Appl Math Optim 53:185–208 (2006)
2006 Springer Science+Business Media, Inc.
Stability of the Minimizers of Least Squares
with a Non-Convex Regularization.
Part I: Local Behavior
and M. Nikolova
LAMFA UMR 6140, Universit´e de Picardie,
33 rue Saint-Leu, 90039 Amien Cedex, France
CMLA UMR 8536, ENS de Cachan,
61 av. du President Wilson, 94235 Cachan Cedex, France
Abstract. Many estimation problems amount to minimizing a piecewise C
jective function, with m ≥ 2, composed of a quadratic data-ﬁdelity term and a
general regularization term. It is widely accepted that the minimizers obtained us-
ing non-convex and possibly non-smooth regularization terms are frequently good
estimates. However, few facts are known on the ways to control properties of
these minimizers. This work is dedicated to the stability of the minimizers of such
objective functions with respect to variations of the data. It consists of two parts:
ﬁrst we consider all local minimizers, whereas in a second part we derive results
on global minimizers. In this part we focus on data points such that every local
minimizer is isolated and results from a C
local minimizer function, deﬁned on
some neighborhood. We demonstrate that all data points for which this fails form a
set whose closure is negligible.
Key Words. Stability analysis, Regularized least squares, Non-smooth analysis,
Non-convex analysis, Signal and image processing.
AMS Classiﬁcation. 26B, 49J, 68U, 94A.
This is the ﬁrst of two papers devoted to the stability of minimizers of regularized least
squares objective functions as customarily used in signal and image reconstruction. In