The classical Canonical Correlation Analysis (CCA) identifies the correlations between two sets of multivariate variables based on their covariance, which has been widely applied in diverse fields such as computer vision, natural language processing, and speech analysis. Despite its popularity, CCA can encounter challenges in explaining correlations between two variable sets within high-dimensional data contexts. Thus, this paper studies Sparse Canonical Correlation Analysis (SCCA) that enhances the interpretability of CCA. We first show that SCCA generalizes three well-known sparse optimization problems, sparse PCA, sparse SVD, and sparse regression, which are all classified as NP-hard problems. This result motivates us to develop strong formulations and efficient algorithms. Our main contributions include (i) the introduction of a combinatorial formulation that captures the essence of SCCA and allows the development of exact and approximation algorithms; (ii) the establishment of the complexity results for two low-rank special cases of SCCA; and (iii) the derivation of an equivalent mixed-integer semidefinite programming model that facilitates a specialized branch-and-cut algorithm with analytical cuts. The effectiveness of our proposed formulations and algorithms is validated through numerical experiments.

Section 4 derives an equivalent mixed-integer SDP (MISDP) reformulation for SCCA.

The classical Canonical Correlation Analysis (CCA) identifies the correlations between two sets of multivariate variables based on their covariance, which has been widely applied in diverse fields such as computer vision, natural language processing, and speech analysis. Despite its popularity, CCA can encounter challenges in explaining correlations between two variable sets within high-dimensional data contexts. Thus, this paper studies Sparse Canonical Correlation Analysis (SCCA) that enhances the interpretability of CCA. We first show that SCCA generalizes three well-known sparse optimization problems, sparse PCA, sparse SVD, and sparse regression, which are all classified as NP-hard problems. This result motivates us to develop strong formulations and efficient algorithms. Our main contributions include (i) the introduction of a combinatorial formulation that captures the essence of SCCA and allows the development of exact and approximation algorithms; (ii) the establishment of the complexity results for two low-rank special cases of SCCA; and (iii) the derivation of an equivalent mixed-integer semidefinite programming model that facilitates a specialized branch-and-cut algorithm with analytical cuts.

Sparse attention as a efficient method can significantly decrease the computation cost, but current sparse attention tend to rely on window self attention which block the global information flow. For this problem, we present Shifted Cross Chunk Attention (SCCA), using different KV shifting strategy to extend respective field in each attention layer. Except, we combine Dilated Attention(DA) and Dilated Neighborhood Attention(DNA) to present Shifted Dilated Attention(SDA). Both SCCA and SDA can accumulate attention results in multi head attention to obtain approximate respective field in full attention. In this paper, we conduct language modeling experiments using different pattern of SCCA and combination of SCCA and SDA. The proposed shifted cross chunk attention (SCCA) can effectively extend large language models (LLMs) to longer context combined with Positional interpolation(PI) and LoRA than current sparse attention. Notably, SCCA adopts LLaMA2 7B from 4k context to 8k in single V100. This attention pattern can provide a Plug-and-play fine-tuning method to extend model context while retaining their original architectures, and is compatible with most existing techniques.

It can be challenging to perform an integrative statistical analysis of multi-view high-dimensional data acquired from different experiments on each subject who participated in a joint study. Canonical Correlation Analysis (CCA) is a statistical procedure for identifying relationships between such data sets. In that context, Structured Sparse CCA (ScSCCA) is a rapidly emerging methodological area that aims for robust modeling of the interrelations between the different data modalities by assuming the corresponding CCA directional vectors to be sparse. Although it is a rapidly growing area of statistical methodology development, there is a need for developing related methodologies in the Bayesian paradigm. In this manuscript, we propose a novel ScSCCA approach where we employ a Bayesian infinite factor model and aim to achieve robust estimation by encouraging sparsity in two different levels of the modeling framework. Firstly, we utilize a multiplicative Half-Cauchy process prior to encourage sparsity at the level of the latent variable loading matrices. Additionally, we promote further sparsity in the covariance matrix by using graphical horseshoe prior or diagonal structure. We conduct multiple simulations to compare the performance of the proposed method with that of other frequently used CCA procedures, and we apply the developed procedures to analyze multi-omics data arising from a breast cancer study.

We present a novel method for solving Canonical Correlation Analysis (CCA) in a sparse convex framework using a least squares approach. The presented method focuses on the scenario when one is interested in (or limited to) a primal representation for the first view while having a dual representation for the second view. Sparse CCA (SCCA) minimises the number of features used in both the primal and dual projections while maximising the correlation between the two views. The method is demonstrated on two paired corpuses of English-French and English-Spanish for mate-retrieval. We are able to observe, in the mate-retreival, that when the number of the original features is large SCCA outperforms Kernel CCA (KCCA), learning the common semantic space from a sparse set of features.