Random features are an important technique that make it possible to rewrite
positive-definite kernels as infinite-dimensional dot products. Over time,
increasingly elaborate random feature representations have been developed in
pursuit of finite approximations with ever lower error. We resolve this arms
race by deriving an optimal sampling policy, and show that under this policy
all random features representations have the same approximation error. This
establishes a lower bound that holds across all random feature representations,
and shows that we are free to choose whatever representation we please,
provided we sample optimally.

根据我们导出的最优取样策略，我们证明了在这一策略下，所有的随机特征表示具有相同的逼近误差，从而建立了一个适用于所有随机特征表示的下界，这意味着我们可以根据需求自由选择任何表示方式。

所有随机特征表示等效

All Random Features Representations are Equivalent

The Koopman operator allows for handling nonlinear systems through a
(globally) linear representation. In general, the operator is
infinite-dimensional - necessitating finite approximations - for which there is
no overarching framework. Although there are principled ways of learning such
finite approximations, they are in many instances overlooked in favor of, often
ill-posed and unstructured methods. Also, Koopman operator theory has
long-standing connections to known system-theoretic and dynamical system
notions that are not universally recognized. Given the former and latter
realities, this work aims to bridge the gap between various concepts regarding
both theory and tractable realizations. Firstly, we review data-driven
representations (both unstructured and structured) for Koopman operator
dynamical models, categorizing various existing methodologies and highlighting
their differences. Furthermore, we provide concise insight into the paradigm's
relation to system-theoretic notions and analyze the prospect of using the
paradigm for modeling control systems. Additionally, we outline the current
challenges and comment on future perspectives.

本篇论文探讨了 Koopman 算子理论在处理非线性系统方面的应用，着重介绍了 Koopman 算子动力学模型中各种现有方法的优缺点，分析了 Koopman 算子理论与系统理论概念的关系及其在控制系统建模中的潜力，同时讨论了当前的挑战和未来的发展方向。