In conclusion, we completely establish the strong law of large numbers for Betti numbers in both the Euclidean and manifold setting in this chapter. Our result for manifolds, i.e., Theorem 3.1, affirms the belief that the dense regime is the only regime appropriate for homology inference. This is because for two manifolds having different homologies, the value of the limiting constant might be the same. Moreover, since the limiting constant in the theorem is
∫
M
βˆk(m)(κ(z), r) dz
≡0 ifk≥m
>0 otherwise,
it seems that the thermodynamic regime can be used for the dimension estimation of the unknown manifold, which is an important problem in manifold learning. For instance, given a data set inRN, one can choose a sequencern for each m < N such that n1/mrn →r, where r should be ‘small’. Then one can check the above criteria, and theoretically, it must satisfy only for a unique m. However, these statements are at a very preliminary stage and should be taken as informal information. This is because the data set is generally finite, and we are checking the criteria with the limiting constant, and moreover, higher order Betti numbers of ˇCech complexes are still not efficiently computable in practice. Nevertheless, some simulations with data sets in R3 as toy examples would be worth trying, which we shall do in the near future.
In this chapter, we also extracted an upper bound for the limiting constant from the work of Bobrowski et al. [11], which we call as the exponential decay of the limiting
88 3.5. CONCLUSION & FUTURE DIRECTIONS constant (see Lemma3.2). This bound captures the behavior of ˆβk(m)(1, r) in two ways:
by fixingm, ˆβk(m)(1, r)→0 asr → ∞and by fixingr >0, ˆβk(m)(1, r)→0 asm→ ∞. Intuitively, the latter behavior, at least fork= 0, might be happening because of the concentration of distances in high dimensions [40] and large n. However, what the bound does not capture is the behavior for small r illustrated in Figure 2.2. In the figure, we see that at a fixed ‘small’ value ofr, the limiting constant for a particular value of k dominates the other limiting constants. Therefore, it will be interesting to find some other (strict) upper or lower bounds, which captures this phenomenon as well. At this point, it seems difficult to give the exact expression for the limiting constant but would be a breakthrough whenever find out.
After the law of large numbers, the next natural question is about the central limit theorem for Betti numbers in the thermodynamic regime. As we have seen in Chapter 2, the central limit theorem in the Euclidean setting has been completely established (Theorem 2.46). However, the central limit theorem in the manifold setting is still open. Furthermore, the central limit theorem for general stationary point processes is also not established yet. There are central limit theorems for functionals of general point processes (satisfying some constraints such as exponential decay of correlations) [6], however, that results are not directly applicable to Betti numbers. Nevertheless, the reference [6] might be a good start for solving the problem for Betti numbers.
Persistent Betti numbers and persistence diagrams
4.1 Introduction
In this chapter, we derive the strong law of large numbers for persistent Betti numbers of random ˇCech filtrations built on scaled binomial point processes and their Pois-sonized versions, in both Euclidean and manifold settings. These settings are same as those in Chapter3. The law of large numbers for persistent Betti numbers is derived by the same approach as used in that for Betti numbers. Furthermore, we also prove the vague convergence of persistence diagrams of random ˇCech filtrations in both the settings, using the similar techniques to the ones used in deriving Theorem2.42. Let us begin by mentioning all the main results of this chapter in the next subsection.
4.1.1 Main results
We first discuss the law of large numbers for persistent Betti numbers.
Law of large numbers for persistent Betti numbers
Let us recall a known result on homogenous Poisson point processes before stating our main results. For 0≤k≤m−1, asL→ ∞, (Theorem 2.40)
βks,t(C(PL(λ)))
L →βˆk(m)(λ, s, t) a.s.
As a consequence of the nerve theorem (Theorem 1.16), we set ˆβk(m)(λ, s, t) ≡ 0, if k ≥m. The scaling property, continuity and positivity of the limiting constant are already mentioned in Section2.4. Moreover, since βks,t(C(PL(λ)))≤βk(C(PL(λ), t)), it follows from Lemma3.2that
βˆk(m)(λ, s, t)≤Cm,ktmkexp(−cmtm),
whereCm,k and cm are positive constants depending only on their subscripts.
89
90 4.1. INTRODUCTION Recall thatZnandQnare the binomial point process and its Poissonized version in manifold setting, respectively. We consider the scaled binomial point process and the corresponding Poissonized version to construct the random ˇCech filtrations, which are denoted by C(nm1Zn) := {C(nm1Zn, r)}r≥0 and C(nm1Qn) := {C(nm1Qn, r)}r≥0
respectively. In manifold setting, we shall prove the following.
