Uniform-in-submodel deterministic linear regression and functional dependence bounds

⚠️ Curated Limitations & Negative Results

• Minimum k-Sparse Eigenvalue Condition RIP(k, Sigma_1 - Sigma_2) < Lambda(k; Sigma_2): Unlike the normal-equation residual formulation in Kuchibhotla et al. (2020, arXiv:1806.04119v1), the deterministic uniform l2 and Bahadur bounds in Theorems 1 and 3 require positive k-sparse minimum eigenvalue Lambda(k; Sigma_2) > 0 and RIP(k, Sigma_1 - Sigma_2) < Lambda(k; Sigma_2) to invert Sigma_1(M) and Sigma_2(M) across all M in M(k).
• Effective Tail Exponent s(alpha) = (1/2 + 1/alpha)^{-1} < 2 and Slow-Decay Penalty Omega_n(nu) Under Functional Dependence: In Theorem 12 (Equation 35), blocking dependent summands yields an effective heavy-tail exponent s(alpha) = (1/2 + 1/alpha)^{-1} strictly smaller than min{alpha, 2}, and when the functional dependence decay parameter satisfies nu < 1/2, the variance-like factor scales with Omega_n(nu) = n^{1/2 - nu}, slowing the convergence rate below n^{-1/2}.
• Primary-Source Typographical Notes in arXiv:1802.05801v3: Verified five self-correcting typographical slips in the primary manuscript: (1) Proof of Theorem 1 (p. 10, display 3) prints 'Lambda(k; Sigma_2) - RIP(k; Sigma_2)' in the denominator instead of 'Lambda(k; Sigma_2) - RIP(k, Sigma_1 - Sigma_2)'; (2) Theorem 6 (p. 19) labels both parts '(a)' ('(a) under Assumption (MExp)' and '(a) under Assumption (JExp)') instead of '(a)' and '(b)'; (3) Theorem 8 (p. 25, second display for RIP_n(k)) prints 'Upsilon_{n,k}^Gamma' under the square root instead of 'Upsilon_{n,k}^Sigma'; (4) Proof of Lemma 2(b) (p. 31, line 1) prints 'If beta > 1/2, then' at the start of the third case where it should read 'If beta < 1/2, then'; and (5) Theorem 12 (p. 31, Equation 35) and p. 36 (first display after Equation 45) print '||{Z}||_{psi_alpha,nu}' in the first (Gaussian) term instead of '||{Z}||_{2,nu}' (cf. Equations 37 and 45), and Proof of Lemma 4 (p. 37, line 1) defines 'delta_{s,r}^{(j)} := ||W_i^{(1)} - W_{i,s}^{(1)}||_r' with superscript '(1)' instead of '(j)'.
• All 41 pages of the full-text PDF of Kuchibhotla, Brown, Buja, George, & Zhao (2021, arXiv:1802.05801v3) were read. Constructed numerical checks in research/discovery/uniform-in-submodel-deterministic-linear-regression-boundaries.py verify (1) Theorem 1 deterministic uniform-in-submodel l2 bound and Equation (24) moving-target bound across all 25 submodels, (2) Theorem 3 deterministic uniform Bahadur linear representation bound and Remark 3.3 factor-of-2 sandwiching, (3) Remark 4.3 identically zero Bahadur remainder under fixed design and Equation (26) sparse-sphere covering bound, and (4) Section 5 / Theorem 12 / Lemma 4 functional dependence norms, effective exponent s(alpha), three Omega_n(nu) regimes, and product persistence.