Optimal transport (OT) formalizes the problem of finding an optimal coupling
between probability measures given a cost matrix. The inverse problem of
inferring the cost given a coupling is Inverse Optimal Transport (IOT). IOT is
less well understood than OT. We formalize and systematically analyze the
properties of IOT using tools from the study of entropy-regularized OT.
Theoretical contributions include characterization of the manifold of
cross-ratio equivalent costs, the implications of model priors, and derivation
of an MCMC sampler. Empirical contributions include visualizations of
cross-ratio equivalent effect on basic examples and simulations validating
theoretical results.

本文通过熵正则化的 Optimal transport（OT）工具，对 Inverse Optimal Transport（IOT）进行正式化和系统分析，包括代价等价成本的流形特性、模型先验的影响以及基于模拟的结果验证等方面。

离散概率逆最优输运

Discrete Probabilistic Inverse Optimal Transport

In many classification tasks, the set of target classes can be organized into
a hierarchy. This structure induces a semantic distance between classes, and
can be summarised under the form of a cost matrix, which defines a finite
metric on the class set. In this paper, we propose to model the hierarchical
class structure by integrating this metric in the supervision of a prototypical
network. Our method relies on jointly learning a feature-extracting network and
a set of class prototypes whose relative arrangement in the embedding space
follows an hierarchical metric. We show that this approach allows for a
consistent improvement of the error rate weighted by the cost matrix when
compared to traditional methods and other prototype-based strategies.
Furthermore, when the induced metric contains insight on the data structure,
our method improves the overall precision as well. Experiments on four
different public datasets - from agricultural time series classification to
depth image semantic segmentation - validate our approach.

本文提出了一种基于成本矩阵和原型网络的层次分类方法，其通过在嵌入空间中排列类原型来表征层次结构，并在四个公共数据集上验证了该方法的效果。

利用类层次结构进行度量引导的原型学习

Leveraging Class Hierarchies with Metric-Guided Prototype Learning

Several recent works have developed methods for training classifiers that are
certifiably robust against norm-bounded adversarial perturbations. These
methods assume that all the adversarial transformations are equally important,
which is seldom the case in real-world applications. We advocate for
cost-sensitive robustness as the criteria for measuring the classifier's
performance for tasks where some adversarial transformation are more important
than others. We encode the potential harm of each adversarial transformation in
a cost matrix, and propose a general objective function to adapt the robust
training method of Wong & Kolter (2018) to optimize for cost-sensitive
robustness. Our experiments on simple MNIST and CIFAR10 models with a variety
of cost matrices show that the proposed approach can produce models with
substantially reduced cost-sensitive robust error, while maintaining
classification accuracy.

本文研究了在实际应用中，某些对抗变换的重要性可能比其他变换更高的情况下，如何衡量分类器在对抗干扰下的鲁棒性，并提出了一般性的目标函数，将 Wong 和 Kolter（2018）的鲁棒训练方法进行了改进。通过实验证明，所提出的方法可以显著降低成本敏感鲁棒错误，同时保持分类准确性。