Shayan Hundrieser, Insung Kong, Johannes Schmidt-Hieber
原文摘要
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 reliably 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 data, where they oftentimes outperform ICNN-based neural optimal transport methods and other baselines across a wide range of setting.
The optimal transport (OT) map provides a geometric transformation for aligning probability distributions and has become a useful tool in machine learning. However, existing estimators of the OT map still exhibit a gap between sharp statistical guarantees and practical parametric estimation based on stable training objectives. Theoretical estimators achieve minimax optimal convergence rates, but they are typically nonparametric and can incur demanding implementation design or inference costs. Practical estimators are parametric and scalable, but their statistical guarantees remain underexplored, and their min-max, adversarial-like training objectives can be sensitive to optimization algorithms. We propose BROT (Barycentric Regression for Optimal Transport), a simple two-step method that first computes the unregularized OT plan and then fits a deep neural network (DNN) to the induced barycentric targets by least-squares regression. Under standard regularity conditions, we prove that the DNN estimator of BROT attains the minimax convergence rate, when the ground-truth OT map is Lipschitz. Numerical studies on synthetic datasets and an image dataset show that BROT provides accurate map estimates, strong target distribution matching, and competitive transport costs, compared to existing estimation methods. Experiments on two downstream tasks, single-cell perturbation prediction and unsupervised domain adaptation, further suggest that the accurate estimation of BROT can translate into stronger task performance.