# Bounded Manifold Completion

arXiv: https://arxiv.org/abs/1912.09026 · HTML: https://mlsanity.com/paper/1912.09026/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Dec 2019.

## Built on by (most cited first, 7 of 7)

- [Recurrent Neural Networks for Dynamical Systems: Applications to Ordinary Differential Equations, Collective Motion, and Hydrological Modeling](https://mlsanity.com/paper/2202.07022/) (2022)
- [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)

## Abstract (arXiv)

Nonlinear dimensionality reduction or, equivalently, the approximation of high-dimensional data using a low-dimensional nonlinear manifold is an active area of research. In this paper, we will present a thematically different approach to detect the existence of a low-dimensional manifold of a given dimension that lies within a set of bounds derived from a given point cloud. A matrix representing the appropriately defined distances on a low-dimensional manifold is low-rank, and our method is based on current techniques for recovering a partially observed matrix from a small set of fully observed entries that can be implemented as a low-rank Matrix Completion (MC) problem. MC methods are currently used to solve challenging real-world problems, such as image inpainting and recommender systems, and we leverage extent efficient optimization techniques that use a nuclear norm convex relaxation as a surrogate for non-convex and discontinuous rank minimization. Our proposed method provides several advantages over current nonlinear dimensionality reduction techniques, with the two most important being theoretical guarantees on the detection of low-dimensional embeddings and robustness to non-uniformity in the sampling of the manifold. We validate the performance of this approach using both a theoretical analysis as well as synthetic and real-world benchmark datasets.