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

Neil D. Lawrence
,  13, 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
@InProceedings{pmlr-v-lawrence-unifying12, title = {A Unifying Probabilistic Perspective for Spectral Dimensionality Reduction: Insights and New Models}, author = {Neil D. Lawrence}, year = {}, editor = {}, volume = {13}, url = {http://inverseprobability.com/publications/lawrence-unifying12.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 Conference Paper %T A Unifying Probabilistic Perspective for Spectral Dimensionality Reduction: Insights and New Models %A Neil D. Lawrence %B %C Proceedings of Machine Learning Research %D %E %F pmlr-v-lawrence-unifying12 %I PMLR %J Proceedings of Machine Learning Research %P -- %U http://inverseprobability.com %V %W PMLR %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 - CPAPER TI - A Unifying Probabilistic Perspective for Spectral Dimensionality Reduction: Insights and New Models AU - Neil D. Lawrence BT - PY - DA - ED - ID - pmlr-v-lawrence-unifying12 PB - PMLR SP - DP - PMLR EP - L1 - UR - http://inverseprobability.com/publications/lawrence-unifying12.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.. (). A Unifying Probabilistic Perspective for Spectral Dimensionality Reduction: Insights and New Models. , in PMLR :-

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