Source clarifications for the linear representation hypothesis

Proposition 9: choosing the coherence scale

Auxiliary scale: replace $\mu=\epsilon/[k(1+\lambda^2)]$ by $\mu=\frac{\epsilon\lambda^2} {4(k+1)(1+\lambda)^2(1+\lambda^2)(1+2\lambda)}, \qquad \lambda=\sqrt{1/\delta-1}.$ The printed choice gives recovery bound $\epsilon(1+\lambda)^2/(1+\lambda^2)>\epsilon$ for $\lambda>0$ . The smaller choice proves recovery and keeps normalization denominators positive. For fixed $0<\epsilon,\delta<1$ , it preserves the $O_{\epsilon,\delta}(k^2\log m)$ dimension rate. This is a sufficient construction, not a counterexample to the proposition.

Source: Proposition 9, cited publication:547-610; scale choice, :580-597.

Theorem 1 (compressed sensing): the finite construction is proved for $1\leq k$ and $2k\leq m$ . The paper identifies $k=O(\log m)$ as its intended regime in Section 1.2; since $k/m\to0$ , the finite cutoff holds for all sufficiently large $m$ . The checked bound therefore proves the stated $d=O(k\log(m/k))$ asymptotic conclusion in that regime.
Theorem 3 (lower bound): use the nontrivial accuracy range $0<\epsilon<1$ , together with the displayed lower restriction on $\epsilon$ . Since features lie in $[-1,1]$ , tolerance $\epsilon>1$ already permits the zero predictor. The endpoint $\epsilon=1$ is outside the checked result.
Proposition 11: $\epsilon>0$ → $0<\epsilon<1$ , making the norm lower bounds used in division positive. This is the range covered by the normalization proof; necessity for every broader formulation is not established.

Source: compressed sensing, cited publication:184-199, intended regime :227; lower bound, :220-227; Proposition 11, :638-680.

Proposition 11: first denominator $1-\epsilon$ → $(1-\epsilon)^2$ .
Theorem 12 and Corollary 13: add a complementary rank argument for the part of $k<\sqrt m$ not covered by the printed submatrix argument.
Lemma 7 (cited Alon theorem): remains an external proof boundary for the lower-bound branch.