# A Nonlinear Dimensionality Reduction Framework Using Smooth Geodesics

arXiv: https://arxiv.org/abs/1707.06757 · HTML: https://mlsanity.com/paper/1707.06757/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Jul 2017.

## Built on by (most cited first, 10 of 10)

- [Data-driven Control of Agent-based Models: an Equation/Variable-free Machine Learning Approach](https://mlsanity.com/paper/2207.05779/) (2022)
- [Recurrent Neural Networks for Dynamical Systems: Applications to Ordinary Differential Equations, Collective Motion, and Hydrological Modeling](https://mlsanity.com/paper/2202.07022/) (2022)
- [Time Series Forecasting Using Manifold Learning](https://mlsanity.com/paper/2110.03625/) (2021)
- [Reconstruction of Fragmented Trajectories of Collective Motion using Hadamard Deep Autoencoders](https://mlsanity.com/paper/2110.10428/) (2021)
- [Geodesic Gramian Denoising Applied to the Images Contaminated With Noise Sampled From Diverse Probability Distributions](https://mlsanity.com/paper/2203.02600/) (2022)
- [Real-time Forecasting of Time Series in Financial Markets Using Sequentially Trained Many-to-one LSTMs](https://mlsanity.com/paper/2205.04678/) (2022)
- [Fraud Detection Using Optimized Machine Learning Tools Under Imbalance Classes](https://mlsanity.com/paper/2209.01642/) (2022)
- [Efficient Image Denoising by Low-Rank Singular Vector Approximations of Geodesics' Gramian Matrix](https://mlsanity.com/paper/2209.13094/) (2022)
- [Efficient and Robust Remote Sensing Image Denoising Using Randomized Approximation of Geodesics' Gramian on the Manifold Underlying the Patch Space](https://mlsanity.com/paper/2504.10820/) (2025)
- [Dimensionality Reduction on Riemannian Manifolds in Data Analysis](https://mlsanity.com/paper/2602.05936/) (2026)

## Abstract (arXiv)

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.