L1-Penalized and Post-Penalized Quantile Regression Process in High-Dimensional Sparse Models

⚠️ Curated Limitations & Negative Results

• Condition D.1 and Condition D.4 require the conditional density $f_{y_i \mid x_i}(y \mid x_i)$ to be bounded strictly away from zero ($f_{y_i \mid x_i}(x_i'\beta(u) \mid x_i) \ge \underline{f} > 0$) and have bounded derivative ($\le \bar{f}'$) in a neighborhood of the true conditional quantile $x_i'\beta(u)$ uniformly over $u \in \mathcal{U}$, so $\mathcal{U} \subset (0, 1)$ must be compact and bounded away from extreme tails $u \to 0$ or $u \to 1$ where $W_{\mathcal{U}} = \sup_{u \in \mathcal{U}} [u(1-u)]^{-1/2}$ diverges and conditional densities vanish.
• Uniform sparsity $\sup_{u \in \mathcal{U}} \|\widehat{\beta}(u)\|_0 \lesssim_P s$ in Theorem 3 requires the penalty multiplier $\lambda$ to exceed a sparse-eigenvalue threshold $\lambda > 2\sqrt{2}\sup_{u \in \mathcal{U}} [\widehat{\mu}(u)/\sqrt{u(1-u)}]\sqrt{\bar{f}\phi_{\max}(n)}$ (Lemma 10, Eq. C.6), and exact support inclusion $T_u \subseteq \widehat{T}_u$ in Theorem 4 requires a beta-min signal strength condition $\inf_{u \in \mathcal{U}} \min_{j \in T_u} |\beta_j(u)| > \sup_{u \in \mathcal{U}} \|\widehat{\beta}(u) - \beta(u)\|_2$.
• Primary-source typographical audit of `arXiv:0904.2931v5`: (1) Section 3.3 Proof of Theorem 2 (p. 20, line 23) writes 'By definition of $\widehat{\beta}(u)$ in (3.16)' instead of '(2.4)' (since Eq. 3.16 is the Symmetrization Lemma display on p. 19); (2) Appendix B Proof of Lemma 6 (p. 29, Eq. B.2) prints '$\le 2\|\widehat{\beta}_{T_u}(u) = \beta(u)\|_{1,n}$' with an equals sign `$=$' instead of a minus sign `$-$' in $\|\widehat{\beta}_{T_u}(u) - \beta(u)\|_{1,n}$; (3) Appendix B Proof of Lemma 7 (p. 31, line 18) writes '$w_i(\delta, u)$' in place of '$w_i(x_i'\delta, u)$'; (4) Appendix C Lemma 9(1) (p. 33, Eq. C.3) states the complementary slackness relation as 'iff' ($\widehat{\beta}_j(u) \ne 0 \iff \mathbb{E}_n[x_{ij} \widehat{a}_i(u)] = \widehat{\sigma}_j \frac{\lambda\sqrt{u(1-u)}}{n}\operatorname{sign}(\widehat{\beta}_j(u))$) whereas in general LP duality necessity ($\implies$) always holds and sufficiency holds almost surely under primal non-degeneracy proved in Step 3 on p. 34; (5) Appendix C Lemma 11(2) (p. 37, line 16) prints duplicated word 'the the' ('with the the same probability'); (6) Appendix C Proof of Lemma 13 (p. 38, line 28) prints a truncated word 'w' ('since $s_u \le s$, w $r_u \le r$'); and (7) Appendix E Proof of Lemma 21 (p. 49, line 26) writes '$x_i'\beta_1^o$' instead of '$x_i'\beta^o$'.