Theorem 4.1(For Manifolds). Assume that the common probability density func-tion κ(z) is supported on an m-dimensional compact C1 manifold M ⊂ RN and for allj∈N, ∫
Mκ(z)jdz <+∞. Then, for any 0≤s≤t <∞, as n→ ∞, βs,tk
(C(nm1Zn) ) n
resp. βks,t
(C(nm1Qn) ) n
→
∫
M
βˆk(m)(κ(z), s, t)dz a.s.
Note that it may be possible that for somem≤k < N,βks,t
(C(nm1Zn) )
>0, but βˆk(m)(λ, s, t)≡0 for allk≥m. Whens=t,βks,t
(C(nm1Zn) )
=βk
(C(nm1Zn, s) )
and so by Theorem3.1, ˆβk(m)(λ, s, t) = ˆβk(m)(λ, s) (the limiting constant in Theorem 3.1).
Now we discuss our result in the Euclidean setting. In this setting, we consider the following random ˇCech filtrations: C(nN1Xn, ρn) :={C(nN1Xn, r, ρn)}r≥0 for binomial point processes andC(nN1 Pn, ρn) :={C(nN1Pn, r, ρn)}r≥0 for Poisson point processes.
Here, ρn is a metric on nN1A defined as ρn(x, y) := nN1ρ(x/nN1, y/nN1 ). We shall prove the following law of large numbers in Euclidean setting.
Theorem 4.2(For Euclidean Spaces). Let(A, ρ)be a metric space, where Ais a Borel subset ofRN withLebN(∂A) = 0 and the metric ρ satisfies the properties (P1) and (P2). Assume that the common probability density function f(x)is supported on Aand for all j∈N, ∫
RNf(x)jdx <+∞. Then, for any0≤s≤t <∞, as n→ ∞, (a)
βks,t
(C(nN1Xn, ρn) )
n →
∫
RN
βˆk(N) (f(x)
D(x), s, t )
D(x)dxa.s.,
(b) βs,tk
(C(nN1 Pn, ρn) )
n →
∫
RN
βˆk(N)
(f(x) D(x), s, t
)
D(x)dx a.s.,
whereD(x) = det(Bx)withBx being the positive definite matrix in the property(P1).
Here, βˆk(N)(λ, s, t) is the limit of persistent Betti numbers in case of homogeneous Poisson point processes onRN (not on Rm).
Remark 4.3. Under the same assumptions as in the above theorems, the convergence in expectation of persistent Betti numbers also hold. For example, in manifold setting and for binomial point processes, under the assumptions as in Theorem4.1,
E[ βks,t
(C(nm1 Zn) )]
n →
∫
M
βˆk(m)(κ(z), s, t)dz.
Vague convergence of persistence diagrams
Again, before stating our main results, we recall a known result on vague convergence of persistence diagrams in case of homogenous Poisson point process. For 0 ≤ k ≤ N−1, asL→ ∞, (Theorem 2.42)
Dgmk(C(PL(λ))) n
→v νk,λ(N) a.s,
where →v denotes the vague convergence of measures (see Definition 1.22). We also recall from Section2.4thatνk,λ(N)is a null measure for allk≥N and for anyB∈ R(∆),
νk,λ(N)(B) =λνk,1(N)(λ1/NB).
Now our results for vague convergence of persistence diagrams of random ˇCech filtrations in manifold and Euclidean settings are as follows:
Theorem 4.4 (For Manifolds). Under the same assumptions as in Theorem 4.1, as n→ ∞,
Dgmk
(C(nm1Zn) ) n
resp.
Dgmk
(C(nm1Qn) ) n
→v νk,κ(m) a.s.,
where for 0≤k≤m−1 and A∈ R(∆), νk,κ(m)(A) =
∫
Mνk,κ(z)(m) (A)dz=
∫
Mνk,1(m)(κ(z)1/mA)dz, and for all k≥m, νk,κ(m) is a null measure.
Theorem 4.5 (For Euclidean spaces). Under the same assumptions as in Theo-rem 4.2, as n→ ∞,
Dgmk
(C(nN1Xn, ρn) ) n
resp.
Dgmk
(C(nN1Pn, ρn) ) n
→v νk,f /D(N) a.s.,
where for 0≤k≤N −1 and A∈ R(∆), νk,f /D(N) (A) =
∫
RNνk,f(x)/D(x)(N) (A)D(x)dx=
∫
RNνk,1(N)
((f(x) D(x)
)1/N
A )
D(x)dx, and for all k≥N, νk,f /D(N) is a null measure.
Remark 4.6. The vague convergence of the expectation of persistence diagrams in both settings also hold. For instance, in the manifold setting and for binomial point processes, under the assumptions as in Theorem4.1, asn→ ∞,
E[ Dgmk
(C(nm1 Zn) )]
n
→v νk,κ(m).
92 4.2. SLLN FOR PERSISTENT BETTI NUMBERS