Title: Maximally Correlated Principal Component Analysis,
Abstract: In the era of big data, reducing data dimensionality is critical in many
areas of science. Widely used Principal Component Analysis (PCA) addresses this
problem by computing a low dimensional data embedding that maximally explain
variance of the data. However, PCA has two major weaknesses. Firstly, it only
considers linear correlations among variables (features), and secondly it is
not suitable for categorical data. We resolve these issues by proposing
Maximally Correlated Principal Component Analysis (MCPCA). MCPCA computes
transformations of variables whose covariance matrix has the largest Ky Fan
norm. Variable transformations are unknown, can be nonlinear and are computed
in an optimization. MCPCA can also be viewed as a multivariate extension of
Maximal Correlation. For jointly Gaussian variables we show that the covariance
matrix corresponding to the identity (or the negative of the identity)
transformations majorizes covariance matrices of non-identity functions. Using
this result we characterize global MCPCA optimizers for nonlinear functions of
jointly Gaussian variables for every rank constraint. For categorical variables
we characterize global MCPCA optimizers for the rank one constraint based on
the leading eigenvector of a matrix computed using pairwise joint
distributions. For a general rank constraint we propose a block coordinate
descend algorithm and show its convergence to stationary points of the MCPCA
optimization. We compare MCPCA with PCA and other state-of-the-art
dimensionality reduction methods including Isomap, LLE, multilayer autoencoders
(neural networks), kernel PCA, probabilistic PCA and diffusion maps on several
synthetic and real datasets. We show that MCPCA consistently provides improved
performance compared to other methods.