Decomposable Regularizer and Restricted Strong Convexity M-Estimator Framework

⚠️ Curated Limitations & Negative Results

• Non-Overlapping Group Structure for Direct Subspace Decomposability: As noted in Section 2.2 (p. 5), the decomposability verification for the block-(1, alpha_vec) group norm in Example 2 relies on disjoint group partitions G = {G_1, ..., G_{N_G}}; overlapping or hierarchical group regularizers require modified subspace constructions.
• Local Radius Restriction for Non-Quadratic Losses and Linear-Sparsity Penalty Refinement: As discussed in Section 2.4 (p. 9) and Section 6 (pp. 17-18; v1 Section 6.2 & 8, pp. 22-27), while quadratic least-squares losses have global second-order Taylor remainders, general non-quadratic losses such as generalized linear models require restricting the RSC condition to the intersection of C(M, M_bar^perp; theta*) with a local ball {||Delta||_2 <= R} via Lipschitz quadratic truncation; moreover, specifying lambda_n via the dual norm R*(nabla L(theta*)) can be suboptimal by a logarithmic factor in the linear-sparsity regime s = Theta(p).
• Primary-Source Typographical Notes Across v3 (Statistical Science 2012) and v1 (45-Page Preprint): Verified four self-correcting typographical details during full-text inspection: (1) in v3 Corollary 1 (p. 11, Equation 25a), the left-hand side is printed unsquared as ||theta_hat_{lambda_n} - theta*|| <= 9 (lambda_n^2 / kappa_L) Psi^2(M_bar) (dropping the square on the left-hand side and on kappa_L relative to Theorem 1 Equation 22 ||theta_hat_{lambda_n} - theta*||^2 <= 9 (lambda_n^2 / kappa_L^2) Psi^2(M_bar)), whereas v1 Corollary 1 (p. 12, Equation 22a) and v3 Corollary 2 (p. 13, Equation 34) use the dimensionally consistent unsquared bound <= (2 lambda_n / kappa_L) Psi(M_bar) and squared bound <= (64 sigma^2 / kappa_L^2) (s log p)/n; (2) in v3 Equation (33) on p. 13 (and v1 Equation 30 on p. 15), the sub-Gaussian tail condition prints P[|<v, w>| >= t] <= 2 exp(-delta^2 / (2 sigma^2)) for all delta > 0, mixing t and delta; (3) on p. 14 of v3 (and p. 17 of v1), the text writes 'Letting S_eta^c denote the complementary set S_eta \ {1, 2, ..., p}' for {1, 2, ..., p} \ S_eta and writes |theta*_i|^q inside sum_{j in S_eta} in Equation (39); and (4) in v1 Appendix B (p. 30, Proof of Lemma 2), the first display writes <= sqrt(|S_tau|) ||Delta||_2 + 4 R_q tau^{1-q} before restoring the factor 4 in the next inequality <= 4 sqrt(R_q) tau^{-q/2} ||Delta||_2 + 4 R_q tau^{1-q}.
• All 20 pages of Negahban et al. (2012, Statistical Science 27(4):538-557, arXiv:1010.2731v3) and all 45 pages of arXiv:1010.2731v1 (including Appendices A-D) were read. Constructed numerical checks in research/discovery/decomposable-regularizer-m-estimator-rsc-boundaries.py verify (1) exact subspace decomposability R(theta + gamma) = R(theta) + R(gamma), dual-norm Holder tightness, and nuclear-norm subspace inclusion dim M(U, V) = 1 < dim M_bar(U, V) = 7 with Psi(M_bar(U, V)) = sqrt(2r) = 1.414214; (2) Lemma 3 deviation inequalities, Lemma 1 cone vs. star-shaped constraint set membership, and Theorem 1 / Corollary 1 deterministic error bounds (||Delta||_2 = 0.127644 <= 0.598244); and (3) Corollary 3 l_q-ball thresholding bounds, Corollary 4 group-norm rate ordering (alpha = 2 rate 0.11912 < alpha = infty rate 0.67912), and Proposition 2 GLM Lipschitz quadratic truncation function phi_tau(u).