Optimal Whitening and Decorrelation and a New Look at Probabilistic Canonical Correlation Analysis

Korbinian Strimmer (University of Manchester)

Friday 27th April, 2018 15:00-16:00 Maths 311B


Whitening, or sphering, is a common preprocessing step in statistical analysis to transform random variables to orthogonality. However, due to rotational freedom there are infinitely many possible whitening procedures. Consequently, there is a diverse range of sphering methods in use, such as based on principal component analysis, Cholesky matrix decomposition and Mahalanobis transformation,
among others.

In my talk I will first provide an overview of the underlying theory to establish some connections among the most common whitening methods [1]. In particular, I will investigate how constraints on the cross-covariance and the cross-correlation matrix between sphered and original variables allows to break the rotational invariance and to identify optimal whitening transformations. As a result, this leads to two recommended whitening transformations: one for for producing sphered variables that are maximally similar to the original variables (ZCA-cor), and a second different one for obtaining maximally compressed whitened variables (PCA-cor).

In the second part of the talk I will then focus on Probabilistic Canonical Correlation Analysis (PCCA) viewed from a whitening perspective. Only relatively recently, the probabilistic latent variable view of CCA has become more prevalent (e.g. [2]). Correspondingly, I will discuss a novel two-layer latent variable probabilistic generative model of CCA rooted in the framework of whitening and multivariate regression [3]. The advantages of this variant of PCCA include non-ambiguity of the latent variables, flexibility to allow non-normal generative variables, possibility of negative canonical correlations, as well as simplicity of interpretation on all levels of the model. Furthermore, this approach is amenable to computationally efficient estimation in high-dimensional settings using regularized inference.

This is joint work with Agnan Kessy, Alexandra Lewin, and Takoua Jendoubi.

[1] https://arxiv.org/abs/1512.00809, http://dx.doi.org/10.1080/00031305.2016.1277159
[2] https://www.stat.berkeley.edu/~jordan/688.pdf
[3] https://arxiv.org/abs/1802.03490

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