Probabilistic verification of neural networks is concerned with formally
analysing the output distribution of a neural network under a probability
distribution of the inputs. Examples of probabilistic verification include
verifying the demographic parity fairness notion or quantifying the safety of a
neural network. We present a new algorithm for the probabilistic verification
of neural networks based on an algorithm for computing and iteratively refining
lower and upper bounds on probabilities over the outputs of a neural network.
By applying state-of-the-art bound propagation and branch and bound techniques
from non-probabilistic neural network verification, our algorithm significantly
outpaces existing probabilistic verification algorithms, reducing solving times
for various benchmarks from the literature from tens of minutes to tens of
seconds. Furthermore, our algorithm compares favourably even to dedicated
algorithms for restricted subsets of probabilistic verification. We complement
our empirical evaluation with a theoretical analysis, proving that our
algorithm is sound and, under mildly restrictive conditions, also complete when
using a suitable set of heuristics.

用具有适当启发式条件的一组算法基于神经网络的输出分布来验证其概率，同时计算和迭代优化神经网络输出概率的下界和上界，并通过应用非概率性神经网络验证中的最先进的边界传播和分支约束技术，显著提高了解决时间。

使用分支界限法对神经网络进行概率验证

Probabilistic Verification of Neural Networks using Branch and Bound

For a $3$-tensor of dimensions $I_1\times I_2\times I_3$, we show that the
nuclear norm of its every matrix flattening is a lower bound of the tensor
nuclear norm, and which in turn is upper bounded by $\sqrt{\min\{I_i : i\neq
j\}}$ times the nuclear norm of the matrix flattening in mode $j$ for all
$j=1,2,3$. The results can be generalized to $N$-tensors with any $N\geq 3$.
Both the lower and upper bounds for the tensor nuclear norm are sharp in the
case $N=3$. A computable criterion for the lower bound being tight is given as
well.

本文研究了 $3$ 维张量的核范数及其矩阵展开的核范数之间的关系，并提出了相应的上下界和计算标准，进一步推广到 $N$ 维张量，并在 $N=3$ 的情况下获得了最优解。