Diffusion Nets

TL;DR

Diffusion Nets leverage deep learning to perform efficient out-of-sample extension and outlier detection in manifold learning, preserving local geometry with provable convergence rates.

Contribution

This paper introduces a novel autoencoder called diffusion net that combines manifold learning with neural networks, enabling efficient out-of-sample extension and outlier detection.

Findings

Effective out-of-sample extension demonstrated

Preserves local geometry with new neural net constraints

Achieves computational efficiency over previous methods

Abstract

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.