Maximum Likelihood Estimation of Intrinsic Dimension
Published 1 December 2004
Elizaveta Levina, Peter J. Bickel
Citations709
Generate an AI Snapshot to get a quick, structured summary of this paper.
Study Snapshot
ObjectiveStudy objective
MethodsResearch methodology
PopulationPopulation studied
Sample sizeSample sizes
OutcomesStudy outcomes here
ResultsStudy results comes here
LimitationsResearch study limitations comes here
A concise AI-generated summary of the paper will appear here once you click Generate AI Snapshot.
TL;DR
A new method for estimating intrinsic dimension of a dataset derived by applying the principle of maximum likelihood to the distances between close neighbors is proposed, which has the best overall performance compared with two other intrinsic dimension estimators.
Abstract
We propose a new method for estimating intrinsic dimension of a dataset derived by applying the principle of maximum likelihood to the distances between close neighbors. We derive the estimator by a Poisson process approximation, assess its bias and variance theo-retically and by simulations, and apply it to a number of simulated and real datasets. We also show it has the best overall performance compared with two other intrinsic dimension estimators. 1
Keywords
Computer Science
ScienceNonlinear Dimensionality Reduction by Locally Linear Embedding
15,035 Citations2000Sam T. Roweis, Lawrence K. Saul
Locally linear embedding (LLE) is introduced, an unsupervised learning algorithm that computes low-dimensional, neighborhood-preserving embeddings of high-dimensional inputs that learns the global structure of nonlinear manifolds.
ScienceA Global Geometric Framework for Nonlinear Dimensionality Reduction
13,740 Citations2000Joshua B. Tenenbaum, Vin de Silva +1 more
An approach to solving dimensionality reduction problems that uses easily measured local metric information to learn the underlying global geometry of a data set and efficiently computes a globally optimal solution, and is guaranteed to converge asymptotically to the true structure.
Physica D Nonlinear PhenomenaMeasuring the strangeness of strange attractors
5,643 Citations1983Peter Grassberger, Itamar Procaccia
The MIT Press eBooksLaplacian Eigenmaps and Spectral Techniques for Embedding and Clustering
4,535 Citations2002Mikhail Belkin, Partha Niyogi
The algorithm provides a computationally efficient approach to nonlinear dimensionality reduction that has locality preserving properties and a natural connection to clustering.
Proceedings of the National Academy of SciencesHessian eigenmaps: Locally linear embedding techniques for high-dimensional data
1,696 Citations2003David L. Donoho, Carrie Grimes
The Hessian-based locally linear embedding method for recovering the underlying parametrization of scattered data (mi) lying on a manifold M embedded in high-dimensional Euclidean space is described, where the isometric coordinates can be recovered up to a linear isometry.
Charting a Manifold
374 Citations2002Matthew Brand
A nonlinear mapping from a high-dimensional sample space to a low-dimensional vector space is constructed, effectively recovering a Cartesian coordinate system for the manifold from which the data is sampled, and is pseudo-invertible.
IEEE Transactions on ComputersAn Algorithm for Finding Intrinsic Dimensionality of Data
346 Citations1971Keinosuke Fukunaga, D.R. Olsen
An algorithm for the analysis of multivariant data is presented along with some experimental results, and an analysis that demonstrates the feasability of this approach.
IEEE Transactions on Signal ProcessingGeodesic Entropic Graphs for Dimension and Entropy Estimation in Manifold Learning
286 Citations2004Jose A. Costa, Alfred O. Hero
This paper focuses on the geodesic-minimal-spanning-tree (GMST) method, which uses the overall lengths of the MSTs to simultaneously estimate manifold dimension and entropy.
Intrinsic Dimension Estimation Using Packing Numbers
224 Citations2002Balázs Kégl
A new algorithm is proposed to estimate the intrinsic dimension of data sets based on geometric properties of the data and requires neither parametric assumptions on the data generating model nor input parameters to set.
IEEE Transactions on Pattern Analysis and Machine IntelligenceEstimating the intrinsic dimension of data with a fractal-based method
222 Citations2002Francesco Camastra, Alessandro Vinciarelli
The problem of estimating the intrinsic dimension of a data set is investigated and a fractal-based approach using the Grassberger-Procaccia algorithm is proposed.
IEEE Transactions on Pattern Analysis and Machine IntelligenceAn Intrinsic Dimensionality Estimator from Near-Neighbor Information
220 Citations1979Karl Pettis, Thomas A. Bailey +2 more
An intuitively appealing, noniterative estimator for intrinsic dimensionality which is based on nearneighbor information is proposed which works well in identifying the true dimensionality for a variety of artificial data sets and is fairly insensitive to the number of samples and to the algorithmic parameters.
Non-linear dimensionality reduction techniques for classification and visualization
177 Citations2002Michail Vlachos, Carlotta Domeniconi +3 more
This paper exploits the low-dimensional mapping provided by these embeddings, to develop new classification techniques, and shows experimentally that the classification accuracy is comparable (albeit using fewer dimensions) to a number of other classification procedures.
IEEE Transactions on Pattern Analysis and Machine IntelligenceAn evaluation of intrinsic dimensionality estimators
158 Citations1995Peter J. Verveer, Robert P. W. Duin
The authors compare the algorithms for two estimators of the intrinsic dimensionality of a given data set and extend their capabilities, finding that both can be applied successfully, but that they might fail in certain cases.
IEEE Transactions on Pattern Analysis and Machine IntelligenceIntrinsic dimensionality estimation with optimally topology preserving maps
125 Citations1998Jörg Brüske, Gerald Sommer
A new method for analyzing the intrinsic dimensionality of low-dimensional manifolds in high-dimensional feature spaces is presented, which has only linear instead of cubic time complexity with respect to thedimensionality of the input space.
Neural Information Processing SystemsUsing manifold structure for partially labelled classification
90 Citations2002Mikhail Belkin, Partha Niyogi
An algorithmic framework to classify a partially labeled data set in a principled manner under the assumption that the data lie on a submanifold in a high dimensional space is developed.