# Vector-Valued Graph Trend Filtering with Non-Convex Penalties

## TL;DR

This paper introduces a non-convex regularization approach for denoising vector-valued graph signals, demonstrating improved recovery performance and providing theoretical error bounds and an efficient algorithm.

## Contribution

It extends graph trend filtering to vector-valued signals with non-convex penalties, offering better denoising and recovery capabilities along with convergence guarantees.

## Key findings

- Non-convex regularizers outperform convex ones in denoising tasks.
- The proposed ADMM algorithm converges reliably.
- Numerical experiments validate improved performance on real-world data.

## Abstract

This work studies the denoising of piecewise smooth graph signals that exhibit inhomogeneous levels of smoothness over a graph, where the value at each node can be vector-valued. We extend the graph trend filtering framework to denoising vector-valued graph signals with a family of non-convex regularizers, which exhibit superior recovery performance over existing convex regularizers. Using an oracle inequality, we establish the statistical error rates of first-order stationary points of the proposed non-convex method for generic graphs. Furthermore, we present an ADMM-based algorithm to solve the proposed method and establish its convergence. Numerical experiments are conducted on both synthetic and real-world data for denoising, support recovery, event detection, and semi-supervised classification.