Higher-Order Total Variation Classes on Grids: Minimax Theory and Trend Filtering Methods
We consider the problem of estimating the values of a function over n nodes of a d-dimensional grid graph (having equal side lengths n 1/d ) from noisy observations. The function is assumed to be smooth, but is allowed to exhibit different amounts of smoothness at different regions in the grid. Such heterogeneity eludes classical measures of smoothness from nonparametric statistics, such as Holder smoothness. Meanwhile, total variation (TV) smoothness classes allow for heterogeneity, but are restrictive in another sense: only constant functions count as perfectly smooth (achieve zero TV). To move past this, we define two new higher-order TV classes, based on two ways of compiling the discrete derivatives of a parameter across the nodes. We relate these two new classes to Holder classes, and derive lower bounds on their minimax errors. We also analyze two naturally associated trend filtering methods; when d = 2, each is seen to be rate optimal over the appropriate class.
This paper considers graph-structured signal denoising  problem. The particular structure enforced involves a total variation type penalty involving higher order discrete derivatives. Optimal rates for penalized least-squares estimator is established and minimax lower bound is established. For the grid filtering case the upper bounds are established under an assumed conjecture. 
The paper is well written. The main proof techniques of the paper are borrowed from previous work but there are some additional technicalities required to establish the current results. I went over the proofs  (not 100% though) and it seems there is no significant issues with it. Overall I find the results of the paper as interesting contributions to trend filtering literature and I recommend to accept the paper. 
Minor: it is not clear why lines 86-87 are there in the supplementary material.