# Prediction bounds for higher order total variation regularized least squares

## TL;DR

This paper derives adaptive prediction bounds for higher order total variation regularized least squares, showing the estimator's ability to adapt to the underlying signal's jump structure for orders 1 through 4.

## Contribution

It introduces a novel approach combining oracle inequalities and interpolating vectors to bound effective sparsity in trend filtering for higher orders.

## Key findings

- Establishes adaptive prediction bounds for orders 1 to 4.
- Demonstrates the estimator's adaptivity to the number of jumps.
- Provides a framework extendable to general order k.

## Abstract

We establish adaptive results for trend filtering: least squares estimation with a penalty on the total variation of $(k-1)^{\rm th}$ order differences. Our approach is based on combining a general oracle inequality for the $\ell_1$-penalized least squares estimator with "interpolating vectors" to upper-bound the "effective sparsity". This allows one to show that the $\ell_1$-penalty on the $k^{\text{th}}$ order differences leads to an estimator that can adapt to the number of jumps in the $(k-1)^{\text{th}}$ order differences of the underlying signal or an approximation thereof. We show the result for $k \in \{1,2,3,4\}$ and indicate how it could be derived for general $k\in \mathbb{N}$.