Quantum-inspired canonical correlation analysis for exponentially large dimensional data.

Canonical correlation analysis (CCA) serves to identify statistical dependencies between pairs of multivariate data. However, its application to high-dimensional data is limited due to considerable computational complexity. As an alternative to the conventional CCA approach that requires polynomial computational time, we propose an algorithm that approximates CCA using quantum-inspired computations with computational time proportional to the logarithm of the input dimensionality. The computational efficiency and performance of the proposed quantum-inspired CCA (qiCCA) algorithm are experimentally evaluated on synthetic and real datasets. Furthermore, the fast computation provided by qiCCA allows directly applying CCA even after nonlinearly mapping raw input data into high-dimensional spaces. The conducted experiments demonstrate that, as a result of mapping raw input data into the high-dimensional spaces with the use of second-order monomials, qiCCA extracts more correlations compared with the linear CCA and achieves comparable performance with state-of-the-art nonlinear variants of CCA on several datasets. These results confirm the appropriateness of the proposed qiCCA and the high potential of quantum-inspired computations in analyzing high-dimensional data.