A Higher Order Unscented Transform

The paper’s Theorem 3.1 proves linear convergence of the deflation scheme with rate r = sqrt(1 − c^2/d^k), using the eigenpair identity ‖T − λ v^{⊗k}‖_F^2 = ‖T‖_F^2 − λ^2 (Lemma 2.14) and the entrywise-to-eigenvalue bound λ ≥ (c/d^{k/2})‖T‖_F derived from the assumption λ ≥ c|T_{i_1…i_k}| for all entries; see Lemma 2.14 and Theorem 3.1 in the uploaded paper . The candidate solution applies exactly the same steps: (i) sum-of-squares to get λ_ℓ^2 ≥ (c^2/d^k)‖T_ℓ‖_F^2; (ii) plug into the identity to obtain a uniform contraction; and (iii) telescope to get T = ∑_{ℓ=1}^L λ_ℓ v_ℓ^{⊗k} + E_L with ‖E_L‖_F = O(r^L). The paper also supplies the needed entrywise lower bounds on λ for orders 3 and 4 (Lemma 3.2, Appendix), confirming the assumption that underpins the contraction . Hence both arguments are correct and essentially the same.