Optimization with sums of exponentials and applications

We present a method for optimization with sums of exponentials subject to positivity constraints and apply it to the modeling of empirical probability distribution functions and to the design of IIR filters with non-negative impulse response. Our approach uses exponents in a sparse arithmetic progression and hence is able to transform the positivity condition to a polynomial form that is computationally tractable. We show how to obtain initial values for the exponents by sparsifying a full progression and then present an iterative optimization procedure using gradient steps. The modeling and design examples indicate a good behavior of our method.