Discrete empirical interpolation in the tensor t-product framework

The paper’s Theorem 3.5 states and proves the t-Q-DEIM error bound ‖f − ftq‖ ≤ ‖(PT ∗ U)−1‖ · ‖(I − U ∗ UT) ∗ f‖, using the interpolatory projector D := U ∗ (PT ∗ U)−1 ∗ PT, the orthogonal projection f∗ = U ∗ UT ∗ f, the identity f − ftq = (I − D) ∗ (I − U ∗ UT) ∗ f, sub-multiplicativity of the t-spectral norm, the projector norm identity ‖I − D‖ = ‖D‖, and the facts ‖U‖ = ‖PT‖ = 1; see the statement and proof around Lemma 3.4 and Theorem 3.5, and Properties 3.1–3.2 (projector/interpolatory) and Lemma 3.1 (sub-multiplicativity) in the paper . The candidate solution reproduces the same construction and steps (definition of D, showing D is a projector and that (I − D) annihilates the U-component, the same norm inequalities, and bounding ‖D‖ ≤ ‖(PT ∗ U)−1‖ using ‖U‖ = ‖PT‖ = 1), arriving at the identical bound; this mirrors the paper’s proof essentially line-for-line . Hence both are correct and substantially the same proof.