Learning gradients on manifolds

TL;DR

This paper introduces an algorithm for learning gradients on manifolds to enable dimension reduction in high-dimensional data, with theoretical guarantees and empirical validation demonstrating effectiveness on simulated and real datasets.

Contribution

It presents a novel method for gradient estimation on manifolds with error bounds, focusing on intrinsic dimension rather than ambient space.

Findings

Convergence rate depends on manifold's intrinsic dimension.

Method outperforms existing dimension reduction techniques.

Generalization error bounds are established.

Abstract

A common belief in high-dimensional data analysis is that data are concentrated on a low-dimensional manifold. This motivates simultaneous dimension reduction and regression on manifolds. We provide an algorithm for learning gradients on manifolds for dimension reduction for high-dimensional data with few observations. We obtain generalization error bounds for the gradient estimates and show that the convergence rate depends on the intrinsic dimension of the manifold and not on the dimension of the ambient space. We illustrate the efficacy of this approach empirically on simulated and real data and compare the method to other dimension reduction procedures.