# Skeleton Regression: A Graph-Based Approach to Estimation with Manifold Structure

arXiv: https://arxiv.org/abs/2303.11786 · HTML: https://mlsanity.com/paper/2303.11786/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 57 references that are in the map.
- First posted on arXiv 19 Mar 2023.

## Builds on (most cited first, 9 of 9)

- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networks](https://mlsanity.com/paper/2103.03212/) (2021)
- [A Tutorial on Kernel Density Estimation and Recent Advances](https://mlsanity.com/paper/1704.03924/) (2017)
- [Adaptive piecewise polynomial estimation via trend filtering](https://mlsanity.com/paper/1304.2986/) (2013)
- [Trend Filtering on Graphs](https://mlsanity.com/paper/1410.7690/) (2014)
- [Minimax Optimal Regression over Sobolev Spaces via Laplacian Regularization on Neighborhood Graphs](https://mlsanity.com/paper/2106.01529/) (2021)
- [A Spectral Series Approach to High-Dimensional Nonparametric Regression](https://mlsanity.com/paper/1602.00355/) (2016)
- [Skeleton Clustering: Dimension-Free Density-based Clustering](https://mlsanity.com/paper/2104.10770/) (2021)
- [Multivariate Regression with Gross Errors on Manifold-valued Data](https://mlsanity.com/paper/1703.08772/) (2017)

## Abstract (arXiv)

We introduce a new regression framework designed to deal with large-scale, complex data that lies around a low-dimensional manifold with noises. Our approach first constructs a graph representation, referred to as the skeleton, to capture the underlying geometric structure. We then define metrics on the skeleton graph and apply nonparametric regression techniques, along with feature transformations based on the graph, to estimate the regression function. We also discuss the limitations of some nonparametric regressors with respect to the general metric space such as the skeleton graph. The proposed regression framework suggests a novel way to deal with data with underlying geometric structures and provides additional advantages in handling the union of multiple manifolds, additive noises, and noisy observations. We provide statistical guarantees for the proposed method and demonstrate its effectiveness through simulations and real data examples.