# Diffusion Nets

arXiv: https://arxiv.org/abs/1506.07840 · HTML: https://mlsanity.com/paper/1506.07840/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Jun 2015.

## Built on by (most cited first, 19 of 19)

- [Manifold Interpolating Optimal-Transport Flows for Trajectory Inference](https://mlsanity.com/paper/2206.14928/) (2022)
- [Learning Signal-Agnostic Manifolds of Neural Fields](https://mlsanity.com/paper/2111.06387/) (2021)
- [Deep autoencoders for physics-constrained data-driven nonlinear materials modeling](https://mlsanity.com/paper/2209.04416/) (2022)
- [LDLE: Low Distortion Local Eigenmaps](https://mlsanity.com/paper/2101.11055/) (2021)
- [Learning shared neural manifolds from multi-subject FMRI data](https://mlsanity.com/paper/2201.00622/) (2021)
- [Semi-Supervised Manifold Learning with Complexity Decoupled Chart Autoencoders](https://mlsanity.com/paper/2208.10570/) (2022)
- [SpecNet2: Orthogonalization-free spectral embedding by neural networks](https://mlsanity.com/paper/2206.06644/) (2022)
- [Tree-Wasserstein Distance for High Dimensional Data with a Latent Feature Hierarchy](https://mlsanity.com/paper/2410.21107/) (2024)
- [BASiS: Batch Aligned Spectral Embedding Space](https://mlsanity.com/paper/2211.16960/) (2022)
- [DeepDiffusion: Unsupervised Learning of Retrieval-adapted Representations via Diffusion-based Ranking on Latent Feature Manifold](https://mlsanity.com/paper/2112.07082/) (2021)
- [Conformal Disentanglement and Latent-Space Curation: A Neural Framework for Perspective Synthesis, Differentiation and Targeted Generation](https://mlsanity.com/paper/2408.15344/) (2024)
- [Generalizable Spectral Embedding with an Application to UMAP](https://mlsanity.com/paper/2501.11305/) (2025)
- [Learning collective variables that preserve transition rates](https://mlsanity.com/paper/2506.01222/) (2025)
- [MIOFlow 2.0: A unified framework for inferring cellular stochastic dynamics from single cell and spatial transcriptomics data](https://mlsanity.com/paper/2603.22564/) (2026)
- [Bi-Lipschitz Autoencoder With Injectivity Guarantee](https://mlsanity.com/paper/2604.06701/) (2026)
- [Voice Activity Detection for Transient Noisy Environment Based on Diffusion Nets](https://mlsanity.com/paper/2106.13763/) (2021)
- [Learning Nonparametric High-Dimensional Generative Models: The Empirical-Beta-Copula Autoencoder](https://mlsanity.com/paper/2309.09916/) (2023)
- [SlepNet: Spectral Subgraph Representation Learning for Neural Dynamics](https://mlsanity.com/paper/2506.16602/) (2025)
- [A Graph Laplacian Eigenvector-based Pre-training Method for Graph Neural Networks](https://mlsanity.com/paper/2509.02803/) (2025)

## Abstract (arXiv)

Non-linear manifold learning enables high-dimensional data analysis, but requires out-of-sample-extension methods to process new data points. In this paper, we propose a manifold learning algorithm based on deep learning to create an encoder, which maps a high-dimensional dataset and its low-dimensional embedding, and a decoder, which takes the embedded data back to the high-dimensional space. Stacking the encoder and decoder together constructs an autoencoder, which we term a diffusion net, that performs out-of-sample-extension as well as outlier detection. We introduce new neural net constraints for the encoder, which preserves the local geometry of the points, and we prove rates of convergence for the encoder. Also, our approach is efficient in both computational complexity and memory requirements, as opposed to previous methods that require storage of all training points in both the high-dimensional and the low-dimensional spaces to calculate the out-of-sample-extension and the pre-image.