Deconvolution of Sparse Positive Spikes

Deconvolution is usually regarded as one of the ill-posed problems in applied mathematics if no constraints on the unknowns are assumed. This article discusses the idea of well-defined statistical models being a counterpart of the notion of well-posedness. We show that constraints on the unknowns such as positivity and sparsity can go a long way towards overcoming the ill-posedness in deconvolution. We show how these issues are dealt with in a parametric deconvolution model introduced recently. From the same perspective we take a fresh look at two familiar deconvolvers: the widely used Jansson method and another one that minimizes the Kullback-Leibler divergence between observations and fitted values. In the latter case, we point out that in the context of deconvolution and the general linear inverse problems with positivity contraints, a counterpart of the EM algorithm exists for the problem of minimizing the Kullback-Leibler divergence. We graphically compare the performance of these deconvolvers using data simulated from a spike-convolution model and DNA sequencing data.