Learning Koopman Eigenfunctions and Invariant Subspaces From Data: Symmetric Subspace Decomposition

Parallel Learning of Koopman Eigenfunctions and Invariant Subspaces For Accurate Long-Term Prediction

“…(A.3). Importantly, all the results for SSD subspaces in [22] are also valid for the new dictionary D. For space reasons, we omit those results here and only mention that under some mild conditions on data sampling, D(x) spans the maximal Koopman-invariant subspace in span(D(x)) and D(x)w is an eigenfunction of the Koopman operator almost surely, given K P-SSD w = λw with λ ∈ C and w ∈ C cols(KSSD) \ {0} (see [22,] for more information).…”

“…When all the data is available at a single processor, the Symmetric Subspace Decomposition (SSD) algorithm proposed in [22], cf. Algorithm 1, is a centralized procedure that identifies D by iteratively pruning the original dictionary to enforce (6) (for reference, Appendix A summarizes the main properties of SSD).…”

“…, M }. Considering the fact that the most time consuming operation in Algorithm 2 is Step 6, one can use [22,Remark 5.2] and deduce that each iteration of the P-SSD algorithm takes O(N i N 3 d ) FLOPs for agent i. Based on Theorem 5.3(b), agent i performs at most O(N i N 3 d d) to find the SSD subspace.…”

“…The latter relies on the fact that EDMD captures all the Koopman eigenfunctions in the span of the dictionary, cf. [22], but not all identified eigenfunctions using EDMD are actually Koopman eigenfunctions. The literature contains several methods based on neural networks that can approximate Koopman-invariant subspaces [23]- [25].…”

“…It is worth mentioning that none of the aforementioned methods provide mathematical guarantees for the identified subspaces to be Koopman invariant. Our recent work [22] provides a necessary and sufficient condition for the identification of functions that evolve linearly according to the dynamics based on the application of EDMD forward and backward in time, and establishes conditions on the density of data sampling to ensure that the identified functions are Koopman eigenfunctions almost surely. The work also introduces Symmetric Subspace Decomposition (SSD) algorithms which, should all data be centrally available, provably find the maximal Koopman-invariant subspace in the span of an arbitrary finite-dimensional functional space.…”