A Unifying Probabilistic Perspective for Spectral Dimensionality Reduction: Insights and New Models

Neil D. Lawrence
Journal of Machine Learning Research, 13(51):1609-1638, 2012.

Abstract

We introduce a new perspective on spectral dimensionality reduction which views these methods as Gaussian Markov random fields (GRFs). Our unifying perspective is based on the maximum entropy principle which is in turn inspired by maximum variance unfolding. The resulting model, which we call maximum entropy unfolding (MEU) is a nonlinear generalization of principal component analysis. We relate the model to Laplacian eigenmaps and isomap. We show that parameter fitting in the locally linear embedding (LLE) is approximate maximum likelihood MEU. We introduce a variant of LLE that performs maximum likelihood exactly: Acyclic LLE (ALLE). We show that MEU and ALLE are competitive with the leading spectral approaches on a robot navigation visualization and a human motion capture data set. Finally the maximum likelihood perspective allows us to introduce a new approach to dimensionality reduction based on L1 regularization of the Gaussian random field via the graphical lasso.

Cite this Paper

BibTeX
@Article{Lawrence-unifying12, title = {A Unifying Probabilistic Perspective for Spectral Dimensionality Reduction: Insights and New Models}, author = {Lawrence, Neil D.}, journal = {Journal of Machine Learning Research}, pages = {1609--1638}, year = {2012}, volume = {13}, number = {51}, pdf = {ftp://ftp.dcs.shef.ac.uk/home/neil/spectral.pdf}, url = {http://inverseprobability.com/publications/a-unifying-probabilistic-perspective-for-spectral-dimensionality-reduction-insights-and-new-models.html}, abstract = {We introduce a new perspective on spectral dimensionality reduction which views these methods as Gaussian Markov random fields (GRFs). Our unifying perspective is based on the maximum entropy principle which is in turn inspired by maximum variance unfolding. The resulting model, which we call maximum entropy unfolding (MEU) is a nonlinear generalization of principal component analysis. We relate the model to Laplacian eigenmaps and isomap. We show that parameter fitting in the locally linear embedding (LLE) is approximate maximum likelihood MEU. We introduce a variant of LLE that performs maximum likelihood exactly: Acyclic LLE (ALLE). We show that MEU and ALLE are competitive with the leading spectral approaches on a robot navigation visualization and a human motion capture data set. Finally the maximum likelihood perspective allows us to introduce a new approach to dimensionality reduction based on L1 regularization of the Gaussian random field via the graphical lasso. } }
Endnote
%0 Journal Article %T A Unifying Probabilistic Perspective for Spectral Dimensionality Reduction: Insights and New Models %A Neil D. Lawrence %J Journal of Machine Learning Research %D 2012 %F Lawrence-unifying12 %P 1609--1638 %U http://inverseprobability.com/publications/a-unifying-probabilistic-perspective-for-spectral-dimensionality-reduction-insights-and-new-models.html %V 13 %N 51 %X We introduce a new perspective on spectral dimensionality reduction which views these methods as Gaussian Markov random fields (GRFs). Our unifying perspective is based on the maximum entropy principle which is in turn inspired by maximum variance unfolding. The resulting model, which we call maximum entropy unfolding (MEU) is a nonlinear generalization of principal component analysis. We relate the model to Laplacian eigenmaps and isomap. We show that parameter fitting in the locally linear embedding (LLE) is approximate maximum likelihood MEU. We introduce a variant of LLE that performs maximum likelihood exactly: Acyclic LLE (ALLE). We show that MEU and ALLE are competitive with the leading spectral approaches on a robot navigation visualization and a human motion capture data set. Finally the maximum likelihood perspective allows us to introduce a new approach to dimensionality reduction based on L1 regularization of the Gaussian random field via the graphical lasso.
RIS
TY - JOUR TI - A Unifying Probabilistic Perspective for Spectral Dimensionality Reduction: Insights and New Models AU - Neil D. Lawrence DA - 2012/01/01 ID - Lawrence-unifying12 VL - 13 IS - 51 SP - 1609 EP - 1638 L1 - ftp://ftp.dcs.shef.ac.uk/home/neil/spectral.pdf UR - http://inverseprobability.com/publications/a-unifying-probabilistic-perspective-for-spectral-dimensionality-reduction-insights-and-new-models.html AB - We introduce a new perspective on spectral dimensionality reduction which views these methods as Gaussian Markov random fields (GRFs). Our unifying perspective is based on the maximum entropy principle which is in turn inspired by maximum variance unfolding. The resulting model, which we call maximum entropy unfolding (MEU) is a nonlinear generalization of principal component analysis. We relate the model to Laplacian eigenmaps and isomap. We show that parameter fitting in the locally linear embedding (LLE) is approximate maximum likelihood MEU. We introduce a variant of LLE that performs maximum likelihood exactly: Acyclic LLE (ALLE). We show that MEU and ALLE are competitive with the leading spectral approaches on a robot navigation visualization and a human motion capture data set. Finally the maximum likelihood perspective allows us to introduce a new approach to dimensionality reduction based on L1 regularization of the Gaussian random field via the graphical lasso. ER -
APA
Lawrence, N.D.. (2012). A Unifying Probabilistic Perspective for Spectral Dimensionality Reduction: Insights and New Models. Journal of Machine Learning Research 13(51):1609-1638 Available from http://inverseprobability.com/publications/a-unifying-probabilistic-perspective-for-spectral-dimensionality-reduction-insights-and-new-models.html.

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