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Estimating Manifold Dimension by Inversion Error

Martin, Shawn and Bäcker, Alex (2005) Estimating Manifold Dimension by Inversion Error. In: SAC '05 Proceedings of the 2005 ACM symposium on Applied computing. ACM , New York, NY, pp. 22-26. ISBN 1-58113-964-0. https://resolver.caltech.edu/CaltechAUTHORS:20161130-150437077

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Abstract

Video and image datasets can often be described by a small number of parameters, even though each image usually consists of hundreds or thousands of pixels. This observation is often exploited in computer vision and pattern recognition by the application of dimensionality reduction techniques. In particular, there has been recent interest in the application of a class of nonlinear dimensionality reduction algorithms which assume that an image dataset has been sampled from a manifold. From this assumption, it follows that estimating the dimension of the manifold is the first step in analyzing an image dataset. Typically, this estimate is obtained either by using a priori knowledge, or by applying one of the various statistical and geometrical methods available. Once an estimate is obtained, it is used as a parameter for the nonlinear dimensionality reduction algorithm. In this paper, we consider reversing this approach. Instead of estimating the dimension of the manifold in order to obtain a low dimensional representation, we consider producing low dimensional representations in order to estimate of the dimensionality of the manifold. By varying the dimensionality parameter, we obtain different low dimensional representations of the original dataset. The dimension of the best representation should then correspond to the actual dimension of the manifold. In order to determine the best representation, we propose a metric based on inversion. In particular, we propose that a good representation should be invertible, in that we should be able to reverse the reduction algorithm's transformation to obtain the original dataset. By coupling this metric with any reduction algorithm, we can estimate the dimensionality of an image manifold. We apply our method in the context of locally linear embedding (LLE) and Isomap to six frequently used examples and two image datasets.


Item Type:Book Section
Related URLs:
URLURL TypeDescription
https://doi.org/10.1145/1066677.1066686DOIArticle
http://dl.acm.org/citation.cfm?doid=1066677.1066686PublisherArticle
Additional Information:© 2005 ACM. This work was supported by the Department of Energy’s Office of Advanced Scientific Computing Research (OASCR) Applied Mathematics program, specifically under Mathematics, Information and Computational Sciences (MICS) project 547 task 02.09. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under Contract DE-AC04-94AL85000.
Funders:
Funding AgencyGrant Number
Department of Energy (DOE)547 task 02.09
Department of Energy (DOE)DE-AC04-94AL85000
DOI:10.1145/1066677.1066686
Record Number:CaltechAUTHORS:20161130-150437077
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20161130-150437077
Official Citation:Shawn Martin and Alex Bäcker. 2005. Estimating manifold dimension by inversion error. In Proceedings of the 2005 ACM symposium on Applied computing (SAC '05), Lorie M. Liebrock (Ed.). ACM, New York, NY, USA, 22-26. DOI: http://dx.doi.org/10.1145/1066677.1066686
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:72469
Collection:CaltechAUTHORS
Deposited By: Kristin Buxton
Deposited On:30 Nov 2016 23:13
Last Modified:11 Nov 2021 05:01

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