We present a post-training quantization algorithm with error estimates
relying on ideas originating from frame theory. Specifically, we use
first-order Sigma-Delta ($\Sigma\Delta$) quantization for finite unit-norm
tight frames to quantize weight matrices and biases in a neural network. In our
scenario, we derive an error bound between the original neural network and the
quantized neural network in terms of step size and the number of frame
elements. We also demonstrate how to leverage the redundancy of frames to
achieve a quantized neural network with higher accuracy.

我们提出了一种基于来自框架理论的思想的后训练量化算法，特别是我们使用一阶 Sigma-Delta 量化来量化神经网络中的权重矩阵和偏差。在我们的情景中，我们推导了原始神经网络和量化神经网络之间的误差界限，其中考虑了步长和框架元素数量。我们还展示了如何利用框架的冗余性来实现更高精度的量化神经网络。

神经网络的框架量化

Frame Quantization of Neural Networks

In this paper, error estimates of classification Random Forests are
quantitatively assessed. Based on the initial theoretical framework built by
Bates et al. (2023), the true error rate and expected error rate are
theoretically and empirically investigated in the context of a variety of error
estimation methods common to Random Forests. We show that in the classification
case, Random Forests' estimates of prediction error is closer on average to the
true error rate instead of the average prediction error. This is opposite the
findings of Bates et al. (2023) which were given for logistic regression. We
further show that this result holds across different error estimation
strategies such as cross-validation, bagging, and data splitting.

分类随机森林的误差估计量进行了定量评估，结果显示在不同的误差估计方法中，随机森林的预测误差估计更接近真实误差率而不是平均预测误差率。

随机森林中的预测误差估计

Prediction Error Estimation in Random Forests

Simulations of QCD are known to suffer from serious critical slowing down
towards the continuum limit. This is particularly prominent in the topological
charge. We investigate the severeness of the problem in the range of lattice
spacings used in contemporary simulations and propose a method to give more
reliable error estimates.

针对 QCD 模拟中拓扑场的严重致命减速问题，本文研究了当今模拟使用的格点间距范围内的问题严重程度，并提出了一种给出更可靠误差估计的方法。

研究 QCD 模拟的临界减速

Investigating the critical slowing down of QCD simulations

Numerical approximation of the long time behavior of a stochastic
differential equation (SDE) is considered. Error estimates for time-averaging
estimators are obtained and then used to show that the stationary behavior of
the numerical method converges to that of the SDE. The error analysis is based
on using an associated Poisson equation for the underlying SDE. The main
advantage of this approach is its simplicity and universality. It works equally
well for a range of explicit and implicit schemes including those with simple
simulation of random variables, and for hypoelliptic SDEs. To simplify the
exposition, we consider only the case where the state space of the SDE is a
torus and we study only smooth test functions. However we anticipate that the
approach can be applied more widely. An analogy between our approach and
Stein's method is indicated. Some practical implications of the results are
discussed.

探讨了随机微分方程的长时间行为的数值逼近方法，得到了时间平均估计器的误差估计，并利用其展示了数值方法的稳态行为趋近于随机微分方程的稳态行为，其误差分析基于底层随机微分方程的泊松方程，并且该方法的主要优点是其简单性和普适性。