The rank of neural networks measures information flowing across layers. It is
an instance of a key structural condition that applies across broad domains of
machine learning. In particular, the assumption of low-rank feature
representations leads to algorithmic developments in many architectures. For
neural networks, however, the intrinsic mechanism that yields low-rank
structures remains vague and unclear. To fill this gap, we perform a rigorous
study on the behavior of network rank, focusing particularly on the notion of
rank deficiency. We theoretically establish a universal monotonic decreasing
property of network rank from the basic rules of differential and algebraic
composition, and uncover rank deficiency of network blocks and deep function
coupling. By virtue of our numerical tools, we provide the first empirical
analysis of the per-layer behavior of network rank in practical settings, i.e.,
ResNets, deep MLPs, and Transformers on ImageNet. These empirical results are
in direct accord with our theory. Furthermore, we reveal a novel phenomenon of
independence deficit caused by the rank deficiency of deep networks, where
classification confidence of a given category can be linearly decided by the
confidence of a handful of other categories. The theoretical results of this
work, together with the empirical findings, may advance understanding of the
inherent principles of deep neural networks.

本文研究神经网络的秩与结构缺陷，通过理论和实验发现了深度神经网络中秩缺陷和分类的独立性缺陷。

深度神经网络中的降维

Rank Diminishing in Deep Neural Networks

Prior work has successfully incorporated optimization layers as the last
layer in neural networks for various problems, thereby allowing joint learning
and planning in one neural network forward pass. In this work, we identify a
weakness in such a set-up where inputs to the optimization layer lead to
undefined output of the neural network. Such undefined decision outputs can
lead to possible catastrophic outcomes in critical real time applications. We
show that an adversary can cause such failures by forcing rank deficiency on
the matrix fed to the optimization layer which results in the optimization
failing to produce a solution. We provide a defense for the failure cases by
controlling the condition number of the input matrix. We study the problem in
the settings of synthetic data, Jigsaw Sudoku, and in speed planning for
autonomous driving, building on top of prior frameworks in end-to-end learning
and optimization. We show that our proposed defense effectively prevents the
framework from failing with undefined output. Finally, we surface a number of
edge cases which lead to serious bugs in popular equation and optimization
solvers which can be abused as well.

该研究分析了神经网络中嵌入 optimization layers 的弱点，提出了一种防御方法，防范了 undefined output 对关键应用造成的潜在威胁。

超越 NaN：在不可行性面前的优化层恢复能力

Beyond NaN: Resiliency of Optimization Layers in The Face of  Infeasibility

The Hessian of a neural network captures parameter interactions through
second-order derivatives of the loss. It is a fundamental object of study,
closely tied to various problems in deep learning, including model design,
optimization, and generalization. Most prior work has been empirical, typically
focusing on low-rank approximations and heuristics that are blind to the
network structure. In contrast, we develop theoretical tools to analyze the
range of the Hessian map, providing us with a precise understanding of its rank
deficiency as well as the structural reasons behind it. This yields exact
formulas and tight upper bounds for the Hessian rank of deep linear networks,
allowing for an elegant interpretation in terms of rank deficiency. Moreover,
we demonstrate that our bounds remain faithful as an estimate of the numerical
Hessian rank, for a larger class of models such as rectified and hyperbolic
tangent networks. Further, we also investigate the implications of model
architecture (e.g.~width, depth, bias) on the rank deficiency. Overall, our
work provides novel insights into the source and extent of redundancy in
overparameterized networks.

本研究通过开发理论工具来分析 Hessian 映射的范围，提供了深度线性网络 Hessian 级别的确切公式和紧密的上限，以及其冗余程度的原因，从而深入探究了过参数化网络中冗余的来源和程度。