arXiv：cs.LG· Mohit Kumar, Somayeh Kargaran·· 4 小时前AI 评分29
几何诱导软状态抽象的预测极限与 Koopman 闭包
Prediction Limits and Koopman Closure of Geometry-Induced Soft State Abstractions
AI 导读
研究软状态表示中类权重建构与状态动态如何共同决定线性预测精度，针对任意固定可测表示，推导出在指定谱范数约束下最小总体均方根预测误差的有限样本置信下界，该下界由独立评估对直接计算而无需拟合预测矩阵。
正文
Abstract: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.
| Subjects: | Artificial Intelligence (cs.AI); Machine Learning (cs.LG); Dynamical Systems (math.DS) |
| Cite as: | arXiv:2609.32652 [cs.AI] |
| (or arXiv:2609.32652v2 [cs.AI] for this version) | |
| https://doi.org/10.48550/arXiv.2609.32652 arXiv-issued DOI via DataCite |
Submission history
From: Mohit Kumar [view email]
[v1]
Sat, 26 Sep 2026 14:10:31 UTC (3,297 KB)
[v2]
Tue, 6 Oct 2026 11:40:42 UTC (503 KB)
来源：arXiv：cs.LG · arxiv.org