# Topological Dictionary Learning

arXiv: https://arxiv.org/abs/2503.11470 · HTML: https://mlsanity.com/paper/2503.11470/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 65 references that are in the map.
- First posted on arXiv 14 Mar 2025.

## Builds on (most cited first, 23 of 23)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Graph Signal Processing: Overview, Challenges and Applications](https://mlsanity.com/paper/1712.00468/) (2017)
- [Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networks](https://mlsanity.com/paper/2103.03212/) (2021)
- [Topological Signal Processing over Simplicial Complexes](https://mlsanity.com/paper/1907.11577/) (2019)
- [Random Walks on Simplicial Complexes and the normalized Hodge 1-Laplacian](https://mlsanity.com/paper/1807.05044/) (2018)
- [Signal Processing on Higher-Order Networks: Livin' on the Edge ... and Beyond](https://mlsanity.com/paper/2101.05510/) (2021)
- [A Unified Framework for Structured Graph Learning via Spectral Constraints](https://mlsanity.com/paper/1904.09792/) (2019)
- [Graph Filters for Signal Processing and Machine Learning on Graphs](https://mlsanity.com/paper/2211.08854/) (2022)
- [Architectures of Topological Deep Learning: A Survey of Message-Passing Topological Neural Networks](https://mlsanity.com/paper/2304.10031/) (2023)
- [Goal-Oriented and Semantic Communication in 6G AI-Native Networks: The 6G-GOALS Approach](https://mlsanity.com/paper/2402.07573/) (2024)
- [Online Graph Dictionary Learning](https://mlsanity.com/paper/2102.06555/) (2021)
- [Signal Processing on Cell Complexes](https://mlsanity.com/paper/2110.05614/) (2021)
- [Topological Signal Processing over Cell Complexes](https://mlsanity.com/paper/2112.06709/) (2021)
- [Two's company, three (or more) is a simplex: Algebraic-topological tools for understanding higher-order structure in neural data](https://mlsanity.com/paper/1601.01704/) (2016)
- [Persformer: A Transformer Architecture for Topological Machine Learning](https://mlsanity.com/paper/2112.15210/) (2021)
- [From Latent Graph to Latent Topology Inference: Differentiable Cell Complex Module](https://mlsanity.com/paper/2305.16174/) (2023)
- [Localized Spectral Graph Filter Frames: A Unifying Framework, Survey of Design Considerations, and Numerical Comparison (Extended Cut)](https://mlsanity.com/paper/2006.11220/) (2020)
- [Learning parametric dictionaries for graph signals](https://mlsanity.com/paper/1401.0887/) (2014)
- [Topological Signal Processing and Learning: Recent Advances and Future Challenges](https://mlsanity.com/paper/2412.01576/) (2024)
- [Hodgelets: Localized Spectral Representations of Flows on Simplicial Complexes](https://mlsanity.com/paper/2109.08728/) (2021)
- [Topological Slepians: Maximally Localized Representations of Signals over Simplicial Complexes](https://mlsanity.com/paper/2210.14758/) (2022)
- [Signal processing on simplicial complexes](https://mlsanity.com/paper/2004.02392/) (2020)
- [Sampling and Recovery of Graph Signals](https://mlsanity.com/paper/1712.09310/) (2017)

## Built on by (most cited first, 4 of 4)

- [Topological Adaptive Least Mean Squares Algorithms over Simplicial Complexes](https://mlsanity.com/paper/2505.23160/) (2025)
- [Topological Kalman Filtering on Cell Complexes](https://mlsanity.com/paper/2605.15955/) (2026)
- [Topological Signal Processing: An Application-Oriented Tutorial](https://mlsanity.com/paper/2605.22853/) (2026)
- [Sheaf-theoretic Signal Processing on Graphs: Spectral Theory, Filtering, and Sampling](https://mlsanity.com/paper/2608.01318/) (2026)

## Abstract (arXiv)

The aim of this paper is to introduce a novel dictionary learning algorithm for sparse representation of signals defined over combinatorial topological spaces, specifically, regular cell complexes. Leveraging Hodge theory, we embed topology into the dictionary structure via concatenated sub-dictionaries, each as a polynomial of Hodge Laplacians, yielding localized spectral topological filter frames. The learning problem is cast to jointly infer the underlying cell complex and optimize the dictionary coefficients and the sparse signal representation. We efficiently solve the problem via iterative alternating algorithms. Numerical results on both synthetic and real data show the effectiveness of the proposed procedure in jointly learning the sparse representations and the underlying relational structure of topological signals.