超越DSA：基于共轭的动力系统比较
Beyond DSA: Conjugacy-based Comparison of Dynamical Systems
AI总结：
研究比较两个动力系统是否实现相同计算的问题，指出DSA中正交对齐对拓扑共轭非必要非充分，提出基于共轭的相似性分析（CSA），证明其拟合对齐是相关合成算子的有限数据投影。
摘要：
比较两个动力系统在坐标或测量不同时是否执行相同计算是神经科学和机器学习中的核心问题。动态相似性分析[DSA；Ostrow等人，2023]通过正交相似变换对齐有限维Koopman近似来解决此问题。本文表明正交对齐对于拓扑共轭既非必要也非充分：共轭系统可能需要DSA无法捕获的非正交基转移矩阵，而非共轭系统可能具有DSA无法区分的正交等价Koopman算子。我们利用这一观察结果制定了基于共轭的相似性分析（CSA），它将对齐限制为由候选状态空间双射而非任意正交矩阵诱导的对齐。我们证明CSA的拟合对齐是与候选双射相关的合成算子的有限数据投影，并使用受控示例展示当从数据中显式或隐式选择可观测字典时这种区别为何重要。这些结果阐明了基于Koopman的相似性度量必须确保什么才能支持识别计算系统之间共轭性的主张。
英文摘要：
Comparing whether two dynamical systems implement the same computation despite differences in coordinates or measurements is a central problem in neuroscience and machine learning. Dynamical Similarity Analysis [DSA; Ostrow et al., 2023] addresses this problem by aligning finite-dimensional Koopman approximations through an orthogonal similarity transformation. Here we show that orthogonal alignment is neither necessary nor sufficient for topological conjugacy: conjugate systems may require a non-orthogonal basis-transfer matrix that DSA cannot capture, while non-conjugate systems may have orthogonally equivalent Koopman operators that DSA fails to distinguish. We use this observation to formulate Conjugacy-based Similarity Analysis (CSA), which restricts alignments to those induced by candidate state-space bijections rather than arbitrary orthogonal matrices. We prove that CSA's fitted alignment is the finite-data projection of the composition operator associated with the candidate bijection, and use controlled examples to show why this distinction matters when observable dictionaries are chosen explicitly or implicitly from data. These results clarify what Koopman-based similarity measures must ensure to support claims of identifying conjugacies between computational systems.
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