This paper explores the null space properties of neural networks. We extend
the null space definition from linear to nonlinear maps and discuss the
presence of a null space in neural networks. The null space of a given neural
network can tell us the part of the input data that makes no contribution to
the final prediction so that we can use it to trick the neural network. This
reveals an inherent weakness in neural networks that can be exploited. One
application described here leads to a method of image steganography. Through
experiments on image datasets such as MNIST, we show that we can use null space
components to force the neural network to choose a selected hidden image class,
even though the overall image can be made to look like a completely different
image. We conclude by showing comparisons between what a human viewer would
see, and the part of the image that the neural network is actually using to
make predictions and, hence, show that what the neural network ``sees'' is
completely different than what we would expect.

本文探讨神经网络的零空间特性，扩展了线性到非线性映射的零空间定义，并讨论了神经网络中的零空间存在。我们利用零空间的部分来欺骗神经网络，揭示了神经网络的内在弱点。我们描述了一种图像隐写术的应用，并通过在 MNIST 等图像数据集上的实验表明，我们可以利用零空间成分来强制神经网络选择一个选定的隐藏图像类别，即使整体图像看起来完全不同。最后，我们比较了人类观察者所看到的图像和神经网络实际用于进行预测的图像部分，证明神经网络 “看到” 的与我们预期完全不同。

神经网络的零空间特性及其在图像隐写术中的应用

Null Space Properties of Neural Networks with Applications to Image  Steganography

We study sparse recovery with structured random measurement matrices having
independent, identically distributed, and uniformly bounded rows and with a
nontrivial covariance structure. This class of matrices arises from random
sampling of bounded Riesz systems and generalizes random partial Fourier
matrices. Our main result improves the currently available results for the null
space and restricted isometry properties of such random matrices. The main
novelty of our analysis is a new upper bound for the expectation of the
supremum of a Bernoulli process associated with a restricted isometry constant.
We apply our result to prove new performance guarantees for the CORSING method,
a recently introduced numerical approximation technique for partial
differential equations (PDEs) based on compressive sensing.

研究利用具有独立同分布和均匀有界行以及具有非平凡协方差结构的结构化随机测量矩阵的稀疏恢复。因为这类矩阵是从边界瑞格系统的随机采样得来的，并且广义化了随机部分傅立叶矩阵。该结果改进了目前对这种随机矩阵的零空间和限制保持特性的可用结果，并将其应用于证明 CORSING 方法的新绩效保证。

带应用于 PDE 数值方法的有界 Riesz 系统的稀疏恢复

Sparse recovery in bounded Riesz systems with applications to numerical  methods for PDEs

Most existing person re-identification (re-id) methods focus on learning the
optimal distance metrics across camera views. Typically a person's appearance
is represented using features of thousands of dimensions, whilst only hundreds
of training samples are available due to the difficulties in collecting matched
training images. With the number of training samples much smaller than the
feature dimension, the existing methods thus face the classic small sample size
(SSS) problem and have to resort to dimensionality reduction techniques and/or
matrix regularisation, which lead to loss of discriminative power. In this
work, we propose to overcome the SSS problem in re-id distance metric learning
by matching people in a discriminative null space of the training data. In this
null space, images of the same person are collapsed into a single point thus
minimising the within-class scatter to the extreme and maximising the relative
between-class separation simultaneously. Importantly, it has a fixed dimension,
a closed-form solution and is very efficient to compute. Extensive experiments
carried out on five person re-identification benchmarks including VIPeR,
PRID2011, CUHK01, CUHK03 and Market1501 show that such a simple approach beats
the state-of-the-art alternatives, often by a big margin.

本研究提出了一种新的人物再识别（re-id）距离度量学习方法，通过匹配训练数据的判别式空间来克服小样本问题，并且在多个数据集上进行的实验表明此方法比现有方法具有更好的识别效果。