We present several exact and approximate mathematical methods and open-source
software to compute the cdf, pdf and inverse cdf of the generalized chi-square
distribution, which appears in Bayesian classification problems. Some methods
are geared for speed, while others are designed to be accurate far into the
tails, using which we can also measure large values of the discriminability
index $d'$ between multinormals. We compare the accuracy and speed of these
methods against the best existing methods.

我们提出了几种精确和近似的数学方法和开源软件来计算广义卡方分布的累积分布函数、概率密度函数和反函数累积分布函数，该分布在贝叶斯分类问题中出现。我们比较了这些方法与现有最佳方法之间的准确性和速度。

计算广义卡方分布的新方法

New methods for computing the generalized chi-square distribution

There is strong interest in developing mathematical methods that can be used
to understand complex neural networks used in image analysis. In this paper, we
introduce techniques from Linear Algebra to model neural network layers as maps
between signal spaces. First, we demonstrate how signal spaces can be used to
visualize weight spaces and convolutional layer kernels. We also demonstrate
how residual vector spaces can be used to further visualize information lost at
each layer. Second, we introduce the concept of invertible networks and an
algorithm for computing input images that yield specific outputs. We
demonstrate our approach on two invertible networks and ResNet18.

利用线性代数中的技术对神经网络层进行建模，并引入可逆网络的概念和算法，实现了特定输出的输入图像的计算。

利用向量空間和逆映射理解影像分析的神經網路系統

Understanding Neural Network Systems for Image Analysis using Vector  Spaces and Inverse Maps

We consider multiple access communication on a binary input additive white
Gaussian noise channel using randomly spread code division. For a general class
of symmetric distributions for spreading coefficients, in the limit of a large
number of users, we prove an upper bound on the capacity, which matches a
formula that Tanaka obtained by using the replica method. We also show
concentration of various relevant quantities including mutual information,
capacity and free energy. The mathematical methods are quite general and allow
us to discuss extensions to other multiuser scenarios.

本文探讨多用户二进制输入加性白噪声信道上的多址通信，运用随机扩展码分复用技术，对于一类对称分布涂抹系数，在大量用户情况下，我们证明了一个容量的上界，与 Tanaka 公式一致，同时还展示了多个相关量的浓度，包括相互信息，容量和自由能，该数学方法相当普适，可扩展至其他多用户场景。