# Incremental Canonical Correlation Analysis
## 1 Introduction  
Canonical correlation analysis, as a mathematical statistical tool, was proposed by [PERSON] [1] in 1936. It is mainly used to analyze the association between two groups of random variables. In 2003, [PERSON] et al. [2] briefly reviewed CCA and provided a generalized framework with optimization and kernel tricks. CCA is relevant to mutual information, thus it was used early in information retrieval. Later on, it was broadly applied to reduce dimension [3; 4; 5], clustering [6; 7], regression [8; 9], word embedding [10; 11; 12], and discriminant learning [13; 14; 15]. In such studies, plain CCA, which was purely used there, ignored the nonlinearity or geometrical structure within data sets. To this end, many variants of CCA emerged in diverse manners. For instance, sparse CCA [16] accounted for feature sparseness for more intuitive interpretation via \(L_{1}\) regularization. Graph-embedded-based CCA (GCCA) [17; 18; 19] integrated local manifold structure within data for image classification by graph-embedded subspace learning with auxiliary methods [20; 21]. More typically, deep CCA (DCCA) [22] trained the neural network under the guidance of the objective of CCA to model the data nonlinearity. In addition, generalized CCA (GCCA) [23] extended CCA with latent common feature representation. Similar to DCCA, deep GCCA [24] also realized CCA with deep architectures usingthe objective GCCA. In the above methods, eigenvalue decomposition, matrix inversion, or both are, as usual, involved. This obstructed the applicability of CCA and its variants in large-scale datasets or online learning tasks. To make CCA efficient, many methods endeavored to explore efficient optimization algorithms, which address the computational inefficiency for the high-dimensional issue of CCA in different settings [25; 26; 27; 28; 29; 30; 31; 32; 33]. [PERSON] et al. [34] showed that CCA was equivalent to the constrained least squares optimization problem. [PERSON] et al. [35] calculated CCA by QR decomposition which denoted that a matrix \(A(A\in R^{m\times n})\) was divided into a product of an orthogonal matrix \(Q\) and an upper triangular matrix \(R\). However, the large scale of the data would still slow the calculation process. Afterwards, [PERSON] et al. [36] proposed a fast CCA algorithm by employing randomized dimensionality reduction transform [37]. Recently, other methods to solve CCA were based on the methods of gradient descent. [PERSON] et al. [33] introduced the enhanced approximate gradient mechanism and further extended it to stochastic optimization problems. In [38; 39], CCA was reformulated into a series of least-square problems, which were solved by fast gradient descent. Such algorithms are still slow in reality. Thus, they are unsuited for online learning tasks. To address this issue, we develop a simple yet efficient incremental canonical correlation analysis (ICCA) to accelerate plain CCA. Particularly, ICCA maintains constant memory storage for both the mean and covariance matrices, etc., in an adaptive manner. This does not require storage of large amounts of history data, thus a certain space complexity remains. To reduce time complexity, we readily avoid matrix inverse in the learning process by using sequential Karhunen-Loeve (SKL) or singular value decomposition (SVD), which is still efficient in the case that the number of samples is sufficiently few. Driven by visual tracking as an online learning task, which locates a given target in a long video sequence, we apply the proposed ICCA for this task through some essential modifications to ensure its efficacy. Extensive experiments on several video sequences show the superiority of ICCA as compared to several classical trackers. Importantly, the efficiency of ICCA is promising.