Learning Bilinear Models of Actuated Koopman Generators from Partially-Observed Trajectories

The candidate’s E/M-step derivation matches Theorem 4.1 exactly: the ELBO splits into three independent blocks (initial-state, observation, transition) as in Proposition A.1, and the maximizers are the closed-form updates for μ0, Σ0, c0, Σv, the stacked Kronecker normal equations for [Ṽ0; …; Ṽdimu], and Σw, respectively. The formulae for c0 (Eq. (37)), Σv (Eq. (38)), the stacked generator system (Eq. (39)), and Σw (Eq. (40)) coincide with the paper’s statements; the candidate’s use of an augmented input vector with u0≡1 and a row-selection matrix J is notation-equivalent to the paper’s formulation of the bilinear HMM (21)–(24) and U_l = I + Δt∑[u_l]_k V_k. The role of the bounded-likelihood assumption ensuring positive definiteness/invertibility is also aligned with the argument in Appendix A. Any differences are cosmetic, not substantive. See Theorem 4.1 and its proof for the precise updates and assumptions, including the definitions of G_l^(m) and H̃_l^(m) and the Kronecker system for Ṽ (35)–(40) and Appendix A’s argument about positive definiteness under bounded likelihood.