# Bias-Variance Tradeoff of Graph Laplacian Regularizer

## TL;DR

This paper analyzes the bias-variance tradeoff of the graph Laplacian regularizer, providing a spectral scaling law for optimal regularization and demonstrating its effectiveness across various graph signal applications.

## Contribution

It introduces a spectral analysis of the regularizer's bias-variance tradeoff and derives a scaling law for the optimal regularization parameter based on graph properties.

## Key findings

- Optimal regularization parameter scales with spectral graph properties.
- Mediocre regularization choices are often suboptimal.
- Experiments confirm near-optimal performance on synthetic and real graphs.

## Abstract

This paper presents a bias-variance tradeoff of graph Laplacian regularizer, which is widely used in graph signal processing and semi-supervised learning tasks. The scaling law of the optimal regularization parameter is specified in terms of the spectral graph properties and a novel signal-to-noise ratio parameter, which suggests selecting a mediocre regularization parameter is often suboptimal. The analysis is applied to three applications, including random, band-limited, and multiple-sampled graph signals. Experiments on synthetic and real-world graphs demonstrate near-optimal performance of the established analysis.