# Graph Laplacian Learning with Exponential Family Noise

arXiv: https://arxiv.org/abs/2306.08201 · HTML: https://mlsanity.com/paper/2306.08201/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 69 references that are in the map.
- First posted on arXiv 14 Jun 2023.

## Builds on (most cited first, 24 of 24)

- [Graph Signal Processing: Overview, Challenges and Applications](https://mlsanity.com/paper/1712.00468/) (2017)
- [How to learn a graph from smooth signals](https://mlsanity.com/paper/1601.02513/) (2016)
- [Learning graphs from data: A signal representation perspective](https://mlsanity.com/paper/1806.00848/) (2018)
- [Connecting the Dots: Identifying Network Structure via Graph Signal Processing](https://mlsanity.com/paper/1810.13066/) (2018)
- [Neural Latents Benchmark '21: Evaluating latent variable models of neural population activity](https://mlsanity.com/paper/2109.04463/) (2021)
- [Network Topology Inference from Spectral Templates](https://mlsanity.com/paper/1608.03008/) (2016)
- [A Unified Framework for Structured Graph Learning via Spectral Constraints](https://mlsanity.com/paper/1904.09792/) (2019)
- [Learning heat diffusion graphs](https://mlsanity.com/paper/1611.01456/) (2016)
- [Large Scale Graph Learning from Smooth Signals](https://mlsanity.com/paper/1710.05654/) (2017)
- [On Graphical Models via Univariate Exponential Family Distributions](https://mlsanity.com/paper/1301.4183/) (2013)
- [A Time-Vertex Signal Processing Framework](https://mlsanity.com/paper/1705.02307/) (2017)
- [Learning Graphs from Smooth and Graph-Stationary Signals with Hidden Variables](https://mlsanity.com/paper/2111.05588/) (2021)
- [Estimation of positive definite M-matrices and structure learning for attractive Gaussian Markov Random fields](https://mlsanity.com/paper/1404.6640/) (2014)
- [Joint Inference of Multiple Graphs from Matrix Polynomials](https://mlsanity.com/paper/2010.08120/) (2020)
- [DELTACON: A Principled Massive-Graph Similarity Function](https://mlsanity.com/paper/1304.4657/) (2013)
- [Accelerated Graph Learning from Smooth Signals](https://mlsanity.com/paper/2110.09677/) (2021)
- [Identifying the Topology of Undirected Networks from Diffused Non-stationary Graph Signals](https://mlsanity.com/paper/1801.03862/) (2018)
- [Variational inference for sparse network reconstruction from count data](https://mlsanity.com/paper/1806.03120/) (2018)
- [Multi-way Graph Signal Processing on Tensors: Integrative analysis of irregular geometries](https://mlsanity.com/paper/2007.00041/) (2020)
- [Sparse Inverse Covariance Matrix Estimation Using Quadratic Approximation](https://mlsanity.com/paper/1306.3212/) (2013)
- [Graph topology inference based on sparsifying transform learning](https://mlsanity.com/paper/1806.01701/) (2018)
- [Kernel-Based Structural Equation Models for Topology Identification of Directed Networks](https://mlsanity.com/paper/1605.03122/) (2016)
- [Graph Laplacian mixture model](https://mlsanity.com/paper/1810.10053/) (2018)
- [Graph learning under sparsity priors](https://mlsanity.com/paper/1707.05587/) (2017)

## Built on by (most cited first, 3 of 3)

- [Learning Cartesian Product Graphs with Laplacian Constraints](https://mlsanity.com/paper/2402.08105/) (2024)
- [Learning Kronecker-Structured Graphs from Smooth Signals](https://mlsanity.com/paper/2505.09822/) (2025)
- [From topology learning to graph generation: A unifying perspective](https://mlsanity.com/paper/2609.02286/) (2026)

## Abstract (arXiv)

Graph signal processing (GSP) is a prominent framework for analyzing signals on non-Euclidean domains. The graph Fourier transform (GFT) uses the combinatorial graph Laplacian matrix to reveal the spectral decomposition of signals in the graph frequency domain. However, a common challenge in applying GSP methods is that in many scenarios the underlying graph of a system is unknown. A solution in such cases is to construct the unobserved graph from available data, which is commonly referred to as graph or network inference. Although different graph inference methods exist, these are restricted to learning from either smooth graph signals or simple additive Gaussian noise. Other types of noisy data, such as discrete counts or binary digits, are rather common in real-world applications, yet are underexplored in graph inference. In this paper, we propose a versatile graph inference framework for learning from graph signals corrupted by exponential family noise. Our framework generalizes previous methods from continuous smooth graph signals to various data types. We propose an alternating algorithm that jointly estimates the graph Laplacian and the unobserved smooth representation from the noisy signals. We also extend our approach to a variational form to account for the inherent stochasticity of the latent smooth representation. Finally, since real-world graph signals are frequently non-independent and temporally correlated, we further adapt our original setting to a time-vertex formulation. We demonstrate on synthetic and real-world data that our new algorithms outperform competing Laplacian estimation methods that suffer from noise model mismatch.