# Learning Sheaf Laplacian Optimizing Restriction Maps

arXiv: https://arxiv.org/abs/2501.19207 · HTML: https://mlsanity.com/paper/2501.19207/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 13 references that are in the map.
- First posted on arXiv 31 Jan 2025.

## Builds on (most cited first, 6 of 6)

- [How to learn a graph from smooth signals](https://mlsanity.com/paper/1601.02513/) (2016)
- [Topological Signal Processing over Simplicial Complexes](https://mlsanity.com/paper/1907.11577/) (2019)
- [Graph signal processing for machine learning: A review and new perspectives](https://mlsanity.com/paper/2007.16061/) (2020)
- [Learning Sparse Graphs Under Smoothness Prior](https://mlsanity.com/paper/1609.03448/) (2016)
- [Multireference Alignment using Semidefinite Programming](https://mlsanity.com/paper/1308.5256/) (2013)
- [Topological Signal Processing over Generalized Cell Complexes](https://mlsanity.com/paper/2201.08993/) (2022)

## Built on by (most cited first, 7 of 7)

- [Learning the Structure of Connection Graphs](https://mlsanity.com/paper/2510.11245/) (2025)
- [Learning Network Sheaves for AI-native Semantic Communication](https://mlsanity.com/paper/2512.03248/) (2025)
- [SheafStain: Sheaf-Theoretic Schr"odinger Bridge for Spatially and Biologically Coherent Virtual Staining](https://mlsanity.com/paper/2606.11846/) (2026)
- [Learned, Relied Upon, or Necessary? Separating Checkpoint Dependence from Task-Level Value in Sheaf GNNs](https://mlsanity.com/paper/2607.25387/) (2026)
- [Sheaf-Based Federated Representation Learning](https://mlsanity.com/paper/2608.10016/) (2026)
- [From topology learning to graph generation: A unifying perspective](https://mlsanity.com/paper/2609.02286/) (2026)
- [Gluing Local Contexts into Global Meaning: A Sheaf-Theoretic Decomposition of Transformer Representations](https://mlsanity.com/paper/2610.17117/) (2026)

## Abstract (arXiv)

The aim of this paper is to propose a novel framework to infer the sheaf Laplacian, including the topology of a graph and the restriction maps, from a set of data observed over the nodes of a graph. The proposed method is based on sheaf theory, which represents an important generalization of graph signal processing. The learning problem aims to find the sheaf Laplacian that minimizes the total variation of the observed data, where the variation over each edge is also locally minimized by optimizing the associated restriction maps. Compared to alternative methods based on semidefinite programming, our solution is significantly more numerically efficient, as all its fundamental steps are resolved in closed form. The method is numerically tested on data consisting of vectors defined over subspaces of varying dimensions at each node. We demonstrate how the resulting graph is influenced by two key factors: the cross-correlation and the dimensionality difference of the data residing on the graph's nodes.