# Prediction bounds for higher order total variation regularized least squares

arXiv: https://arxiv.org/abs/1904.10871 · HTML: https://mlsanity.com/paper/1904.10871/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Apr 2019.

## Built on by (most cited first, 7 of 7)

- [Optimal Dynamic Regret in Exp-Concave Online Learning](https://mlsanity.com/paper/2104.11824/) (2021)
- [Optimal Dynamic Regret in Proper Online Learning with Strongly Convex Losses and Beyond](https://mlsanity.com/paper/2201.08905/) (2022)
- [Multivariate Trend Filtering for Lattice Data](https://mlsanity.com/paper/2112.14758/) (2021)
- [Variance estimation in graphs with the fused lasso](https://mlsanity.com/paper/2207.12638/) (2022)
- [A Cross Validation Framework for Signal Denoising with Applications to Trend Filtering, Dyadic CART and Beyond](https://mlsanity.com/paper/2201.02654/) (2022)
- [Denoising and change point localisation in piecewise-constant high-dimensional regression coefficients](https://mlsanity.com/paper/2110.14298/) (2021)
- [A new approach to locally adaptive polynomial regression](https://mlsanity.com/paper/2412.19802/) (2024)

## Abstract (arXiv)

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 ℓ1-penalized least squares estimator with "interpolating vectors" to upper-bound the "effective sparsity". This allows one to show that the ℓ1-penalty on the kth order differences leads to an estimator that can adapt to the number of jumps in the (k-1)th order differences of the underlying signal or an approximation thereof. We show the result for k ∈ 1,2,3,4 and indicate how it could be derived for general k∈ mathbbN.