# Sparsest Piecewise-Linear Regression of One-Dimensional Data

arXiv: https://arxiv.org/abs/2003.10112 · HTML: https://mlsanity.com/paper/2003.10112/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Mar 2020.

## Built on by (most cited first, 12 of 12)

- [Gradient flow dynamics of shallow ReLU networks for square loss and orthogonal inputs](https://mlsanity.com/paper/2206.00939/) (2022)
- [Improving Lipschitz-Constrained Neural Networks by Learning Activation Functions](https://mlsanity.com/paper/2210.16222/) (2022)
- [Deep Learning Meets Sparse Regularization: A Signal Processing Perspective](https://mlsanity.com/paper/2301.09554/) (2023)
- [Penalising the biases in norm regularisation enforces sparsity](https://mlsanity.com/paper/2303.01353/) (2023)
- [Noisy Interpolation Learning with Shallow Univariate ReLU Networks](https://mlsanity.com/paper/2307.15396/) (2023)
- [Intrinsic dimensionality and generalization properties of the R-norm inductive bias](https://mlsanity.com/paper/2206.05317/) (2022)
- [Measuring Complexity of Learning Schemes Using Hessian-Schatten Total Variation](https://mlsanity.com/paper/2112.06209/) (2021)
- [Minimum norm interpolation by perceptra: Explicit regularization and implicit bias](https://mlsanity.com/paper/2311.06138/) (2023)
- [Controlled Learning of Pointwise Nonlinearities in Neural-Network-Like Architectures](https://mlsanity.com/paper/2408.13114/) (2024)
- [Delaunay-Triangulation-Based Learning with Hessian Total-Variation Regularization](https://mlsanity.com/paper/2208.07787/) (2022)
- [The Effects of Multi-Task Learning on ReLU Neural Network Functions](https://mlsanity.com/paper/2410.21696/) (2024)
- [Global Minimizers of ℓp-Regularized Objectives Yield the Sparsest ReLU Neural Networks](https://mlsanity.com/paper/2505.21791/) (2025)

## Abstract (arXiv)

We study the problem of one-dimensional regression of data points with total-variation (TV) regularization (in the sense of measures) on the second derivative, which is known to promote piecewise-linear solutions with few knots. While there are efficient algorithms for determining such adaptive splines, the difficulty with TV regularization is that the solution is generally non-unique, an aspect that is often ignored in practice. In this paper, we present a systematic analysis that results in a complete description of the solution set with a clear distinction between the cases where the solution is unique and those, much more frequent, where it is not. For the latter scenario, we identify the sparsest solutions, i.e., those with the minimum number of knots, and we derive a formula to compute the minimum number of knots based solely on the data points. To achieve this, we first consider the problem of exact interpolation which leads to an easier theoretical analysis. Next, we relax the exact interpolation requirement to a regression setting, and we consider a penalized optimization problem with a strictly convex data-fidelity cost function. We show that the underlying penalized problem can be reformulated as a constrained problem, and thus that all our previous results still apply. Based on our theoretical analysis, we propose a simple and fast two-step algorithm, agnostic to uniqueness, to reach a sparsest solution of this penalized problem.