Let $X$ and $Z$ be random vectors, and $Y=g(X,Z)$. In this paper, on the one
hand, for the case that $X$ and $Z$ are continuous, by using the ideas from the
total variation and the flux of $g$, we develop a point of view in causal
inference capable of dealing with a broad domain of causal problems. Indeed, we
focus on a function, called Probabilistic Easy Variational Causal Effect
(PEACE), which can measure the direct causal effect of $X$ on $Y$ with respect
to continuously and interventionally changing the values of $X$ while keeping
the value of $Z$ constant. PEACE is a function of $d\ge 0$, which is a degree
managing the strengths of probability density values $f(x|z)$. On the other
hand, we generalize the above idea for the discrete case and show its
compatibility with the continuous case. Further, we investigate some properties
of PEACE using measure theoretical concepts. Furthermore, we provide some
identifiability criteria and several examples showing the generic capability of
PEACE. We note that PEACE can deal with the causal problems for which
micro-level or just macro-level changes in the value of the input variables are
important. Finally, PEACE is stable under small changes in $\partial
g_{in}/\partial x$ and the joint distribution of $X$ and $Z$, where $g_{in}$ is
obtained from $g$ by removing all functional relationships defining $X$ and
$Z$.

本文研究了因果推断中连续和离散情况下的概率易变因果效应（PEACE），并探讨了 PEACE 的可识别性准则及其对输入变量值的微小变化的稳定性。

概率易于变分因果效应

Probabilistic Easy Variational Causal Effect

Strong theoretical guarantees of robustness can be given for ensembles of
classifiers generated by input randomization. Specifically, an $\ell_2$ bounded
adversary cannot alter the ensemble prediction generated by an additive
isotropic Gaussian noise, where the radius for the adversary depends on both
the variance of the distribution as well as the ensemble margin at the point of
interest. We build on and considerably expand this work across broad classes of
distributions. In particular, we offer adversarial robustness guarantees and
associated algorithms for the discrete case where the adversary is $\ell_0$
bounded. Moreover, we exemplify how the guarantees can be tightened with
specific assumptions about the function class of the classifier such as a
decision tree. We empirically illustrate these results with and without
functional restrictions across image and molecule datasets.

采用输入随机化生成的分类器的集合可以具有很强的鲁棒性，该方法可扩展到更广泛的分布，特别是在离散情况下保证敌对的鲁棒性，并给出了相应的算法，此外，该研究有助于通过假设分类器的函数类来提高保证，并且具有图像和分子数据集上的实际应用。

对于随机平滑分类器的严格对抗鲁棒性证明

Tight Certificates of Adversarial Robustness for Randomly Smoothed  Classifiers

First steps towards a mathematical theory of deep convolutional neural
networks for feature extraction were made---for the continuous-time case---in
Mallat, 2012, and Wiatowski and B\"olcskei, 2015. This paper considers the
discrete case, introduces new convolutional neural network architectures, and
proposes a mathematical framework for their analysis. Specifically, we
establish deformation and translation sensitivity results of local and global
nature, and we investigate how certain structural properties of the input
signal are reflected in the corresponding feature vectors. Our theory applies
to general filters and general Lipschitz-continuous non-linearities and pooling
operators. Experiments on handwritten digit classification and facial landmark
detection---including feature importance evaluation---complement the
theoretical findings.

本文针对离散情况，引入新的卷积神经网络结构，并为其分析提出了数学框架，具体包括局部和全局的变形与平移敏感结果，以及研究输入信号的某些结构性质在相应特征向量中的反映，理论适用于一般的滤波器和 Lipschitz 连续的非线性和池化运算符，并通过手写数字分类和面部关键点检测的实验结果对理论研究进行了补充。