# Deep nets for local manifold learning

arXiv: https://arxiv.org/abs/1607.07110 · HTML: https://mlsanity.com/paper/1607.07110/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Jul 2016.

## Built on by (most cited first, 26 of 26)

- [Besov Function Approximation and Binary Classification on Low-Dimensional Manifolds Using Convolutional Residual Networks](https://mlsanity.com/paper/2109.02832/) (2021)
- [Why Robust Generalization in Deep Learning is Difficult: Perspective of Expressive Power](https://mlsanity.com/paper/2205.13863/) (2022)
- [Sample Complexity of Nonparametric Off-Policy Evaluation on Low-Dimensional Manifolds using Deep Networks](https://mlsanity.com/paper/2206.02887/) (2022)
- [Understanding CNN Fragility When Learning With Imbalanced Data](https://mlsanity.com/paper/2210.09465/) (2022)
- [Deep Nonparametric Estimation of Intrinsic Data Structures by Chart Autoencoders: Generalization Error and Robustness](https://mlsanity.com/paper/2303.09863/) (2023)
- [Deep Neural Networks with ReLU-Sine-Exponential Activations Break Curse of Dimensionality in Approximation on H"older Class](https://mlsanity.com/paper/2103.00542/) (2021)
- [Double-well Net for Image Segmentation](https://mlsanity.com/paper/2401.00456/) (2023)
- [Side Effects of Learning from Low-dimensional Data Embedded in a Euclidean Space](https://mlsanity.com/paper/2203.00614/) (2022)
- [Benefits of Overparameterized Convolutional Residual Networks: Function Approximation under Smoothness Constraint](https://mlsanity.com/paper/2206.04569/) (2022)
- [Estimation of a regression function on a manifold by fully connected deep neural networks](https://mlsanity.com/paper/2107.09532/) (2021)
- [Solving parametric partial differential equations with deep rectified quadratic unit neural networks](https://mlsanity.com/paper/2203.06973/) (2022)
- [Transformers for Learning on Noisy and Task-Level Manifolds: Approximation and Generalization Insights](https://mlsanity.com/paper/2505.03205/) (2025)
- [Learning on manifolds without manifold learning](https://mlsanity.com/paper/2402.12687/) (2024)
- [Deep Federated Anomaly Detection for Multivariate Time Series Data](https://mlsanity.com/paper/2205.04041/) (2022)
- [Deep neural networks can stably solve high-dimensional, noisy, non-linear inverse problems](https://mlsanity.com/paper/2206.00934/) (2022)
- [Linear Regression on Manifold Structured Data: the Impact of Extrinsic Geometry on Solutions](https://mlsanity.com/paper/2307.02478/) (2023)
- [A manifold learning approach for gesture recognition from micro-Doppler radar measurements](https://mlsanity.com/paper/2110.01670/) (2021)
- [Geometric Neural Operators (GNPs) for Data-Driven Deep Learning of Non-Euclidean Operators](https://mlsanity.com/paper/2404.10843/) (2024)
- [Convergence Analysis for Deep Sparse Coding via Convolutional Neural Networks](https://mlsanity.com/paper/2408.05540/) (2024)
- [Approximation by non-symmetric networks for cross-domain learning](https://mlsanity.com/paper/2305.03890/) (2023)
- [Contrastive Local Manifold Learning for No-Reference Image Quality Assessment](https://mlsanity.com/paper/2406.19247/) (2024)
- [Sparse-Aware Neural Networks for Nonlinear Functionals: Mitigating the Exponential Dependence on Dimension](https://mlsanity.com/paper/2604.06774/) (2026)
- [Shallow neural network representation of polynomials](https://mlsanity.com/paper/2208.08138/) (2022)
- [Local transfer learning from one data space to another](https://mlsanity.com/paper/2302.00160/) (2023)
- [Invariant Manifolds of Discrete-time Dynamical Systems with Nonlinear Exosystems via Hybrid Physics-Informed Neural Networks](https://mlsanity.com/paper/2506.13950/) (2025)
- [Extending Neural Operators: Robust Handling of Functions Beyond the Training Set](https://mlsanity.com/paper/2603.03621/) (2026)

## Abstract (arXiv)

The problem of extending a function f defined on a training data C on an unknown manifold mathbbX to the entire manifold and a tubular neighborhood of this manifold is considered in this paper. For mathbbX embedded in a high dimensional ambient Euclidean space ℝD, a deep learning algorithm is developed for finding a local coordinate system for the manifold bf without eigen--decomposition, which reduces the problem to the classical problem of function approximation on a low dimensional cube. Deep nets (or multilayered neural networks) are proposed to accomplish this approximation scheme by using the training data. Our methods do not involve such optimization techniques as back--propagation, while assuring optimal (a priori) error bounds on the output in terms of the number of derivatives of the target function. In addition, these methods are universal, in that they do not require a prior knowledge of the smoothness of the target function, but adjust the accuracy of approximation locally and automatically, depending only upon the local smoothness of the target function. Our ideas are easily extended to solve both the pre--image problem and the out--of--sample extension problem, with a priori bounds on the growth of the function thus extended.