  This manuscript seeks to bridge two seemingly disjoint paradigms of
nonparametric regression estimation based on smoothness assumptions and shape
constraints. The proposed approach is motivated by a conceptually simple
observation: Every Lipschitz function is a sum of monotonic and linear
functions. This principle is further generalized to the higher-order
monotonicity and multivariate covariates. A family of estimators is proposed
based on a sample-splitting procedure, which inherits desirable methodological,
theoretical, and computational properties of shape-restricted estimators. Our
theoretical analysis provides convergence guarantees of the estimator under
heteroscedastic and heavy-tailed errors, as well as adaptive properties to the
complexity of the true regression function. The generality of the proposed
decomposition framework is demonstrated through new approximation results, and
extensive numerical studies validate the theoretical properties and empirical
evidence for the practicalities of the proposed estimation framework.
