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PolyCF: Towards the Optimal Spectral Graph Filters for Collaborative Filtering

Yifang Qin, Wei Ju, Xiao Luo, Yiyang Gu, Zhiping Xiao, Ming Zhang

TL;DR

The paper addresses the expressiveness limits of embedding-based collaborative filtering by reframing CF as a graph signal processing task. It introduces PolyCF, a graph-filter-based model built on a generalized Gram convolution framework that aggregates multiple eigenspaces and can approximate the optimal polynomial filter for recovering missing interactions. The method couples a graph-optimization objective with Bayesian ranking to learn a flexible polynomial kernel and a low-pass enhancement, yielding state-of-the-art results on three real-world datasets. This approach offers robust performance in sparse settings and provides interpretable insights via spectral-filter visualizations, suggesting a practical path forward for spectral CF models.

Abstract

Collaborative Filtering (CF) is a pivotal research area in recommender systems that capitalizes on collaborative similarities between users and items to provide personalized recommendations. With the remarkable achievements of node embedding-based Graph Neural Networks (GNNs), we explore the upper bounds of expressiveness inherent to embedding-based methodologies and tackle the challenges by reframing the CF task as a graph signal processing problem. To this end, we propose PolyCF, a flexible graph signal filter that leverages polynomial graph filters to process interaction signals. PolyCF exhibits the capability to capture spectral features across multiple eigenspaces through a series of Generalized Gram filters and is able to approximate the optimal polynomial response function for recovering missing interactions. A graph optimization objective and a pair-wise ranking objective are jointly used to optimize the parameters of the convolution kernel. Experiments on three widely adopted datasets demonstrate the superiority of PolyCF over current state-of-the-art CF methods. Moreover, comprehensive studies empirically validate each component's efficacy in the proposed PolyCF.

PolyCF: Towards the Optimal Spectral Graph Filters for Collaborative Filtering

TL;DR

The paper addresses the expressiveness limits of embedding-based collaborative filtering by reframing CF as a graph signal processing task. It introduces PolyCF, a graph-filter-based model built on a generalized Gram convolution framework that aggregates multiple eigenspaces and can approximate the optimal polynomial filter for recovering missing interactions. The method couples a graph-optimization objective with Bayesian ranking to learn a flexible polynomial kernel and a low-pass enhancement, yielding state-of-the-art results on three real-world datasets. This approach offers robust performance in sparse settings and provides interpretable insights via spectral-filter visualizations, suggesting a practical path forward for spectral CF models.

Abstract

Collaborative Filtering (CF) is a pivotal research area in recommender systems that capitalizes on collaborative similarities between users and items to provide personalized recommendations. With the remarkable achievements of node embedding-based Graph Neural Networks (GNNs), we explore the upper bounds of expressiveness inherent to embedding-based methodologies and tackle the challenges by reframing the CF task as a graph signal processing problem. To this end, we propose PolyCF, a flexible graph signal filter that leverages polynomial graph filters to process interaction signals. PolyCF exhibits the capability to capture spectral features across multiple eigenspaces through a series of Generalized Gram filters and is able to approximate the optimal polynomial response function for recovering missing interactions. A graph optimization objective and a pair-wise ranking objective are jointly used to optimize the parameters of the convolution kernel. Experiments on three widely adopted datasets demonstrate the superiority of PolyCF over current state-of-the-art CF methods. Moreover, comprehensive studies empirically validate each component's efficacy in the proposed PolyCF.
Paper Structure (31 sections, 2 theorems, 21 equations, 6 figures, 3 tables)

This paper contains 31 sections, 2 theorems, 21 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related Works
  3. Collaborative Filtering
  4. Graph Signal Processing
  5. Preliminary
  6. Graph-based Collaborative Filtering
  7. Graph Signal Processing
  8. Methodology
  9. From Embedding to Filter-based Models
  10. Generalized Gram Convolution
  11. Generalized Gram Filter
  12. Polynomial Graph Convolution
  13. Low-pass Enhancement
  14. Comprehensive Polynomial Filter
  15. Model Optimization
  16. ...and 16 more sections

Key Result

theorem 1

For any polynomial graph filter $P(\tilde{A})$, there exists $L,V\in\mathbb{R}^{n\times d}$ that satisfy the relationship: where both $L$ and $V$ are linear combinations of $E_I$ and $\tilde{R}E_U$. The computation of these two matrices depends solely on the polynomial coefficients $\alpha_k$ and the item Gram matrix $\tilde{G}_I$.

Figures (6)

  • Figure 1: Comparison between node embedding-based method (left) and graph filter-based methods (right).
  • Figure 2: A general illustration of the training framework of PolyCF.
  • Figure 3: Model performance with different settings.
  • Figure 4: Model performance w.r.t. cut-off frequency $s$.
  • Figure 5: PolyCF's generalization performance.
  • ...and 1 more figures

Theorems & Definitions (2)

  • theorem 1
  • theorem 2