A Nonlinear Dimensionality Reduction Framework Using Smooth Geodesics

TL;DR

This paper introduces a nonlinear dimensionality reduction framework that constructs smooth geodesics using a network of nearest neighbors and spline fitting, effectively handling sparse and noisy high-dimensional data.

Contribution

The proposed method generates smooth geodesics for dimensionality reduction, improving robustness over noisy and sparse datasets compared to traditional techniques.

Findings

Effective on synthetic datasets

Robust to noise and sparsity

Produces faithful low-dimensional embeddings

Abstract

Existing dimensionality reduction methods are adept at revealing hidden underlying manifolds arising from high-dimensional data and thereby producing a low-dimensional representation. However, the smoothness of the manifolds produced by classic techniques over sparse and noisy data is not guaranteed. In fact, the embedding generated using such data may distort the geometry of the manifold and thereby produce an unfaithful embedding. Herein, we propose a framework for nonlinear dimensionality reduction that generates a manifold in terms of smooth geodesics that is designed to treat problems in which manifold measurements are either sparse or corrupted by noise. Our method generates a network structure for given high-dimensional data using a nearest neighbors search and then produces piecewise linear shortest paths that are defined as geodesics. Then, we fit points in each geodesic by a smoothing spline to emphasize the smoothness. The robustness of this approach for sparse and noisy datasets is demonstrated by the implementation of the method on synthetic and real-world datasets.