提出新型凸神经网络，高效学习凸函数并用于最优传输建模。
Hyper Input Convex Neural Networks for Shape Constrained Learning and Optimal Transport
- 结合Maxout与输入凸网络思想，保证输入空间始终凸。
- 逼近二次函数时参数量比ICNN少指数级，训练更稳定。
- 适用于高维最优传输，在基因数据等场景表现优越。
我们提出超输入凸神经网络（HyCNNs），一种用于学习凸函数的新架构。HyCNNs融合Maxout网络与输入凸神经网络（ICNNs）的原理，确保网络在输入上始终为凸函数，理论上可利用深度优势，且在大规模训练中比ICNNs更具可靠性。具体而言，我们证明了在给定精度下，HyCNNs逼近二次函数所需的参数量比ICNNs呈指数级减少。通过一系列合成实验，我们验证其在凸回归和插值任务中优于现有ICNNs与MLPs。进一步地，我们将HyCNNs应用于高维最优传输映射的学习，涵盖合成数据及单细胞RNA测序数据，结果表明其在多种设置下均优于基于ICNN的神经最优传输方法及其他基线模型。
原文摘要 · Abstract (English)
We introduce Hyper Input Convex Neural Networks (HyCNNs), a novel neural network architecture designed for learning convex functions. HyCNNs combine the principles of Maxout networks with input convex neural networks (ICNNs) to create a neural network that is always convex in the input, theoretically capable of leveraging depth, and performs reliable when trained at scale compared to ICNNs. Concretely, we prove that HyCNNs require exponentially fewer parameters than ICNNs to approximate quadratic functions up to a given precision. Throughout a series of synthetic experiments, we demonstrate that HyCNNs outperform existing ICNNs and MLPs in terms of predictive performance for convex regression and interpolation tasks. We further apply HyCNNs to learn high-dimensional optimal transport maps for synthetic examples and for single-cell RNA sequencing data, where they oftentimes outperform ICNN-based neural optimal transport methods and other baselines across a wide range of settings.
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