Deep Nets for Local Manifold Learning

Charles Kam-Tai CHUI, Hrushikesh N. Mhaskar*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

8 Citations (Scopus)

Abstract

The problem of extending a function f defined on a training data (Formula presented.) on an unknown manifold X to the entire manifold and a tubular neighborhood of this manifold is considered in this paper. For X embedded in a high dimensional ambient Euclidean space RD, a deep learning algorithm is developed for finding a local coordinate system for the manifold 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.

Original languageEnglish
Article number12
JournalFrontiers in Applied Mathematics and Statistics
Volume4
DOIs
Publication statusPublished - 29 May 2018

Scopus Subject Areas

  • Applied Mathematics
  • Statistics and Probability

User-Defined Keywords

  • deep learning
  • function approximation
  • local approximation
  • manifold learning
  • neural networks

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