based on graph spectral filtering. Instead of a learned embedding, it takes a user's raw interaction history to utilize the full spectrum of signals contained in it. Also, it adopts Chebyshev interpolation to effectively approximate a flexible non-linear graph filter, and further enhances it by using an additional ideal pass filter and degree-based normalization. Through extensive experiments, we verify that ChebyCF overcomes the aforementioned bottlenecks and achieves state-of-the-art performance across multiple benchmarks and reasonably fast inference. Our code is available at https://github.com/chanwoo0806/ChebyCF.
Paper Structure (20 sections, 3 theorems, 20 equations, 9 figures, 5 tables)

This paper contains 20 sections, 3 theorems, 20 equations, 9 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem Formulation
  4. Spectral Graph Theory
  5. GCN-based CF
  6. Motivation
  7. Methodology
  8. Graph and Signal
  9. Chebyshev Filter
  10. Filter Enhancement
  11. The Proposed Method: ChebyCF
  12. Experiments
  13. Experimental Setup
  14. Overall Performance
  15. Effectiveness Analysis
  16. ...and 5 more sections

Key Result

theorem 1

Consider the Linear Graph Convolutional Network (LGCN) heLightGCNSimplifyingPowering2020, where propagation is performed via the neighborhood aggregation as in eq:gcn-cf-(ii), and the embedding matrices $\hat{\mathbf{E}}_\mathcal{U} \in \mathbb{R}^{\mathcal{|U|} \times d }$ and $\hat{\mathbf{E}}_\m with $d \le \min(|\mathcal{U}|, |\mathcal{I}|)$ as the embedding dimension and $\Tilde{\mathbf{R}}$

Figures (9)

  • Figure 1: Graph spectral view in movie recommendation. Red denotes movies watched by the user.
  • Figure 2: In the spectral view, LGCN's neighborhood aggregation with an embedding layer corresponds to linear low-pass filtering with a cutoff.
  • Figure 3: An illustration of the proposed ChebyCF.
  • Figure 4: Visualization of the linear (left) and plateau (right) transfer functions. (Frequency $\lambda$ is rescaled to $[-1,1]$ for plateau as described in \ref{['eq:cheby_filter']}.)
  • Figure 5: The best transfer functions found for ChebyCF.
  • ...and 4 more figures

Theorems & Definitions (7)

  • definition 1: Graph Filter
  • theorem 1
  • definition 2: Chebyshev Polynomials
  • definition 3: Chebyshev Nodes
  • definition 4: Chebyshev Interpolation
  • lemma 1: Eckart-Young-Mirsky Theorem eckartApproximationOneMatrix1936
  • lemma 2