  Beside the minimization of the prediction error, two of the most desirable
properties of a regression scheme are stability and interpretability. Driven by
these principles, we propose continuous-domain formulations for one-dimensional
regression problems. In our first approach, we use the Lipschitz constant as a
regularizer, which results in an implicit tuning of the overall robustness of
the learned mapping. In our second approach, we control the Lipschitz constant
explicitly using a user-defined upper-bound and make use of a
sparsity-promoting regularizer to favor simpler (and, hence, more
interpretable) solutions. The theoretical study of the latter formulation is
motivated in part by its equivalence, which we prove, with the training of a
Lipschitz-constrained two-layer univariate neural network with rectified linear
unit (ReLU) activations and weight decay. By proving representer theorems, we
show that both problems admit global minimizers that are continuous and
piecewise-linear (CPWL) functions. Moreover, we propose efficient algorithms
that find the sparsest solution of each problem: the CPWL mapping with the
least number of linear regions. Finally, we illustrate numerically the outcome
of our formulations.
