  In this paper, we investigate the recovery of the sparse representation of
data in general infinite-dimensional optimization problems regularized by
convex functionals. We show that it is possible to define a suitable
non-degeneracy condition on the minimal-norm dual certificate, extending the
well-established non-degeneracy source condition (NDSC) associated with total
variation regularized problems in the space of measures, as introduced in
(Duval and Peyr\'e, FoCM, 15:1315-1355, 2015). In our general setting, we need
to study how the dual certificate is acting, through the duality product, on
the set of extreme points of the ball of the regularizer, seen as a metric
space. This justifies the name Metric Non-Degenerate Source Condition (MNDSC).
More precisely, we impose a second-order condition on the dual certificate,
evaluated on curves with values in small neighbourhoods of a given collection
of n extreme points. By assuming the validity of the MNDSC, together with the
linear independence of the measurements on these extreme points, we establish
that, for a suitable choice of regularization parameters and noise levels, the
minimizer of the minimization problem is unique and is uniquely represented as
a linear combination of n extreme points. The paper concludes by obtaining
explicit formulations of the MNDSC for three problems of interest. First, we
examine total variation regularized deconvolution problems, showing that the
classical NDSC implies our MNDSC, and recovering a result similar to (Duval and
Peyr\'e, FoCM, 15:1315-1355, 2015). Then, we consider 1-dimensional BV
functions regularized with their BV-seminorm and pairs of measures regularized
with their mutual 1-Wasserstein distance. In each case, we provide explicit
versions of the MNDSC and formulate specific sparse representation recovery
results.
