Neural Set Function Extensions: Learning with Discrete Functions in High Dimensions

• URL: http://arxiv.org/abs/2208.04055v1
• Date: Mon, 8 Aug 2022 10:58:02 GMT
• Title: Neural Set Function Extensions: Learning with Discrete Functions in High Dimensions
• Authors: Nikolaos Karalias, Joshua Robinson, Andreas Loukas, Stefanie Jegelka
• Abstract summary: We develop a framework for extending set functions onto low-dimensional continuous domains. Our framework subsumes many well-known extensions as special cases. We convert low-dimensional neural network bottlenecks into representations in high-dimensional spaces.
• Score: 63.21838830509772
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Integrating functions on discrete domains into neural networks is key to developing their capability to reason about discrete objects. But, discrete domains are (1) not naturally amenable to gradient-based optimization, and (2) incompatible with deep learning architectures that rely on representations in high-dimensional vector spaces. In this work, we address both difficulties for set functions, which capture many important discrete problems. First, we develop a framework for extending set functions onto low-dimensional continuous domains, where many extensions are naturally defined. Our framework subsumes many well-known extensions as special cases. Second, to avoid undesirable low-dimensional neural network bottlenecks, we convert low-dimensional extensions into representations in high-dimensional spaces, taking inspiration from the success of semidefinite programs for combinatorial optimization. Empirically, we observe benefits of our extensions for unsupervised neural combinatorial optimization, in particular with high-dimensional representations.

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