# Tensor Canonical Correlation Analysis for Multi-view Dimension Reduction

arXiv: https://arxiv.org/abs/1502.02330 · HTML: https://mlsanity.com/paper/1502.02330/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Feb 2015.

## Built on by (most cited first, 14 of 14)

- [Exploring and Exploiting Uncertainty for Incomplete Multi-View Classification](https://mlsanity.com/paper/2304.05165/) (2023)
- [An Online Riemannian PCA for Stochastic Canonical Correlation Analysis](https://mlsanity.com/paper/2106.07479/) (2021)
- [Simultaneous Dimensionality Reduction: A Data Efficient Approach for Multimodal Representations Learning](https://mlsanity.com/paper/2310.04458/) (2023)
- [Evidential Deep Partial Multi-View Classification With Discount Fusion](https://mlsanity.com/paper/2408.13123/) (2024)
- [Tensor-based Multi-view Spectral Clustering via Shared Latent Space](https://mlsanity.com/paper/2207.11559/) (2022)
- [Multi-Tensor Network Representation for High-Order Tensor Completion](https://mlsanity.com/paper/2109.04022/) (2021)
- [Tensor Generalized Canonical Correlation Analysis](https://mlsanity.com/paper/2302.05277/) (2023)
- [KNH: Multi-View Modeling with K-Nearest Hyperplanes Graph for Misinformation Detection](https://mlsanity.com/paper/2102.07857/) (2021)
- [Attentive Multi-View Deep Subspace Clustering Net](https://mlsanity.com/paper/2112.12506/) (2021)
- [Cross-view Graph Contrastive Representation Learning on Partially Aligned Multi-view Data](https://mlsanity.com/paper/2211.04906/) (2022)
- [Multi-view Fuzzy Representation Learning with Rules based Model](https://mlsanity.com/paper/2309.11473/) (2023)
- [Towards Robust Uncertainty-Aware Incomplete Multi-View Classification](https://mlsanity.com/paper/2409.06270/) (2024)
- [Higher Order Correlation Analysis for Multi-View Learning](https://mlsanity.com/paper/2201.11949/) (2022)
- [Adaptive incomplete multi-view learning via tensor graph completion](https://mlsanity.com/paper/2208.03710/) (2022)

## Abstract (arXiv)

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.