Predicting the behavior of AI-driven agents is particularly challenging
without a preexisting model. In our paper, we address this by treating AI
agents as nonlinear dynamical systems and adopting a probabilistic perspective
to predict their statistical behavior using the Perron-Frobenius (PF) operator.
We formulate the approximation of the PF operator as an entropy minimization
problem, which can be solved by leveraging the Markovian property of the
operator and decomposing its spectrum. Our data-driven methodology
simultaneously approximates the PF operator to perform prediction of the
evolution of the agents and also predicts the terminal probability density of
AI agents, such as robotic systems and generative models. We demonstrate the
effectiveness of our prediction model through extensive experiments on
practical systems driven by AI algorithms.

通过将人工智能代理视为非线性动力系统，并采用概率视角，我们提出了一种预测其统计行为的方法，通过近似 Perron-Frobenius（PF）算子来解决这一特别具有挑战性的问题。我们将 PF 算子的近似建模为一个熵最小化问题，并利用算子的马尔可夫性质和分解其谱来求解。我们的数据驱动方法同时近似 PF 算子以预测代理的演变，并预测人工智能代理（如机器人系统和生成模型）的终端概率密度。通过对由 AI 算法驱动的实际系统进行广泛实验，我们展示了我们预测模型的有效性。

通过近似 Perron-Frobenius 算子预测 AI 代理行为

Predicting AI Agent Behavior through Approximation of the  Perron-Frobenius Operator

Reproducing kernel Hilbert $C^*$-module (RKHM) is a generalization of
reproducing kernel Hilbert space (RKHS) by means of $C^*$-algebra, and the
Perron-Frobenius operator is a linear operator related to the composition of
functions. Combining these two concepts, we present deep RKHM, a deep learning
framework for kernel methods. We derive a new Rademacher generalization bound
in this setting and provide a theoretical interpretation of benign overfitting
by means of Perron-Frobenius operators. By virtue of $C^*$-algebra, the
dependency of the bound on output dimension is milder than existing bounds. We
show that $C^*$-algebra is a suitable tool for deep learning with kernels,
enabling us to take advantage of the product structure of operators and to
provide a clear connection with convolutional neural networks. Our theoretical
analysis provides a new lens through which one can design and analyze deep
kernel methods.

通过结合再生核 Hilbert $C^*$- 模与 Perron-Frobenius 算子来设计和分析深度再生核方法，提出了适用于核方法的深度学习框架 deep RKHM，基于 $C^*$ 代数导出了一种新的 Rademacher 广义界，并通过 Perron-Frobenius 算子对无害过拟合进行了理论解释。

通过 RKHM 和 Perron-Frobenius 算子进行核深度学习

Deep Learning with Kernels through RKHM and the Perron-Frobenius  Operator

In this paper, we study the stability problem of a stochastic, nonlinear,
discrete-time system. We introduce a linear transfer operator-based Lyapunov
measure as a new tool for stability verification of stochastic systems. Weaker
set-theoretic notion of almost everywhere stochastic stability is introduced
and verified, using Lyapunov measure-based stochastic stability theorems.
Furthermore, connection between Lyapunov functions, a popular tool for
stochastic stability verification, and Lyapunov measures is established. Using
the duality property between the linear transfer Perron-Frobenius and Koopman
operators, we show the Lyapunov measure and Lyapunov function used for the
verification of stochastic stability are dual to each other. Set-oriented
numerical methods are proposed for the finite dimensional approximation of the
Perron-Frobenius operator; hence, Lyapunov measure is proposed. Stability
results in finite dimensional approximation space are also presented. Finite
dimensional approximation is shown to introduce further weaker notion of
stability referred to as coarse stochastic stability. The results in this paper
extend our earlier work on the use of Lyapunov measures for almost everywhere
stability verification of deterministic dynamical systems ("Lyapunov Measure
for Almost Everywhere Stability", {\it IEEE Trans. on Automatic Control}, Vol.
53, No. 1, Feb. 2008).

本文在研究中引入基于线性转移算子的李亚普诺夫测度作为随机系统稳定性验证的新工具，并建立了李亚普诺夫函数和测度之间的联系以及引入了用于有限维近似的集合定向数值方法，给出了有限维空间中的稳定性结果。