Soluciones a las problemáticas del Patrimonio Cultural

The iterative formula created in Chapter 3 for the calculation of moments of a self-similar measure serves another purpose. Namely, as we extracted a contraction mapping on ₁`∞ — under certain conditions — we recover another form of the Collage Theorem for moments.

Theorem 4.32. Let M be an IFS with probabilities of the form given in either Theorem 3.53 or 3.55. Then, its associated vector of moments M_x,y∗ obeys the inequality:

kM −M_x,y∗ kg ≤

kΦM−Mk_g

1−λ ,

where M ∈₁`∞ and λis the contractivity factor of the operator Φ : (1`∞,k · kg)→(1`∞,k · kg).

This Theorem has the following interpretation for modelling: given a vector of momentsM, we want to find a contractive operator Φ whose fixed pointM∗ is close toM. The bound states that ifM is almost a fixed point of Φ, then the actual fixed point of Φ is nearM in thek · k_g-norm sense.

Proposition 4.33. Let M be an IFS of the form of the above, then under the norm k · kg we have Mx,y∈c0: the space of convergent sequences in (1`∞,k · kg) approaching zero.

Proof. Select the k · kg norm such that the operator Φ is contractive and the attractor of the

IFSF_g◦f◦g−1 is (strictly) contained in thek · k∞-unit ball — such a choice is always possible as the scaling of the functiong can be arbitrarily close to zero and continue to make Φ contractive. We may either viewMx,y ∈(1`∞,k · kg) as the vector of moments forµAorMx,y ∈(1`∞,k · k∞) as the vector of moments for µA◦g−1 in light of Lemma 3.48. Then for n1, n2∈N0,

Z xn1_yn2_dµ g(A)(x, y) _∞ ≤ Z kxn1_yn2_k ∞dµg(A)(x, y)≤ max x,y∈g(A) kxn1_yn2_k ∞ n1+n2 →∞ −−−−−−−→0,

where we have used µA is normalised and any points on g(A) are less than one. Through the

(graded lexicographic) ordering chosen in the vectorisation mapv, the claim is given.

Corollary 4.34. Let M be of the form assumed above. Let M_x,y∗ be the vector of moments of the invariant measureµAthat is in1`∞. LetM ∈1`∞andMn be itsnthlevel truncation chosen such that n=bnct, then we gain the following chain of bounds:

kM_x,y∗ −Mk_g ≤ kΦMM−Mkg

1−λ ≤

kΦM,nMn−Mnkg

1−λ +ε(n)≤C(λ)kΦM,nMn−Mnk2+ε(n). for some constants ε, C >0 that depends on the truncation level n and contractivityλ. Where ε can be made small with increasingn.

Remark 4.35. The above Corollary firstly uses the Collage Theorem for moments, then Propo- sition 3.52 that states we may appropriately re-scale the space such that the measures considered have moments that tend to zero. This allows for truncation to finitely many moment termsMn

4.3. THE COLLAGE THEOREM FOR MOMENTS 69 plus some error termε >0 which can be made arbitrarily small with increasingnby Proposition 4.33. Finally, as Mn is finite dimensional, we may swap to the L2 norm and have some con-

stant that depends on the contractivity of the operator Φ. This creates a feasible computational problem.

The Corollary above gives justification to truncating finitely many moment equations and finding a least squares solution that does not solve our problem; it simply must be a local-minima of the moment equations. For instance letM21 be the vector of the first 21 moments for the Steemson triangle in Example 2.10. A 21-moment approximation of a Steemson triangle found through a least squares optimisation, wherekΦM,21M21−M21k2 ≈6·10−6 for the affine approximant and ≈9·10−5 for the similitude approximant. In this approximation, the maps were intentionally given the incorrect orientations as to not be an exact reconstruction.

Figure 4.3: Affine (left) and similitude (right) moment approximation of a Steemson triangle.

In particular, we have created an approximation method that does not require the object that we are approximating to be strictly self-similar but may find an exact solution if it exists (see Figure 4.5). This bound also makes sense of the trivial solutions spotted in the algebraic approach to solving these moment equations, seen through the following example.

Example 4.36. Recall we had the trivial solutions to the polynomial system in one dimension:

n ai, bi, pi ∈R ai ∈ {−1,1}, bi= 0, ∀i∈ {1, . . . , N}; N X i=1 aipi= 1 o .

In one dimension we have that|ai|=λi. The bound given by the Collage Theorem for moments

is meaningless for these trivial solutions as the constantC(λ)−−−−−−−→ ∞maxi|ai|→1 . This can be seen through identifying that the contractivity of Φ is λ = max_i∈[N]{λi} in one-dimension, where λi are the contractivity factors of the maps fi in the IFS. In the two-dimensional case, trivial

solutions to the moment equations are in even more abundance and may potentially ruin any numerical experiment run if not appropriately controlled. These solutions cannot be generally written out as knowing precisely these solutions would imply knowing when an affine IFS is contractive with respect to its map parameters.

The above shows one caveat that is often left unmentioned in much of the literature, that being that the quality of the bound given from the Collage Theorem is highly dependent on the operator Φ. In practice, this is not commonly a problem as the model fit is sufficiently constrained to, consciously or unconsciously, make sure this bound does not blow up. If this problem is addressed, then we have the following interpretation of this bound. The vector equation ΦM−M

represents our system of non-linear equations, for which (in practical use), is unreasonable to assume that an exact solution exists. The bound given states that kΦM−Mk_g being small for fixed IFS parameters, fixing Φ will yield an approximate solution, or more explicitly a local- minima of kΦM −Mk_g may produce a ‘good’ approximate. Thus, in a synonymous fashion to the ‘standard’ Collage Theorem, we explore this interpretation computationally in the next section.