AI 中文总结
本研究为几何诱导软状态抽象建立了可计算的预测误差下界，并证明在确定性动力学下闭包矩阵可诱导Koopman特征函数，实验验证了软坐标预测的有效性。
AI 中文摘要
我们研究了几何诱导的软状态抽象何时能够支持精确的有限维线性动力学。每个状态由从类别特定的核仿射包机（KAHM）重构得分获得的单纯形值坐标表示，并使用一个矩阵来预测下一状态的坐标。我们的主要结果是一个可计算的下置信界，用于在所有满足指定谱范数限制的矩阵中，最小均方根预测误差的下界。该界将后继坐标的类内变化与软坐标偏离其独热参考标签的程度相结合，并且可以在不拟合预测矩阵的情况下，从独立的（状态，后继）对中评估。对于固定的坐标和评估分布，随着样本量的增长，该证书几乎必然收敛到总体下界；低于此极限的任何容差最终都会被证明无法达到。重构得分间隔进一步控制软到硬分配误差。在确定性动力学和精确坐标闭包下，闭包矩阵及其约化转置的特征向量分别诱导Koopman和伴随Koopman特征函数。一个四状态KAHM构造表明，相同的软坐标在一个动力学映射下可以允许精确闭包，而在另一个映射下却导致正预测误差。在Duffing、Van der Pol、CartPole、MountainCar和Acrobot上的实验将直接软坐标预测与状态空间DMD/EDMD基线进行比较，并报告预测、表示变化和谱诊断。基准测试评估了拟合模型，但未对排除证书进行数值评估。
英文摘要
A soft state representation assigns each state a vector of nonnegative class weights that sum to one. We study how the construction of these weights and the state dynamics jointly determine the accuracy of linear prediction. For any fixed measurable representation, we derive a finite-sample lower confidence bound on the smallest population root-mean-square prediction error among matrices with a specified spectral-norm limit. The bound compares variation in successor coordinates within each reference class with the improvement that soft inputs could provide. It is computed from independent evaluation pairs without fitting a prediction matrix. A bound above a chosen tolerance rules out that tolerance for the entire matrix class; a zero bound is inconclusive. For coordinates constructed using Kernel Affine Hull Machines, reconstruction-score margins control disagreement with reference labels and enter bounds on prediction error. Under exact deterministic linear evolution, we also establish the Koopman and reproducing-kernel Hilbert-space adjoint interpretation, accounting for redundant coefficient vectors. A four-state study compares the confidence bound with analytically known optima across 117,000 reported replicate datasets. A Van der Pol representation selected on pilot data is then evaluated on 32 independent datasets under each of two transition laws. The reported bounds are positive at the fitted matrix norm, but can become zero at larger norm limits. Further forecasting studies examine coordinate variation, common prediction targets, and long-horizon error. The results distinguish agreement with reconstruction classes, attainable prediction accuracy, and exact operator closure.
