Large-Scale Approximate Kernel Canonical Correlation Analysis

Abstract

Kernel canonical correlation analysis (KCCA) is a nonlinear multi-view representation learning technique with broad applicability in statistics and machine learning. Although there is a closed-form solution for the KCCA objective, it involves solving an $N\times N$ eigenvalue system where $N$ is the training set size, making its computational requirements in both memory and time prohibitive for large-scale problems. Various approximation techniques have been developed for KCCA. A commonly used approach is to first transform the original inputs to an $M$-dimensional random feature space so that inner products in the feature space approximate kernel evaluations, and then apply linear CCA to the transformed inputs. In many applications, however, the dimensionality $M$ of the random feature space may need to be very large in order to obtain a sufficiently good approximation; it then becomes challenging to perform the linear CCA step on the resulting very high-dimensional data matrices. We show how to use a stochastic optimization algorithm, recently proposed for linear CCA and its neural-network extension, to further alleviate the computation requirements of approximate KCCA. This approach allows us to run approximate KCCA on a speech dataset with $1.4$ million training samples and a random feature space of dimensionality $M=100000$ on a typical workstation.