Multivariate extensions of isotonic regression and total variation denoising via entire monotonicity and Hardy-Krause variation

Understand

We consider the problem of nonparametric regression when the covariate is $d$-dimensional, where $d \geq 1$.

• In this paper we introduce and study two nonparametric least squares estimators (LSEs) in this setting---the entirely monotonic LSE and the constrained Hardy-Krause variation LSE.
• We show that these two LSEs are natural generalizations of univariate isotonic regression and univariate total variation denoising, respectively, to multiple dimensions.
• We discuss the characterization and computation of these two LSEs obtained from $n$ data points.