# Learning Graphs With Monotone Topology Properties and Multiple Connected Components

@article{Pavez2017LearningGW, title={Learning Graphs With Monotone Topology Properties and Multiple Connected Components}, author={Eduardo Pavez and Hilmi E. Egilmez and Antonio Ortega}, journal={IEEE Transactions on Signal Processing}, year={2017}, volume={66}, pages={2399-2413} }

Recent papers have formulated the problem of learning graphs from data as an inverse covariance estimation problem with graph Laplacian constraints. While such problems are convex, existing methods cannot guarantee that solutions will have specific graph topology properties (e.g., being a tree), which are desirable for some applications. The problem of learning a graph with topology properties is in general non-convex. In this paper, we propose an approach to solve these problems by decomposing…

## 44 Citations

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

- Computer Science
- 2017

This paper shows 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, which enables us to efficiently construct a tree graph from data, even if there is only a single data sample available.

### ON LEARNING LAPLACIANS OF TREE STRUCTURED GRAPHS

- Computer Science2018 IEEE Data Science Workshop (DSW)
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This paper shows that when the target graph topology is known and does not contain any cycle, i.e., it is a tree, then the optimal Laplacian has a closed form in terms of the empirical covariance matrix.

### Learning graphs with monotone topology properties

- Computer Science, Mathematics2017 IEEE Global Conference on Signal and Information Processing (GlobalSIP)
- 2017

This work proposes a tractable algorithm that finds the generalized Laplacian matrix of a graph with the desired type of topology by solving a combinatorial optimization problem to find a graph topology that satisfies the desired structural property.

### A Unified Framework for Structured Graph Learning via Spectral Constraints

- Computer ScienceJ. Mach. Learn. Res.
- 2020

This paper introduces a unified graph learning framework lying at the integration of Gaussian graphical models and spectral graph theory, and develops an optimization framework that leverages graph learning with specific structures via spectral constraints on graph matrices.

### Optimization Algorithms for Graph Laplacian Estimation via ADMM and MM

- Computer ScienceIEEE Transactions on Signal Processing
- 2019

Numerical experiments show that the inclusion of a nominal eigensubspace significantly improves the estimation of the graph Laplacian, which is more evident when the sample size is smaller than or comparable to the problem dimension.

### Joint Network Topology Inference via Structured Fusion Regularization

- Computer Science
- 2021

A general graph estimator based on a novel structured fusion regularization that enables us to jointly learn multiple graph Laplacian matrices with such complex topological patterns, and enjoys both high computational efficiency and rigorous theoretical guarantee is proposed.

### Nonconvex Sparse Graph Learning under Laplacian Constrained Graphical Model

- Computer ScienceNeurIPS
- 2020

This paper proposes a nonconvex penalized maximum likelihood estimation method, and establishes the order of the statistical error, and proves that a large regularization parameter will surprisingly lead to a solution representing a complete graph.

### On Sparse Graph Estimation Under Statistical and Laplacian Constraints

- Computer Science2021 Asia-Pacific Signal and Information Processing Association Annual Summit and Conference (APSIPA ASC)
- 2021

This work augments the objective function of Kalofolias (2016) which is motivated by a signal smoothness viewpoint and imposes a Laplacian constraint, with a penalized log-likelihood objective function with a lasso constraint, motivated from a statistical viewpoint.

### Bayesian Topology Learning and noise removal from network data

- Computer ScienceDiscover Internet of Things
- 2021

A graph topology inference approach is proposed to learn the underlying graph structure from a given set of noisy multi-variate observations, which are modeled as graph signals generated from a Gaussian Markov Random Field (GMRF) process.

### Graph Learning From Filtered Signals: Graph System and Diffusion Kernel Identification

- Computer ScienceIEEE Transactions on Signal and Information Processing over Networks
- 2019

A novel graph signal processing framework for building graph-based models from classes of filtered signals that reduces to a graph Laplacian estimation problem and demonstrates that the proposed algorithm outperforms the current state-of-the-art methods.

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