Beyond DSA: Conjugacy-based Comparison of Dynamical Systems

Summary · qwen2.5:32b

The study introduces Conjugacy-based Similarity Analysis (CSA) to compare dynamical systems, showing that orthogonal alignment methods like Dynamical Similarity Analysis (DSA) are insufficient for identifying topological conjugacies due to their limitations in capturing non-orthogonal basis-transfer matrices. CSA restricts alignments to state-space bijections and is proven to be the finite-data projection of composition operators linked with candidate bijections, which is crucial when observable dictionaries are selected from data explicitly or implicitly.

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Exploring a new method for comparing dynamical systems in neuroscience and machine learning, Beyond DSA, could revolutionize real-time rendering and simulation.

Excerpt

arXiv:2607.04493v1 Announce Type: cross Abstract: 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.