Data-Driven Model Order Reduction for T-Product-Based Dynamical Systems

The paper’s Proposition 4 states the T-BT H∞ error bound ∥G − Gred∥∞ ≤ 2 max_i Σ_{j=n−k+1}^n σ_j^(i) and proves it by (i) balancing via the T-SVD of the Hankel tensor, (ii) partitioning the balanced system, (iii) block-diagonalizing the block-circulant lift in the Fourier domain, and (iv) applying the classical BT bound per block and the max property of the H∞ norm for block-diagonal transfer matrices. This is explicitly laid out in Definition 11 and Proposition 4, together with the proof steps using Fourier block-diagonalization and per-block BT bounds . The paper also establishes decoupling of the T-Lyapunov equations/Gramians in the Fourier domain, supporting the per-block view used in the proof . The candidate solution follows the same structure: (1) DFT block-diagonalization of ξ(·) to obtain s independent MIMO LTI blocks; (2) decoupling of Gramians and T-SVD into slicewise SVDs; (3) balancing and truncation per block; (4) application of the classical per-block BT bound and the max property of the H∞ norm. The only minor imprecision is the model’s phrase that truncating k singular tuples corresponds to truncating “the k smallest” singular values in each block; strictly, the k truncated tube singular tuples correspond to indices j = n−k+1,…,n across blocks, not necessarily the k smallest per-block singular values. This is a cosmetic issue and does not affect the bound. Net: both are correct, and the proofs are essentially the same.