Discriminant Dynamic Mode Decomposition for Labeled Spatio-Temporal Data Collections

The paper precisely defines the dynamic-mode subspace (DMS), the projection kernel kDMS(W_{Θ1,X1}, W_{Θ2,X2}) via orthonormal bases, and the discriminant DMD objective J(Θ1:n) = (1/n)∑_i fDMD(Θ_i)/(fKFD(Θ1:n)^α+ε), with fKFD = Q1·Q2 taken from kernel Fisher discriminant optimization, and fDMD the optimized-DMD fitting loss; it proposes gradient-based optimization and provides derivative expressions and an algorithmic summary (see the problem setting and formulation, including (3.11)–(3.13), and the DMS kernel definitions in (3.1)–(3.3), along with derivative formulas (3.4)–(3.6) and Algorithm 3.1) . These are consistent and standard, aligning with background on KFD’s Q1 and Q2 (2.21)–(2.23) and the projection kernel on subspaces . The candidate solution largely mirrors the paper (e.g., uses kDMS via projectors tr(P_iP_j), kernel-only formulations for Q1/Q2, and optimized DMD), but it contains a concrete error: the displayed gradient of J with respect to W_i mis-differentiates the ratio, replacing the numerator’s gradient ∇_{W_i}fDMD(Θ_i) by terms involving ∂fKFD/∂K and using an incorrect denominator for the second term (it should be −α N fKFD^{α−1}/(fKFD^α+ε)^2 · ∇fKFD, not −α N/(fKFD^{α+1}+ε fKFD) · ∇fKFD). In addition, it overclaims convergence (“cluster points are first‑order stationary for the constrained problem; any such stationary point constitutes a (possibly local) minimizer”), which is not justified by the provided analysis. The paper itself does not make such convergence guarantees and only advocates quasi-Newton/gradient methods with gradients in an appendix . Hence, the paper’s argument is sound for what it claims, whereas the model’s solution contains a key calculus error and an unsupported convergence claim.