Minmax Trend Filtering: Generalizations of Total Variation Denoising via a Local Minmax/Maxmin Formula

Abstract

Total Variation Denoising (TVD) is a fundamental denoising and smoothing method. In this article, we identify a new local minmax/maxmin formula producing two estimators which sandwich the univariate TVD estimator at every point. Operationally, this formula gives a local definition of TVD as a minmax/maxmin of a simple function of local averages. Moreover we find that this minmax/maxmin formula is generalizeable and can be used to define other TVD like estimators. In this article we propose and study higher order polynomial versions of TVD which are defined pointwise lying between minmax and maxmin optimizations of penalized local polynomial regressions over intervals of different scales. These appear to be new nonparametric regression methods, different from usual Trend Filtering and any other existing method in the nonparametric regression toolbox. We call these estimators Minmax Trend Filtering (MTF). We show how the proposed local definition of TVD/MTF estimator makes it tractable to bound pointwise estimation errors in terms of a local bias variance like trade-off. This type of local analysis of TVD/MTF is new and arguably simpler than existing analyses of TVD/Trend Filtering. In particular, apart from minimax rate optimality over bounded variation and piecewise polynomial classes, our pointwise estimation error bounds also enable us to derive local rates of convergence for (locally) Holder Smooth signals. These local rates offer a new pointwise explanation of local adaptivity of TVD/MTF instead of global (MSE) based justifications.