Estimate of the Neural Network Dimension Using Algebraic Topology and Lie Theory

Melodia L, Lenz R (2021)

Publication Language: English

Publication Type: Conference contribution, Conference Contribution

Publication year: 2021

Publisher: Springer Nature

Edited Volumes: Pattern Recognition and Information Forensics

Series: Image Mining. Theory and Applications

City/Town: Schweiz

Book Volume: 7

Conference Proceedings Title: Pattern Recognition and Information Forensics

Event location: Mailand IT

ISBN: 978-3-030-68820-2

URI: https://www.springer.com/gp/book/9783030688202

DOI: 10.1007/978-3-030-68821-9_2

Abstract

In this paper we present an approach to determine the smallest possible number of neurons in a layer of a neural network in such a way that the topology of the input space can be learned sufficiently well. We introduce a general procedure based on persistent homology to investigate topological invariants of the manifold on which we suspect the data set. We specify the required dimensions precisely, assuming that there is a smooth manifold on or near which the data are located. Furthermore, we require that this space is connected and has a commutative group structure in the mathematical sense. These assumptions allow us to derive a decomposition of the underlying space whose topology is well known. We use the representatives of the k-dimensional homology groups from the persistence landscape to determine an integer dimension for this decomposition.This number is the dimension of the embedding that is capable of capturing the topology of the data manifold. We derive the theory and validate it experimentally on toy data sets.

Authors with CRIS profile

Luciano Melodia Professur für evolutionäres Datenmanagement Richard Lenz Professur für evolutionäres Datenmanagement

Related research project(s)

Schema Inference and Machine Learning (SIML) Aug. 1, 2018 - Aug. 1, 2021

How to cite

APA:

Melodia, L., & Lenz, R. (2021). Estimate of the Neural Network Dimension Using Algebraic Topology and Lie Theory. In Alberto Del Bimbo, Rita Cucchiara, Stan Sciaroff, Giovanni Maria Farinella, Tao Mei, Marco Bertini, Hugo Jair Escalante, Roberto Vezzani (Eds.), Pattern Recognition and Information Forensics. Mailand, IT: Schweiz: Springer Nature.

MLA:

Melodia, Luciano, and Richard Lenz. "Estimate of the Neural Network Dimension Using Algebraic Topology and Lie Theory." Proceedings of the Image Mining. Theory and Applications VII, Mailand Ed. Alberto Del Bimbo, Rita Cucchiara, Stan Sciaroff, Giovanni Maria Farinella, Tao Mei, Marco Bertini, Hugo Jair Escalante, Roberto Vezzani, Schweiz: Springer Nature, 2021.

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