Sparse interpolation, also called Prony's method or exponential analysis, consists in fitting a linear combination of exponential functions. A drawback of the method is the fact that the interpolation data need to be collected equidistantly. 

Variable projection applies to so-called separable problems, in which data are fitted by a linear combination of simple functions characterised by some nonlinear parameters, such as the exponential functions. The nonlinear parameters are computed separately through optimisation and the linear coefficients are the solution of a least squares problem. A drawback is that the method, when applied to higher-dimensional problems, easily gets stuck in a local minimum, unless one can supply a quite accurate starting point for the optimisation.

We consider some low-dimensional separable least squares problems of a Prony-like type, which offer the advantage that the objective function can be written down analytically.