# Linear Regression on Manifold Structured Data: the Impact of Extrinsic Geometry on Solutions

arXiv: https://arxiv.org/abs/2307.02478 · HTML: https://mlsanity.com/paper/2307.02478/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 49 references that are in the map.
- First posted on arXiv 5 Jul 2023.

## Builds on (most cited first, 13 of 13)

- [The Intrinsic Dimension of Images and Its Impact on Learning](https://mlsanity.com/paper/2104.08894/) (2021)
- [Intrinsic dimension of data representations in deep neural networks](https://mlsanity.com/paper/1905.12784/) (2019)
- [Understanding Deep Neural Networks with Rectified Linear Units](https://mlsanity.com/paper/1611.01491/) (2016)
- [Deep ReLU network approximation of functions on a manifold](https://mlsanity.com/paper/1908.00695/) (2019)
- [Provable approximation properties for deep neural networks](https://mlsanity.com/paper/1509.07385/) (2015)
- [On the Intrinsic Dimensionality of Image Representations](https://mlsanity.com/paper/1803.09672/) (2018)
- [Besov Function Approximation and Binary Classification on Low-Dimensional Manifolds Using Convolutional Residual Networks](https://mlsanity.com/paper/2109.02832/) (2021)
- [Deep nets for local manifold learning](https://mlsanity.com/paper/1607.07110/) (2016)
- [LDMNet: Low Dimensional Manifold Regularized Neural Networks](https://mlsanity.com/paper/1711.06246/) (2017)
- [GMLS-Nets: A framework for learning from unstructured data](https://mlsanity.com/paper/1909.05371/) (2019)
- [CURE: Curvature Regularization For Missing Data Recovery](https://mlsanity.com/paper/1901.09548/) (2019)
- [Meshfree Methods on Manifolds for Hydrodynamic Flows on Curved Surfaces: A Generalized Moving Least-Squares (GMLS) Approach](https://mlsanity.com/paper/1905.10469/) (2019)
- [Side Effects of Learning from Low-dimensional Data Embedded in a Euclidean Space](https://mlsanity.com/paper/2203.00614/) (2022)

## Built on by (most cited first, 3 of 3)

- [High-dimensional analysis of ridge regression for non-identically distributed data with a variance profile](https://mlsanity.com/paper/2403.20200/) (2024)
- [Data-induced multiscale losses and efficient multirate gradient descent schemes](https://mlsanity.com/paper/2402.03021/) (2024)
- [Gaussian Linear Functional Manifold Method for Massive Point Cloud Data](https://mlsanity.com/paper/2609.05744/) (2026)

## Abstract (arXiv)

In this paper, we study linear regression applied to data structured on a manifold. We assume that the data manifold is smooth and is embedded in a Euclidean space, and our objective is to reveal the impact of the data manifold's extrinsic geometry on the regression. Specifically, we analyze the impact of the manifold's curvatures (or higher order nonlinearity in the parameterization when the curvatures are locally zero) on the uniqueness of the regression solution. Our findings suggest that the corresponding linear regression does not have a unique solution when the embedded submanifold is flat in some dimensions. Otherwise, the manifold's curvature (or higher order nonlinearity in the embedding) may contribute significantly, particularly in the solution associated with the normal directions of the manifold. Our findings thus reveal the role of data manifold geometry in ensuring the stability of regression models for out-of-distribution inferences.