In this appendix, as a complementary to Theorems 1–2, we provide additional theorems, Theorems 3–4, which further illustrate the two adv
… (see more)antages of the discretization process by considering an abstract model with the discretization bottleneck. For the advantage on the sensitivity, the error due to potential noise and perturbation without discretization — the third term ξ(w, r′,M′, d) > 0 in Theorem 4 — is shown to be minimized to zero with discretization in Theorems 3. For the second advantage, the underlying dimensionality of N(M′,d′)(r,H) + ln(N(M,d)(r,Θ)/δ) without discretization (in the bound of Theorem 4) is proven to be reduced to the typically much smaller underlying dimensionality of L + ln(N(M,d)(r, E ×Θ) with discretization in Theorems 3. Here, for any metric space (M, d) and subset M ⊆ M, the r-converging number of M is defined by N(M,d)(r,M) = min { |C| : C ⊆ M,M ⊆ ∪c∈CB(M,d)[c, r]} where the (closed) ball of radius r at centered at c is denoted by B(M,d)[c, r] = {x ∈M : d(x, c) ≤ r}. See Appendix C.1 for a simple comparison between the bound of Theorem 3 and that of Theorem 4 when the metric spaces (M, d) and (M′, d′) are chosen to be Euclidean spaces.