Mapa 3. Sedes distribuídas entre los ocho Resguardos Indígena asociados
4.2 TRANS-APORTANDO
4.2.3 El Francisco de Orellana, pedagogía intercultural
is the posterior contraction rate of the Gaussian prior considered in this section. By direct calculation, it can be shown that when for all 1 ≤ i ≤ m,
αi=
2H(β) + m 2H(β)
βi,
the rates r1and r2in (10.28) are balanced to r1= r2= m/H(β) and the posterior
contraction rate reaches the minimax rate (see [53]), i.e. εn' n−
H(β) 2H(β)+m.
10.3.2.4 Proof of Theorem 10.5
After the long preparation, the proof of Theorem 10.5 simply follows from assem- bling all the results obtained up to now.
Recall {ϕek,l}(k,l)∈Nd×N= {ϕk⊗ ψl}(k,l)∈Nd×Nis a fixed basis of L2(DT), satis-
fying the assumptions in Section 9.5. For a function f in L2(D
T), its coefficients
in the basis {ϕek,l}(k,l)∈Nd×N are denoted by ef = {fk,l}(k,l)∈Nd×N. Recall that for
the norms k·kL2, k·kHβ, k·k`2, k·khβ from Section 2.3, we have the isometries
kf kL2= k ef k`2, kf kHβ= k ef khβ,
implying that it is sufficient to show the convergence of the coefficients in the sequence space.
Consider the change of variables bf = {|kβ∗|f
k,l}(k,l)∈Nd×N. The model (10.26)
can be rewritten into,
b
Xk,l(n)= bfk,l+
1 √
nzk,l, for k ∈ k, l}(k,l)∈Nd×N,
where zk,l are independent standard Gaussian random variables. Notice that the
prior of f induces a prior of bf in `2
(Nd× N). Therefore, using Lemma 10.13, we
obtain the posterior contraction rate of the induced prior in k·k`2(d+1).
For the preceding change of variables, an isometry k bf k`2= k ef kh
β∗ holds, given
e
f is in hβ∗. The isometry implies that rates of bf relative to k·k`2 can be translated
to the rates of ef relative to k·khβ∗. Consequently, the result can be translated to
the rate in k·kHβ∗ and the proof is complete.
10.4
Entropy Number with Non-Polynomial Rates
In Section 2.4, we have shown the estimate of metric numbers of the embedding ι : Hs+t→ Ht with the scale. The same argument can be applied to ι : eH → Ht
where eH is another Hilbert space contained in the smoothness class {Ht}t, see the
lemma below.
Lemma 10.14. Given HG from (10.13) and the isotropic Sobolev spaces {Hs}s∈R
defined in Section 2.3.1. Then for the canonical embedding ι : HG → Hs, when j
is large enough, the entropy number is of the order e−c1jν/(ν+d) ≤ e
j(ι : HG→ Hs) ≤ e−c2j
ν/(ν+d)
where c1, c2 are universal positive constants.
Proof. The proof follows the same argument from Section 2.4 (also see the Ap- pendix B in [43]). The singular values of ι : HG → Hsare of order e−T j
ν/d
j−(α−s)/d, for which the upper bound is obtained using the same argument in (10.15) and the lower bound is obtained by taking the unit vector ϕk such that ki ' j1/d.
Consequently, the approximation numbers aj(ι : HG→ Hs) have the same order.
In particular, aj(ι : HG → Hs) = O(e−T j
ν/d
). By the second example in Section 3 of [103], we obtain the final statement of the lemma.
Because ε 7→ H(ε, ι) is the inverse mapping of j 7→ ej(ι), we obtain the corollary
below.
Corollary 10.15 (Metric entropy). The metric entropy of the unit ball of HG, given
in (10.13), in isotropic Sobolev spaces {Hs}s∈R is given by
H(ε, ι) := log N ε, {g ∈ HG: kgkHG≤ 1}, k·kHs ∼ log1 ε ν+dν , as ε ↓ 0.
