Residual Component Analysis

Alfredo A. KalaitzisNeil D. Lawrence
,  29, 2012.

Abstract

Probabilistic principal component analysis (PPCA) seeks a low dimensional representation of a data set in the presence of independent spherical Gaussian noise, $\Sigma = \sigma^2\mathbf{I}$. The maximum likelihood solution for the model is an eigenvalue problem on the sample covariance matrix. In this paper we consider the situation where the data variance is already partially explained by other factors, e.g. conditional dependencies between the covariates, or temporal correlations leaving some residual variance. We decompose the residual variance into its components through a generalised eigenvalue problem, which we call residual component analysis (RCA). We explore a range of new algorithms that arise from the framework, including one that factorises the covariance of a Gaussian density into a low-rank and a sparse-inverse component. We illustrate the ideas on the recovery of a protein-signaling network, a gene expression time-series data set and the recovery of the human skeleton from motion capture 3-D cloud data.

Cite this Paper


BibTeX
@InProceedings{pmlr-v-kalaitzis-rca12, title = {Residual Component Analysis}, author = {Alfredo A. Kalaitzis and Neil D. Lawrence}, year = {}, editor = {}, volume = {29}, address = {San Francisco, CA}, url = {http://inverseprobability.com/publications/kalaitzis-rca12.html}, abstract = {Probabilistic principal component analysis (PPCA) seeks a low dimensional representation of a data set in the presence of independent spherical Gaussian noise, $\Sigma = \sigma^2\mathbf{I}$. The maximum likelihood solution for the model is an eigenvalue problem on the sample covariance matrix. In this paper we consider the situation where the data variance is already partially explained by other factors, e.g. conditional dependencies between the covariates, or temporal correlations leaving some residual variance. We decompose the residual variance into its components through a generalised eigenvalue problem, which we call residual component analysis (RCA). We explore a range of new algorithms that arise from the framework, including one that factorises the covariance of a Gaussian density into a low-rank and a sparse-inverse component. We illustrate the ideas on the recovery of a protein-signaling network, a gene expression time-series data set and the recovery of the human skeleton from motion capture 3-D cloud data.} }
Endnote
%0 Conference Paper %T Residual Component Analysis %A Alfredo A. Kalaitzis %A Neil D. Lawrence %B %C Proceedings of Machine Learning Research %D %E %F pmlr-v-kalaitzis-rca12 %I PMLR %J Proceedings of Machine Learning Research %P -- %U http://inverseprobability.com %V %W PMLR %X Probabilistic principal component analysis (PPCA) seeks a low dimensional representation of a data set in the presence of independent spherical Gaussian noise, $\Sigma = \sigma^2\mathbf{I}$. The maximum likelihood solution for the model is an eigenvalue problem on the sample covariance matrix. In this paper we consider the situation where the data variance is already partially explained by other factors, e.g. conditional dependencies between the covariates, or temporal correlations leaving some residual variance. We decompose the residual variance into its components through a generalised eigenvalue problem, which we call residual component analysis (RCA). We explore a range of new algorithms that arise from the framework, including one that factorises the covariance of a Gaussian density into a low-rank and a sparse-inverse component. We illustrate the ideas on the recovery of a protein-signaling network, a gene expression time-series data set and the recovery of the human skeleton from motion capture 3-D cloud data.
RIS
TY - CPAPER TI - Residual Component Analysis AU - Alfredo A. Kalaitzis AU - Neil D. Lawrence BT - PY - DA - ED - ID - pmlr-v-kalaitzis-rca12 PB - PMLR SP - DP - PMLR EP - L1 - UR - http://inverseprobability.com/publications/kalaitzis-rca12.html AB - Probabilistic principal component analysis (PPCA) seeks a low dimensional representation of a data set in the presence of independent spherical Gaussian noise, $\Sigma = \sigma^2\mathbf{I}$. The maximum likelihood solution for the model is an eigenvalue problem on the sample covariance matrix. In this paper we consider the situation where the data variance is already partially explained by other factors, e.g. conditional dependencies between the covariates, or temporal correlations leaving some residual variance. We decompose the residual variance into its components through a generalised eigenvalue problem, which we call residual component analysis (RCA). We explore a range of new algorithms that arise from the framework, including one that factorises the covariance of a Gaussian density into a low-rank and a sparse-inverse component. We illustrate the ideas on the recovery of a protein-signaling network, a gene expression time-series data set and the recovery of the human skeleton from motion capture 3-D cloud data. ER -
APA
Kalaitzis, A.A. & Lawrence, N.D.. (). Residual Component Analysis. , in PMLR :-

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