# Learning Polynomial Representations of Physical Objects with Application to Certifying Correct Packing Configurations

arXiv: https://arxiv.org/abs/2312.06791 · HTML: https://mlsanity.com/paper/2312.06791/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 50 references that are in the map.
- First posted on arXiv 11 Dec 2023.

## Builds on (most cited first, 20 of 20)

- [Surface Reconstruction from Point Clouds by Learning Predictive Context Priors](https://mlsanity.com/paper/2204.11015/) (2022)
- [Surface Reconstruction from Point Clouds: A Survey and a Benchmark](https://mlsanity.com/paper/2205.02413/) (2022)
- [How to Certify Machine Learning Based Safety-critical Systems? A Systematic Literature Review](https://mlsanity.com/paper/2107.12045/) (2021)
- [Topology-Aware Surface Reconstruction for Point Clouds](https://mlsanity.com/paper/1811.12543/) (2018)
- [Learning Dynamical Systems with Side Information](https://mlsanity.com/paper/2008.10135/) (2020)
- [Learning Physically Realizable Skills for Online Packing of General 3D Shapes](https://mlsanity.com/paper/2212.02094/) (2022)
- [Planning Irregular Object Packing via Hierarchical Reinforcement Learning](https://mlsanity.com/paper/2211.09382/) (2022)
- [SDF-Pack: Towards Compact Bin Packing with Signed-Distance-Field Minimization](https://mlsanity.com/paper/2307.07356/) (2023)
- [When to Trust AI: Advances and Challenges for Certification of Neural Networks](https://mlsanity.com/paper/2309.11196/) (2023)
- [Shape-aware Surface Reconstruction from Sparse 3D Point-Clouds](https://mlsanity.com/paper/1602.08425/) (2016)
- [Sum-of-Squares Geometry Processing](https://mlsanity.com/paper/2110.08451/) (2021)
- [Simple Approximations of Semialgebraic Sets and their Applications to Control](https://mlsanity.com/paper/1509.04200/) (2015)
- [Geometry of 3D Environments and Sum of Squares Polynomials](https://mlsanity.com/paper/1611.07369/) (2016)
- [Sublevel Set Approximation in The Hausdorff and Volume Metric with Application to Path Planning and Obstacle Avoidance](https://mlsanity.com/paper/2303.06778/) (2023)
- [Convolutional Occupancy Models for Dense Packing of Complex, Novel Objects](https://mlsanity.com/paper/2308.00091/) (2023)
- [Sparse Identification of Nonlinear Dynamics with Side Information (SINDy-SI)](https://mlsanity.com/paper/2310.04227/) (2023)
- [Using SOS for Optimal Semialgebraic Representation of Sets: Finding Minimal Representations of Limit Cycles, Chaotic Attractors and Unions](https://mlsanity.com/paper/1809.10308/) (2018)
- [Inner and Outer Approximations of Star-Convex Semialgebraic Sets](https://mlsanity.com/paper/2203.13071/) (2022)
- [Closed-Form Minkowski Sum Approximations for Efficient Optimization-Based Collision Avoidance](https://mlsanity.com/paper/2203.15977/) (2022)
- [Category-level Shape Estimation for Densely Cluttered Objects](https://mlsanity.com/paper/2302.11983/) (2023)

## Abstract (arXiv)

This paper introduces a novel approach for learning polynomial representations of physical objects. Given a point cloud data set associated with a physical object, we solve a one-class classification problem to bound the data points by a polynomial sublevel set while harnessing Sum-of-Squares (SOS) programming to enforce prior shape knowledge constraints. By representing objects as polynomial sublevel sets we further show it is possible to construct a secondary SOS program to certify whether objects are packed correctly, that is object boundaries do not overlap and are inside some container set. While not employing reinforcement learning (RL) in this work, our proposed secondary SOS program does provide a potential surrogate reward function for RL algorithms, autonomously rewarding agents that propose object rotations and translations that correctly pack objects within a given container set.