Tensor Canonical Correlation Analysis for Multi-view Dimension Reduction (1502.02330v1)

Abstract: Canonical correlation analysis (CCA) has proven an effective tool for two-view dimension reduction due to its profound theoretical foundation and success in practical applications. In respect of multi-view learning, however, it is limited by its capability of only handling data represented by two-view features, while in many real-world applications, the number of views is frequently many more. Although the ad hoc way of simultaneously exploring all possible pairs of features can numerically deal with multi-view data, it ignores the high order statistics (correlation information) which can only be discovered by simultaneously exploring all features. Therefore, in this work, we develop tensor CCA (TCCA) which straightforwardly yet naturally generalizes CCA to handle the data of an arbitrary number of views by analyzing the covariance tensor of the different views. TCCA aims to directly maximize the canonical correlation of multiple (more than two) views. Crucially, we prove that the multi-view canonical correlation maximization problem is equivalent to finding the best rank-1 approximation of the data covariance tensor, which can be solved efficiently using the well-known alternating least squares (ALS) algorithm. As a consequence, the high order correlation information contained in the different views is explored and thus a more reliable common subspace shared by all features can be obtained. In addition, a non-linear extension of TCCA is presented. Experiments on various challenge tasks, including large scale biometric structure prediction, internet advertisement classification and web image annotation, demonstrate the effectiveness of the proposed method.

• The paper introduces TCCA to maximize canonical correlations across multiple views by converting the problem into a rank-1 tensor approximation.
• It employs Alternating Least Squares for optimization and extends to non-linear mappings via Kernel TCCA for enhanced feature projection.
• Experimental results on various tasks demonstrate that TCCA outperforms traditional bi-view CCA methods, capturing richer multi-view relationships.

Evaluation of Tensor Canonical Correlation Analysis for Multi-view Dimension Reduction

This paper presents an advancement in canonical correlation analysis (CCA) by extending its capacity to process data from multiple views through the development of Tensor Canonical Correlation Analysis (TCCA). The classical CCA is adept at finding correlations between two sets of variables, making it a fundamental method for dimension reduction in bi-view data sets. However, this method's limitation arises in scenarios where data is derived from more than two sources—common in many real-world applications.

Methodological Advancements

The primary contribution of this paper is the formulation of TCCA, which enables direct maximization of the canonical correlation among multiple views by analyzing their covariance tensor—the high-order covariance structure that encompasses all the views. TCCA transforms the problem of multi-view canonical correlation maximization into a rank-1 approximation problem of the data covariance tensor. This approach effectively integrates high-order statistics, thereby harnessing more comprehensive correlations than traditional pairwise methods.

To solve this optimization problem, TCCA utilizes the Alternating Least Squares (ALS) algorithm—a well-established technique in tensor decomposition. The paper also proposes a non-linear extension of TCCA (Kernel TCCA, KTCCA), allowing for non-linear mapping of features into higher dimensions using kernel methods.

Experimental Evaluation

The effectiveness of the proposed methodology is rigorously tested across various applications, including biometric structure prediction, internet advertisement classification, and web image annotation. In these tasks, the TCCA demonstrated superior performance over several benchmark methods: traditional two-view CCA, CCA-LS, and other state-of-the-art multi-view and dimension reduction techniques such as DSE and SSMVD. Notably, TCCA maintained robust results even as the dimensionality of the common subspace increased, showcasing its ability to capture richer multi-view correlations than previous approaches.

Implications and Future Work

The theoretical and practical implications of TCCA are substantial. From a theoretical standpoint, TCCA provides a comprehensive framework for incorporating high-order statistics into dimension reduction tasks, which can be particularly beneficial in scenarios with complex data structures. Practically, this enhancement promises improved efficiency and performance in machine learning tasks involving multi-view data, from classification to clustering and beyond.

Future research could focus on optimizing the computational efficiency of the TCCA and KTCCA methods. As indicated in the analysis of computational complexity, TCCA has a higher time and memory cost due to tensor decomposition. Thus, advances in this area could make TCCA even more applicable to large-scale problems. Additionally, exploring parallel computing possibilities or developing more efficient decomposition algorithms could significantly enhance TCCA's applicability in diverse and growing datascapes.

In conclusion, TCCA represents a substantial advancement in multi-view dimension reduction, capable of handling complex data relationships more effectively than existing CCA methods. This development can open new pathways in data-intensive applications where multi-source data integration is crucial.