Local strong homogeneity of a regularized estimator

This paper deals with regularized pointwise estimation of discrete signals which contain large strongly homogeneous zones, where typically they are constant, or linear, or more generally satisfy a linear equation. The estimate is defined as the minimizer of an objective function combining a quadratic data-fidelity term and a regularization prior term. The latter term is the sum of the values obtained by applying a potential function (PF) to each component, called a difference, of a linear transform of the signal. Minimizers of functions of this form arise in various settings in statistics and optimization. The features exhibited by such an estimate are closely related to the shape of the PF. Our goal is to determine estimators providing solutions which involve large strongly homogeneous zones where more precisely the differences are null in spite of the noise corrupting the data. To this end, we require that the strongly homogeneous zones, recovered by the estimator, be insensitive to any variation of the data inside a small open ball. More generally, this requirement is addressed to any local or global minimizer of the objective function whose local behavior with respect to the data gives rise to a locally continuous minimizer function. On the one hand, we show that if the PF is smooth at zero, then all the data, yielding minimizers with large, strongly homogeneous zones, are contained in a closed, negligible set. The chance that noisy data generate such minimizers is null. In contrast, if the PF is nonsmooth at zero, then for almost all data, the strongly homogeneous zones recovered by a minimizer function are preserved constant under any small perturbation of the data. The data domain is thus organized into volumes whose elements yield minimizers which share the same strongly homogeneous zones. This explains why the solutions, obtained using nonsmooth-at-zero PFs, exhibit strongly homogeneous zones. These theoretical results are illustrated using a numerical example. Our analysis can be extended to general functions combining smooth and nonsmooth terms.