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# 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* :-

#### Related Material

BibTeX

```
@InProceedings{/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 /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 - /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* :-