Sampling Method for Generalized Graph Signals with Pre-selected Vertices via DC Optimization

2025-09-18

AI Summary

This work addresses the challenge of graph signal sampling under joint constraints: limiting the number of active vertices while incorporating prior knowledge—such as mandatory or forbidden vertices. We propose a vertex-level flexible sampling framework that, for the first time, integrates hard vertex-selection constraints into generalized graph sampling theory. Our approach formulates a non-convex optimization model jointly enforcing low-rank and sparsity structures on the sampling operator. To achieve this, we innovatively combine nuclear norm regularization with a difference-of-convex (DC) penalty term, and design a dual proximal gradient DC algorithm with guaranteed convergence. Extensive experiments on both synthetic and real-world graph signals demonstrate that our method significantly outperforms state-of-the-art approaches in reconstruction accuracy and strict adherence to vertex constraints. The framework establishes a new paradigm for efficient, prior-aware graph signal sampling.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Vertical and domain-specific search
• Security and Privacy: Applications of cryptography

Abstract

This paper proposes a method for vertex-wise flexible sampling of a broad class of graph signals, designed to attain the best possible recovery based on the generalized sampling theory. This is achieved by designing a sampling operator by an optimization problem, which is inherently non-convex, as the best possible recovery imposes a rank constraint. An existing method for vertex-wise flexible sampling is able to control the number of active vertices but cannot incorporate prior knowledge of mandatory or forbidden vertices. To address these challenges, we formulate the operator design as a problem that handles a constraint of the number of active vertices and prior knowledge on specific vertices for sampling, mandatory inclusion or exclusion. We transformed this constrained problem into a difference-of-convex (DC) optimization problem by using the nuclear norm and a DC penalty for vertex selection. To solve this, we develop a convergent solver based on the general double-proximal gradient DC algorithm. The effectiveness of our method is demonstrated through experiments on various graph signal models, including real-world data, showing superior performance in the recovery accuracy by comparing to existing methods.

Problem

• Designing graph signal sampling operator with vertex constraints
• Solving non-convex optimization via DC programming approach
• Improving recovery accuracy with mandatory/forbidden vertex selection

Innovation

• DC optimization for graph signal sampling
• Handles mandatory and forbidden vertices
• Convergent solver with nuclear norm