Diffusion nets

Gal Mishne, Uri Shaham, Alexander Cloninger, Israel Cohen

Research output: Contribution to journalArticlepeer-review

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 to 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 preserve 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 of new points.

Original languageEnglish
Pages (from-to)259-285
Number of pages27
JournalApplied and Computational Harmonic Analysis
Volume47
Issue number2
DOIs
StatePublished - Sep 2019
Externally publishedYes

Keywords

  • Autoencoder
  • Deep learning
  • Diffusion maps
  • Manifold learning
  • Out-of-sample extension

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