# Convex and Nonconvex Sublinear Regression with Application to Data-driven Learning of Reach Sets

arXiv: https://arxiv.org/abs/2210.01919 · HTML: https://mlsanity.com/paper/2210.01919/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 36 references that are in the map.
- First posted on arXiv 4 Oct 2022.

## Builds on (most cited first, 13 of 13)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Optimal Control Via Neural Networks: A Convex Approach](https://mlsanity.com/paper/1805.11835/) (2018)
- [Data-Driven Reachability Analysis with Christoffel Functions](https://mlsanity.com/paper/2104.13902/) (2021)
- [Data-Driven Reachability Analysis Using Matrix Zonotopes](https://mlsanity.com/paper/2011.08472/) (2020)
- [Fitting Tractable Convex Sets to Support Function Evaluations](https://mlsanity.com/paper/1903.04194/) (2019)
- [On Suboptimality of Least Squares with Application to Estimation of Convex Bodies](https://mlsanity.com/paper/2006.04046/) (2020)
- [On the Parameterized Computation of Minimum Volume Outer Ellipsoid of Minkowski Sum of Ellipsoids](https://mlsanity.com/paper/1803.08157/) (2018)
- [Learning Approximate Forward Reachable Sets Using Separating Kernels](https://mlsanity.com/paper/2011.09678/) (2020)
- [Smallest Ellipsoid Containing p-Sum of Ellipsoids with Application to Reachability Analysis](https://mlsanity.com/paper/1806.07621/) (2018)
- [Anytime Ellipsoidal Over-approximation of Forward Reach Sets of Uncertain Linear Systems](https://mlsanity.com/paper/2103.04545/) (2021)

## Abstract (arXiv)

We consider estimating a compact set from finite data by approximating the support function of that set via sublinear regression. Support functions uniquely characterize a compact set up to closure of convexification, and are sublinear (convex as well as positive homogeneous of degree one). Conversely, any sublinear function is the support function of a compact set. We leverage this property to transcribe the task of learning a compact set to that of learning its support function. We propose two algorithms to perform the sublinear regression, one via convex and another via nonconvex programming. The convex programming approach involves solving a quadratic program (QP). The nonconvex programming approach involves training a input sublinear neural network. We illustrate the proposed methods via numerical examples on learning the reach sets of controlled dynamics subject to set-valued input uncertainties from trajectory data.