Closed Form Solutions of Combinatorial Graph Laplacian Estimation under Acyclic Topology Constraints

TL;DR

This paper derives a closed-form solution for estimating acyclic graph Laplacians from data, enabling efficient tree graph construction and improved image denoising performance.

Contribution

It provides a closed-form solution for acyclic graph Laplacian estimation, reducing computational complexity and enhancing graph-based image denoising methods.

Findings

Closed-form solution for acyclic graph Laplacian estimation.

Efficient construction of tree graphs from limited data.

Improved image denoising results using learned graph weights.

Abstract

How to obtain a graph from data samples is an important problem in graph signal processing. One way to formulate this graph learning problem is based on Gaussian maximum likelihood estimation, possibly under particular topology constraints. To solve this problem, we typically require iterative convex optimization solvers. In this paper, we show that when the target graph topology does not contain any cycle, then the solution has a closed form in terms of the empirical covariance matrix. This enables us to efficiently construct a tree graph from data, even if there is only a single data sample available. We also provide an error bound of the objective function when we use the same solution to approximate a cyclic graph. As an example, we consider an image denoising problem, in which for each input image we construct a graph based on the theoretical result. We then apply low-pass graph filters based on this graph. Experimental results show that the weights given by the graph learning solution lead to better denoising results than the bilateral weights under some conditions.