  Canonical Correlation Analysis (CCA) is a linear representation learning
method that seeks maximally correlated variables in multi-view data. Non-linear
CCA extends this notion to a broader family of transformations, which are more
powerful in many real-world applications. Given the joint probability, the
Alternating Conditional Expectation (ACE) algorithm provides an optimal
solution to the non-linear CCA problem. However, it suffers from limited
performance and an increasing computational burden when only a finite number of
samples is available. In this work we introduce an information-theoretic
compressed representation framework for the non-linear CCA problem (CRCCA),
which extends the classical ACE approach. Our suggested framework seeks compact
representations of the data that allow a maximal level of correlation. This way
we control the trade-off between the flexibility and the complexity of the
model. CRCCA provides theoretical bounds and optimality conditions, as we
establish fundamental connections to rate-distortion theory, the information
bottleneck and remote source coding. In addition, it allows a soft
dimensionality reduction, as the compression level is determined by the mutual
information between the original noisy data and the extracted signals. Finally,
we introduce a simple implementation of the CRCCA framework, based on lattice
quantization.
