# Fast and Flexible ADMM Algorithms for Trend Filtering

arXiv: https://arxiv.org/abs/1406.2082 · HTML: https://mlsanity.com/paper/1406.2082/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Jun 2014.

## Built on by (most cited first, 10 of 10)

- [Data fission: splitting a single data point](https://mlsanity.com/paper/2112.11079/) (2021)
- [A Priori Denoising Strategies for Sparse Identification of Nonlinear Dynamical Systems: A Comparative Study](https://mlsanity.com/paper/2201.12683/) (2022)
- [Multivariate Trend Filtering for Lattice Data](https://mlsanity.com/paper/2112.14758/) (2021)
- [Short term prediction of demand for ride hailing services: A deep learning approach](https://mlsanity.com/paper/2212.03956/) (2022)
- [Converting ADMM to a Proximal Gradient for Efficient Sparse Estimation](https://mlsanity.com/paper/2104.10911/) (2021)
- [Heterogeneous Federated Learning on a Graph](https://mlsanity.com/paper/2209.08737/) (2022)
- [Inverse Models for Estimating the Initial Condition of Spatio-Temporal Advection-Diffusion Processes](https://mlsanity.com/paper/2302.04134/) (2023)
- [Mitigating Data Consistency Induced Discrepancy in Cascaded Diffusion Models for Sparse-view CT Reconstruction](https://mlsanity.com/paper/2403.09355/) (2024)
- [Mesh-Based Solutions for Nonparametric Penalized Regression](https://mlsanity.com/paper/2112.03428/) (2021)
- [Trend Filtered Mixture of Experts for Automated Gating of High-Frequency Flow Cytometry Data](https://mlsanity.com/paper/2504.12287/) (2025)

## Abstract (arXiv)

This paper presents a fast and robust algorithm for trend filtering, a recently developed nonparametric regression tool. It has been shown that, for estimating functions whose derivatives are of bounded variation, trend filtering achieves the minimax optimal error rate, while other popular methods like smoothing splines and kernels do not. Standing in the way of a more widespread practical adoption, however, is a lack of scalable and numerically stable algorithms for fitting trend filtering estimates. This paper presents a highly efficient, specialized ADMM routine for trend filtering. Our algorithm is competitive with the specialized interior point methods that are currently in use, and yet is far more numerically robust. Furthermore, the proposed ADMM implementation is very simple, and importantly, it is flexible enough to extend to many interesting related problems, such as sparse trend filtering and isotonic trend filtering. Software for our method is freely available, in both the C and R languages.