Appendix A
Mathematical Tools
In this appendix we collect the mathematical elements, mainly from operator the- ory, that serve as the underlying language and building blocks for this thesis. They are from well established fields and can be found in textbooks and monographs. Hence, results will be present, and proofs are referred to the literature.
Operators are ubiquitous in this thesis, as one main component of the Gaussian linear model, the transform A, is an operator. In particular, compact operators is of great importance, which is demonstrated by the following examples. First, the ill-posedness in a large class of linear inverse problems is characterised as the compactness of transform operators. Second, a Gaussian measure is a proper probability measure (instead of a generalised stochastic process) only when its covariance operator is of trace class, which is necessarily compact. Third, an element in a compact space can be well approximated by a finite-dimensional subspace, and the error estimate is closely related to the compactness. Besides the aforementioned cases, there are other places where the compactness is leveraged.
In this section, we collect the necessary information on operator theory, with special attention to compact operators. All the materials are standard and can be found in many textbooks, e.g. [102].
First let us summarize the common notations for operators. Let X, Y be normed spaces over the field R. A linear operator T from X to Y is a linear mapping from the domain of T , i.e. a subspace of X denoted by Dom T , into Y . The image of T is called range, i.e. Ran T = T (Dom T ) = {T f : f ∈ Dom T }. A linear operator from X to R is a linear functional. The notation T : X → Y is understood as Dom T = X and Ran T ⊆ Y , unless the domain is given explicitly. An operator is injective precisely when T f = 0 implies f = 0. For an injective operator, the inverse T−1 of T is given by
Dom T−1= Ran T , T−1g = f, for g = T f ∈ Ran T .
The space of bounded linear operators form X to Y is denoted as B(X, Y ), i.e.
B(X, Y ) := (
T : X → Y | linear and kT kX→Y:= sup h∈X:khk≤1
kT hkY< ∞
) ,
where k·kX→Y is the operator norm and k·k may be used if no danger. If X = Y ,
we write L(X).
Definition A.1 (Adjoints). Let X, Y be Banach spaces. The adjoint T∗of a densely defined (not necessarily bounded) linear operator T : X → Y is the operator uniquely determined by
T∗y∗= x∗,
hy∗, T xi = hx∗, xi , ∀x ∈ Dom T .
An densely defined operator S : X → X is self-adjoint if Dom S = Dom S∗ and hSh, gi = hh, S∗gi, for all h, g ∈ Dom S.
Remark A.2. If X and Y are Hilbert spaces, the dual spaces X∗ and Y∗ can be identified with the original space by Riesz representation theorem. If the operator T : X → Y is bounded, then the definition above is equivalent to the standard definition of adjoints on Hilbert spaces, that there exists a unique operator T∗ : Y → X such that
hT x, yiY = hx, T∗yiX, for all x ∈ X and y ∈ Y .
Definition A.3 (Positivity). An operator T on a Hilbert space is called positive, denoted by T ≥ 0, if hT h, hi ≥ 0, for all h ∈ Dom T . For two positive operators S, T , we write S ≥ T if Dom S ⊂ Dom T and S − T ≥ 0 on Dom S. We also write S = T if S ≥ T and T ≥ S.
If the above properties hold up to independent constants, then we use the notations S. T and S ' T .
A.1
Miscellaneous Lemmas
In this section, we collect a few useful lemmas.
The following lemma is known the bounded linear transform (BLT) theo- rem,(see Theorem I.7, [81]).
Lemma A.4 (BLT theorem). Let T be a bounded linear operator from (X, k·kX)
to a complete normed space Y . Then there exists a unique bounded extension eT of T from the completion of X under k·kX to Y .
The following lemma is a direct consequence of Hahn-Banach theorem. Lemma A.5. Given a normed space (E, k·k) with its topological dual E∗, the fol- lowing holds
kxk= sup
f ∈U (E∗)
|hf, xi|.
