标题： 神经网络的图充分性视角 显示英文标题

标题： A Graph Sufficiency Perspective for Neural Networks

摘要： 本文通过图变量和统计充分性分析神经网络。 我们将神经网络层解释为基于图的变换，其中神经元作为输入和学习到的锚点之间的成对函数。 在此框架下，我们建立了层输出对于层输入是充分的条件，即每一层保留给定输入变量的目标变量的条件分布。 在密集锚点假设下，我们证明了在无限宽度极限下渐近充分性成立，并在整个训练过程中得到保持。 为了更接近实际架构，我们进一步表明，通过假设区域分离的输入分布并构建适当的锚点，有限宽度网络也可以实现充分性。 我们的框架涵盖了全连接层、一般成对函数、ReLU和sigmoid激活函数以及卷积神经网络。 这项工作将统计充分性、图论表示和深度学习联系起来，为神经网络提供了一种新的统计理解。 显示英文摘要

摘要： This paper analyzes neural networks through graph variables and statistical sufficiency. We interpret neural network layers as graph-based transformations, where neurons act as pairwise functions between inputs and learned anchor points. Within this formulation, we establish conditions under which layer outputs are sufficient for the layer inputs, that is, each layer preserves the conditional distribution of the target variable given the input variable. Under dense anchor point assumptions, we prove that asymptotic sufficiency holds in the infinite-width limit and is preserved throughout training. To align more closely with practical architectures, we further show that sufficiency can be achieved with finite-width networks by assuming region-separated input distributions and constructing appropriate anchor points. Our framework covers fully connected layers, general pairwise functions, ReLU and sigmoid activations, and convolutional neural networks. This work bridges statistical sufficiency, graph-theoretic representations, and deep learning, providing a new statistical understanding of neural networks.