# Mesh-Based Solutions for Nonparametric Penalized Regression

arXiv: https://arxiv.org/abs/2112.03428 · HTML: https://mlsanity.com/paper/2112.03428/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 2 of its 12 references that are in the map.
- First posted on arXiv 7 Dec 2021.

## Builds on (most cited first, 2 of 2)

- [Adaptive piecewise polynomial estimation via trend filtering](https://mlsanity.com/paper/1304.2986/) (2013)
- [Fast and Flexible ADMM Algorithms for Trend Filtering](https://mlsanity.com/paper/1406.2082/) (2014)

## Abstract (arXiv)

It is often of interest to estimate regression functions non-parametrically. Penalized regression (PR) is one statistically-effective, well-studied solution to this problem. Unfortunately, in many cases, finding exact solutions to PR problems is computationally intractable. In this manuscript, we propose a mesh-based approximate solution (MBS) for those scenarios. MBS transforms the complicated functional minimization of NPR, to a finite parameter, discrete convex minimization; and allows us to leverage the tools of modern convex optimization. We show applications of MBS in a number of explicit examples (including both uni- and multi-variate regression), and explore how the number of parameters must increase with our sample-size in order for MBS to maintain the rate-optimality of NPR. We also give an efficient algorithm to minimize the MBS objective while effectively leveraging the sparsity inherent in MBS.