Title: The Scaling Limit of High-Dimensional Online Independent Component Analysis,
Abstract: We analyze the dynamics of an online algorithm for independent component
analysis in the high-dimensional scaling limit. As the ambient dimension tends
to infinity, and with proper time scaling, we show that the time-varying joint
empirical measure of the target feature vector and the estimates provided by
the algorithm will converge weakly to a deterministic measured-valued process
that can be characterized as the unique solution of a nonlinear PDE. Numerical
solutions of this PDE, which involves two spatial variables and one time
variable, can be efficiently obtained. These solutions provide detailed
information about the performance of the ICA algorithm, as many practical
performance metrics are functionals of the joint empirical measures. Numerical
simulations show that our asymptotic analysis is accurate even for moderate
dimensions. In addition to providing a tool for understanding the performance
of the algorithm, our PDE analysis also provides useful insight. In particular,
in the high-dimensional limit, the original coupled dynamics associated with
the algorithm will be asymptotically "decoupled", with each coordinate
independently solving a 1-D effective minimization problem via stochastic
gradient descent. Exploiting this insight to design new algorithms for
achieving optimal trade-offs between computational and statistical efficiency
may prove an interesting line of future research.