From exponential analysis to Padé approximation and tensor decomposition, in one and more dimensions

Abstract

Exponential analysis in signal processing is essentially what is known as sparse interpolation in computer algebra. We show how exponential analysis from regularly spaced samples is reformulated as Padé approximation from approximation theory and tensor decomposition from multilinear algebra. The univariate situation is briefly recalled and discussed in Sect. 1. The new connections from approximation theory and tensor decomposition to the multivariate generalization are the subject of Sect. 2. These connections immediately allow for some generalization of the sampling scheme, not covered by the current multivariate theory. An interesting computational illustration of the above in blind source separation is presented in Sect. 3.