While neural networks can enjoy an outstanding flexibility and exhibit
unprecedented performance, the mechanism behind their behavior is still not
well-understood. To tackle this fundamental challenge, researchers have tried
to restrict and manipulate some of their properties in order to gain new
insights and better control on them. Especially, throughout the past few years,
the concept of \emph{bi-Lipschitzness} has been proved as a beneficial
inductive bias in many areas. However, due to its complexity, the design and
control of bi-Lipschitz architectures are falling behind, and a model that is
precisely designed for bi-Lipschitzness realizing a direct and simple control
of the constants along with solid theoretical analysis is lacking. In this
work, we investigate and propose a novel framework for bi-Lipschitzness that
can achieve such a clear and tight control based on convex neural networks and
the Legendre-Fenchel duality. Its desirable properties are illustrated with
concrete experiments. We also apply this framework to uncertainty estimation
and monotone problem settings to illustrate its broad range of applications.

我们研究和提出了一个新颖的针对 bi-Lipschitzness 的框架，该框架可以基于凸神经网络和 Legendre-Fenchel 对偶性实现对常数的直接和简单控制，并通过具体实验展示了其期望的性质。我们还将该框架应用于不确定性估计和单调问题设置，以展示其广泛的应用范围。

通过整体双 Lipschitz 性的直接参数化，对神经网络的灵敏度进行可证控制

A provable control of sensitivity of neural networks through a direct  parameterization of the overall bi-Lipschitzness

Wasserstein barycenters provide a geometric notion of the weighted average of
probability measures based on optimal transport. In this paper, we present a
scalable algorithm to compute Wasserstein-2 barycenters given sample access to
the input measures, which are not restricted to being discrete. While past
approaches rely on entropic or quadratic regularization, we employ input convex
neural networks and cycle-consistency regularization to avoid introducing bias.
As a result, our approach does not resort to minimax optimization. We provide
theoretical analysis on error bounds as well as empirical evidence of the
effectiveness of the proposed approach in low-dimensional qualitative scenarios
and high-dimensional quantitative experiments.

该论文提出了一种可扩展的算法，用于计算 Wasserstein-2 重心，针对输入测量，其不仅限于离散形式，并使用输入凸神经网络和周期一致性正则化以避免引入偏差，并提供了误差界的理论分析，以及在低维定性情景和高维定量实验中提出的方法的实证证据。