Using positivity, we have another characterisation of operator norms on Hilbert spaces.
A.1. Miscellaneous Lemmas
Lemma A.6. Let T be an positive element in B(H). Then, kT k= sup
kxk≤1
hT x, xi.
The following result provides the soundness to Gelfand triples.
Lemma A.7. Let G and H be two Banach spaces such that G is a dense subset of H, and the embedding ι : G → H, g 7→ g is continuous. Then, the following hold.
(i) The inclusion mappingeι : H∗ → G∗, ` 7→ `|
G, where `|G is the restriction
of ` to set G, is continuous. In particular,
h`, giH∗×H = h`|G, giG∗×G, ∀` ∈ H∗, ∀g ∈ G. (A.1)
(ii) H∗ is dense in G∗, if G is reflexive.
In particular, if H is a Hilbert space and G is reflexive, we have G ⊂ H = H∗⊂ G∗.
Proof. First we show the continuity ofeι. Notice that for all g ∈ G, kgkH. kgkG,
because of the continuity of ι. For any ` ∈ H∗, we have |`(g)|. khkHkgkG.
Let e` be the restriction of ` to the subset G ⊂ H. Then, e` ∈ G∗such that
e
`(g) = `(g), ∀g ∈ G, (A.2) and
ke`kG∗≤ k`kX∗, ∀` ∈ H∗.
In addition, e` = 0 implies ` = 0. This is because of (A.2) and the density of G in H. Hence the inclusion mappingeι : ` → e` is injective and continuous, and (A.1) holds.
Now we are going to show that H∗ is dense in G∗ by contradiction. If the statement is not true, then the closure of H∗ in G∗ is a proper closed subspace of G∗. By Hahn-Banach theorem, there exists a non-zero functional ϕg ∈ (G∗)∗
such that ϕg(e`) = 0 for all e` ∈eι(H
∗) ⊂ G∗. Because of reflexivity, the functional
can be identified with an element g ∈ G, such that ϕg(e`) = e`(g) = 0, for all
e
` ∈eι(H∗) ⊂ G∗. Due to (A.1), `(g) = 0, for all ` ∈ H∗. Since g ∈ G ⊂ H, it
implies that g = 0, which contradicts to ϕg 6= 0.
An embedding of Hilbert spaces naturally gives rise to an isometric isomor- phism, which is useful in several occasions in this thesis. Meanwhile, it also shares some similar flavour of Lemma A.7.
Lemma A.8. Assume that Hilbert space H is a dense subspace of Hilbert space X such that khkH≥ khkX, for all h ∈ H, and let the canonical embedding be
ι : H → X, h 7→ h. Then,
U = (ιι∗)−1/2 : Dom U ⊂ X → X,
where Dom U = H, is an isometric isomorphism, i.e. kU hkX= khkH.
Proof. This proof is adopted from Theorem IV.1.12, [63].
Since ι is compact, so is S = ιι∗. Furthermore, S : X → X is self-adjoint and positive, and Ran S ⊂ H. We can define a self-adjoint operator T = S−1 on domain Dom T = Ran S ⊂ H, such that
hh, giH = hT h, giX, (A.3) for all h ∈ Dom T and g ∈ H.
Using spectral theorem, define an operator U = (T )1/2, whose domain Dom U is the closure of Dom T with respect to the norm
kU hkX=
q
hT h, hiX= khkH.
The domain Dom U a closed set in H. We are going to show in fact Dom U = H by contradiction. Assume Dom U ( H. Then by Hahn-Banach theorem, there exists an element h0 ∈ H such that hg, h0iH = 0 for all g ∈ Dom U , and in particular,
all g ∈ Dom T . Due to (A.3), we have hU g, h0iX = 0, for all g ∈ Dom U . Since
Ran U = X, we conclude that h0= 0, which leads to a contradiction.