  Nonparametric methods are widely applicable to statistical inference
problems, since they rely on a few modeling assumptions. In this context, the
fresh look advocated here permeates benefits from variable selection and
compressive sampling, to robustify nonparametric regression against outliers -
that is, data markedly deviating from the postulated models. A variational
counterpart to least-trimmed squares regression is shown closely related to an
L0-(pseudo)norm-regularized estimator, that encourages sparsity in a vector
explicitly modeling the outliers. This connection suggests efficient solvers
based on convex relaxation, which lead naturally to a variational M-type
estimator equivalent to the least-absolute shrinkage and selection operator
(Lasso). Outliers are identified by judiciously tuning regularization
parameters, which amounts to controlling the sparsity of the outlier vector
along the whole robustification path of Lasso solutions. Reduced bias and
enhanced generalization capability are attractive features of an improved
estimator obtained after replacing the L0-(pseudo)norm with a nonconvex
surrogate. The novel robust spline-based smoother is adopted to cleanse load
curve data, a key task aiding operational decisions in the envisioned smart
grid system. Computer simulations and tests on real load curve data corroborate
the effectiveness of the novel sparsity-controlling robust estimators.
