Effective Neural Approximations for Geometric Optimization Problems

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.