Effective Neural Approximations for Geometric Optimization Problems

Samantha Chen, Oren Ciolli, Anastasios Sidiropoulos, Yusu Wang

Keywords: geometric deep learning, computational geometry, neural networks, point clouds

TL;DR: We introduce neural approximation frameworks for approximating a family of geometric shape-fitting problems.

Abstract:

Neural networks offer a promising data-driven approach to tackle computationally challenging optimization problems.
In this work, we introduce neural approximation frameworks for a family of geometric "extent measure" problems, including shape-fitting descriptors (e.g. minimum enclosing ball or annulus).
Central to our approach is the \textit{alignment} of our neural model with a new variant of the classical $\varepsilon$-kernel technique from computational geometry.
In particular, we develop a new relaxed-$\varepsilon$-kernel theory that maintains the approximation guarantees of the classical $\varepsilon$-kernels but with the crucial benefit that it can be easily implemented with \textit{bounded model complexity} (i.e, constant number of parameters) by the simple SumFormer neural network.
This leads to a simple neural model to approximate objects such as the directional width of any input point set, and empirically shows excellent out-of-distribution generalization.
Many geometric extent measures, such as the minimum enclosing spherical shell, cannot be directly captured by $\varepsilon$-kernels.
To this end, we show that an encode-process-decode framework with our kernel approximating NN used as the ``process'' module can approximate such extent measures, again, with bounded model complexity where parameters scale only with the approximation error $\varepsilon$ and not the size of the input set.
Empirical results on diverse point‐cloud datasets demonstrate the practical performance of our models.

Primary Area: Deep learning (e.g., architectures, generative models, optimization for deep networks, foundation models, LLMs)

Submission Number: 11611