# Learning with latent group sparsity via heat flow dynamics on networks

arXiv: https://arxiv.org/abs/2201.08326 · HTML: https://mlsanity.com/paper/2201.08326/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 73 references that are in the map.
- First posted on arXiv 20 Jan 2022.

## Builds on (most cited first, 14 of 14)

- [Community Detection and Stochastic Block Models](https://mlsanity.com/paper/1703.10146/) (2017)
- [Exact Recovery in the Stochastic Block Model](https://mlsanity.com/paper/1405.3267/) (2014)
- [Network Lasso: Clustering and Optimization in Large Graphs](https://mlsanity.com/paper/1507.00280/) (2015)
- [Complex energy landscapes in spiked-tensor and simple glassy models: ruggedness, arrangements of local minima and phase transitions](https://mlsanity.com/paper/1804.02686/) (2018)
- [Learning Some Popular Gaussian Graphical Models without Condition Number Bounds](https://mlsanity.com/paper/1905.01282/) (2019)
- [On the low-rank approach for semidefinite programs arising in synchronization and community detection](https://mlsanity.com/paper/1602.04426/) (2016)
- [Maximum likelihood estimation in Gaussian models under total positivity](https://mlsanity.com/paper/1702.04031/) (2017)
- [Prediction models for network-linked data](https://mlsanity.com/paper/1602.01192/) (2016)
- [Determinantal point processes based on orthogonal polynomials for sampling minibatches in SGD](https://mlsanity.com/paper/2112.06007/) (2021)
- [Likelihood landscape and maximum likelihood estimation for the discrete orbit recovery model](https://mlsanity.com/paper/2004.00041/) (2020)
- [Fast Mixing Markov Chains for Strongly Rayleigh Measures, DPPs, and Constrained Sampling](https://mlsanity.com/paper/1608.01008/) (2016)
- [High-dimensional Gaussian graphical model for network-linked data](https://mlsanity.com/paper/1907.02443/) (2019)
- [Gaussian Determinantal Processes: a new model for directionality in data](https://mlsanity.com/paper/2111.09990/) (2021)
- [Fractal Gaussian Networks: A sparse random graph model based on Gaussian Multiplicative Chaos](https://mlsanity.com/paper/2008.03038/) (2020)

## Built on by (most cited first, 2 of 2)

- [Bayes optimal learning in high-dimensional linear regression with network side information](https://mlsanity.com/paper/2306.05679/) (2023)
- [Learning Networks from Gaussian Graphical Models and Gaussian Free Fields](https://mlsanity.com/paper/2308.02344/) (2023)

## Abstract (arXiv)

Group or cluster structure on explanatory variables in machine learning problems is a very general phenomenon, which has attracted broad interest from practitioners and theoreticians alike. In this work we contribute an approach to learning under such group structure, that does not require prior information on the group identities. Our paradigm is motivated by the Laplacian geometry of an underlying network with a related community structure, and proceeds by directly incorporating this into a penalty that is effectively computed via a heat flow-based local network dynamics. In fact, we demonstrate a procedure to construct such a network based on the available data. Notably, we dispense with computationally intensive pre-processing involving clustering of variables, spectral or otherwise. Our technique is underpinned by rigorous theorems that guarantee its effective performance and provide bounds on its sample complexity. In particular, in a wide range of settings, it provably suffices to run the heat flow dynamics for time that is only logarithmic in the problem dimensions. We explore in detail the interfaces of our approach with key statistical physics models in network science, such as the Gaussian Free Field and the Stochastic Block Model. We validate our approach by successful applications to real-world data from a wide array of application domains, including computer science, genetics, climatology and economics. Our work raises the possibility of applying similar diffusion-based techniques to classical learning tasks, exploiting the interplay between geometric, dynamical and stochastic structures underlying the data.