# Topologically penalized regression on manifolds

arXiv: https://arxiv.org/abs/2110.13749 · HTML: https://mlsanity.com/paper/2110.13749/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 65 references that are in the map.
- First posted on arXiv 26 Oct 2021.

## Builds on (most cited first, 10 of 10)

- [An Introduction to Matrix Concentration Inequalities](https://mlsanity.com/paper/1501.01571/) (2015)
- [A Topology Layer for Machine Learning](https://mlsanity.com/paper/1905.12200/) (2019)
- [Error estimates for spectral convergence of the graph Laplacian on random geometric graphs towards the Laplace--Beltrami operator](https://mlsanity.com/paper/1801.10108/) (2018)
- [A Topological Regularizer for Classifiers via Persistent Homology](https://mlsanity.com/paper/1806.10714/) (2018)
- [Trend Filtering on Graphs](https://mlsanity.com/paper/1410.7690/) (2014)
- [Persistent homology detects curvature](https://mlsanity.com/paper/1905.13196/) (2019)
- [Lipschitz regularity of graph Laplacians on random data clouds](https://mlsanity.com/paper/2007.06679/) (2020)
- [Concentration in unbounded metric spaces and algorithmic stability](https://mlsanity.com/paper/1309.1007/) (2013)
- [Estimating the reach of a manifold via its convexity defect function](https://mlsanity.com/paper/2001.08006/) (2020)
- [Construction of Tight Frames on Graphs and Application to Denoising](https://mlsanity.com/paper/1408.4012/) (2014)

## Built on by (most cited first, 5 of 5)

- [Adaptive and non-adaptive minimax rates for weighted Laplacian-eigenmap based nonparametric regression](https://mlsanity.com/paper/2311.00140/) (2023)
- [Nonsmooth Nonparametric Regression via Fractional Laplacian Eigenmaps](https://mlsanity.com/paper/2402.14985/) (2024)
- [Mix-GENEO: A Flexible Filtration for Multiparameter Persistent Homology Detects Digital Images](https://mlsanity.com/paper/2401.04332/) (2024)
- [Changes from Classical Statistics to Modern Statistics and Data Science](https://mlsanity.com/paper/2211.03756/) (2022)
- [TopoFisher: Learning Topological Summary Statistics by Maximizing Fisher Information](https://mlsanity.com/paper/2605.07720/) (2026)

## Abstract (arXiv)

We study a regression problem on a compact manifold M. In order to take advantage of the underlying geometry and topology of the data, the regression task is performed on the basis of the first several eigenfunctions of the Laplace-Beltrami operator of the manifold, that are regularized with topological penalties. The proposed penalties are based on the topology of the sub-level sets of either the eigenfunctions or the estimated function. The overall approach is shown to yield promising and competitive performance on various applications to both synthetic and real data sets. We also provide theoretical guarantees on the regression function estimates, on both its prediction error and its smoothness (in a topological sense). Taken together, these results support the relevance of our approach in the case where the targeted function is ''topologically smooth''.