[Paper Review] An Information-Theoretic Framework for Non-linear Canonical Correlation Analysis.

Amichai Painsky, Meir Feder+1 more|arXiv (Cornell University)|Oct 31, 2018

Face and Expression Recognition58 references4 citations

TL;DR

This paper proposes ITCCA, an information-theoretic framework for non-linear canonical correlation analysis that optimizes compressed representations to maximize correlation while controlling complexity via mutual information. It achieves superior performance with reduced computational cost compared to ACE and non-linear CCA, linking the method to rate-distortion theory and the information bottleneck.

ABSTRACT

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 for many real-world applications. Given the joint probability, the Alternating Conditional Expectation (ACE) 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 observations is available. In this work we introduce an information-theoretic framework for the non-linear CCA problem (ITCCA), which extends the classical ACE approach. Our suggested framework seeks compressed 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 representation. Our approach demonstrates favorable performance at a reduced computational burden, compared to non-linear alternatives, in a finite sample size regime. Further, ITCCA 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 implies a soft dimensionality reduction, as the compression level is measured (and governed) by the mutual information between the original noisy data and the signals that we extract.

Motivation & Objective

• To address the limitations of non-linear CCA methods like ACE, which suffer from high computational cost and poor finite-sample performance.
• To develop a framework that controls the trade-off between representation flexibility and complexity in non-linear CCA.
• To establish theoretical connections between non-linear CCA and information-theoretic principles such as rate-distortion theory and the information bottleneck.
• To enable soft dimensionality reduction by governing compression through mutual information between original data and extracted signals.

Proposed method

• ITCCA formulates non-linear CCA as an optimization problem that maximizes correlation between transformed views under a mutual information constraint.
• The method uses mutual information as a proxy for representation complexity, enabling control over the trade-off between correlation and model simplicity.
• It extends the ACE approach by embedding it within an information-theoretic framework, ensuring optimality under finite sample conditions.
• The framework draws theoretical parallels to rate-distortion theory and remote source coding, providing principled bounds on performance.
• It introduces a soft dimensionality reduction mechanism where the compression level is explicitly governed by the mutual information between input data and extracted features.
• The optimization process is designed to be computationally efficient, reducing burden compared to traditional non-linear CCA methods.

Experimental results

Research questions

• RQ1 How can non-linear CCA be formulated within an information-theoretic framework to improve finite-sample performance?
• RQ2 What is the role of mutual information in controlling the complexity of non-linear representations in CCA?
• RQ3 How does ITCCA relate to established information-theoretic principles such as the information bottleneck and rate-distortion theory?
• RQ4 Can ITCCA achieve higher correlation with lower computational cost than ACE and other non-linear CCA methods?
• RQ5 How does the mutual information between original data and extracted signals enable soft dimensionality reduction in multi-view learning?

Key findings

• ITCCA achieves higher correlation between views than ACE and other non-linear CCA methods under finite sample conditions.
• The method reduces computational burden compared to traditional non-linear CCA approaches, making it more scalable.
• ITCCA provides theoretical bounds and optimality conditions by linking the problem to rate-distortion theory and the information bottleneck.
• The framework enables soft dimensionality reduction, where the compression level is explicitly controlled by mutual information.
• The approach establishes a principled connection between non-linear CCA and remote source coding, enhancing theoretical grounding.
• Empirical results demonstrate favorable performance in terms of correlation and efficiency, especially in low-data regimes.