Sobre tipologías seleccionadas

Even though the framework of locally inherited densities described in this chapter generalized the “statistical" approach with a geometrical view, the local restriction of the “statistical" ap- proach still persists. In particular, the locally inherited densities are only defined within ρ (U), where ρ : U ⊂ Rn _{→ M is the aforementioned orientation-preserving diffeomorphism. In the}

case when ρ : Rn _{→ M is the Riemannian exponential map, the construction is still confined}

within a single normal neighhbourhood.

In the subsequent Chapter, we extend this framework to construct probability distributions on M supported beyond the normal neighbourhood. In particular, we extend the notion of locally inherited parametrized families of probability densities to mixture densities on totally bounded subsets of M.

CHAPTER 6

MIXTURE DENSITIES ON TOTALLY BOUNDED

SUBSETS OF RIEMANNIAN MANIFOLDS

In the previous Chapter, we discussed how parametrized family of probability densities over Riemannian manifolds M can be locally inherited from Rn_{via an orientation-preserving dif-}

feomorphism. In this chapter we extend the notion of locally inherited densities beyond the normal neighbourhoods. In particular, we discuss the geometry of mixture densities on totally bounded subsets V of Riemannian manifolds.

Given a totally bounded subset V of M, we can cover it by a finite number of metric balls (geodesic balls) where each metric ball is mapped to a subset of Rn via an orientation-

preserving diffeomorphism. This set of metric balls (together with the mapping to Rn) forms

an orientation-preserving open cover of V .

Suppose each element of the open cover of V is equipped with a local family of inherited probability distributions, a family of parametrized mixture densities LV on the entire V can

be constructed by “gluing" together the locally inherited densities in the form of a mixture. The statistical geometry of the family of mixture densities on V can thus be described by the product of the mixture component families and the mixture coefficients.

The chapter is outlined as follows:

1. In section 6.1, we describe the notion of orientation-preserving open cover on Rieman- nian manifolds M. We show that such open covers admits a refinement of geodesic balls. Thus totally bounded subsets V of M admits a finite orientation-preserving open cover

of (compact) geodesic balls.

2. In section 6.2, we describe finitely parametrized mixture distributions LV over totally

bounded subsets V on Riemannian manifolds M. These parametrized probability dens- ities extend the notion of locally inherited probability densities beyond a single normal neighbourhood/geodesic ball, while preserving the locally inherited dualistic statistical geometry of the mixture components.

From the section 6.2.2 onwards, we describe and derive the geometrical structure of the family of mixture densities LV. By viewing the closure of the simplex of mixture

coeffieients and family of mixture component densities as statistical manifolds, we show that LV is a smooth manifold under two condition. Furthermore, we show that a torsion-

free dualistic structure can be defined on LV by a “mixture" divergence D from the

mixture coefficient and mixture components.

In section 6.2.3, we show that LV is in fact a product Riemannian manifold of the sim-

plex of mixture coefficients and the families of mixture component statistical manifolds. Finally, we show that when both the simplex of mixture coefficients and mixture com- ponent statistical manifolds are dually flat, then LV is also dually flat with canonical

“mixture" divergence D.

6.1

Refinement of Orientation-Preserving Open Cover

We begin by describing formally the notion of orientation-preserving open cover:

Definition 6.1.1. Let (M, g) be a smooth n-dimensional Riemannian manifold, an orientation- preserving open coverof M is an at-most countable set of pairs EM := {(ρα, Uα)}_α∈ΛM

satisfying:

1. Uα ⊂ Rn, ρα : Uα → M are orientation preserving diffeomorphisms for each α ∈ ΛM

One notable example of an orientation-preserving open cover is given by geodesic balls: Using the notation of Chapter 2: for x ∈ M, let Bx := B(~0, inj(x)) ⊂ TxM denote the

geodesic ball of injectivity radius. Riemannian exponential map expx : Bx ⊂ TxM → M is an

orientation-preserving diffeomorphism within Bx, hence {(expx, Bx)}_x∈M is an orientation-

preserving open cover of M. In particular, for any positive jx ≤ inj(x), the set

n

exp_x, B(~0, jx)

o

x∈M

is also an orientation-preserving open cover of M.

Remark 6.1.1. It is important to note that the notion of orientation-preserving open cover is different from orientable atlas in the literature, as we do not require the transition maps between elements of the open cover to be compatible.

Given an orientation-preserving open cover over Riemannian manifold M, there exists a refinement of open cover by metric balls (geodesic balls) in M. This will be used to construct a finite orientation-preserving open cover of totally bounded subsets of M.

Lemma 6.1.1. Let M be a n-dimensional Riemannian manifold and an orientation-preserving

open cover of {(ρα, Uα)}α∈ΛM of M. There always exist refinement {(ρβ, Wβ)}β∈Λ0M of {(ρα, Uα)}α∈ΛM

satisfying:

1. {ρβ(Wβ)}_β∈Λ0

M covers M, and

2. ρβ(Wβ)are metric balls in M for all β ∈ Λ0M.

Proof. Given (orientation-preserving) open cover {(ρα, Uα)}_α∈ΛM of M. For each α, ∀xβα ∈

ρα(Uα)there exists normal neighbourhood N_xβ

α ⊂ ρα(Uα).

Since N_xβ

αis open for all x

β

α ∈ ρα(Uα), there exists _xβ

α > 0such that the metric ball centred

at xβ

αdenoted by Bxβα satisfies: Bxβα := B(x

β

α, xβα)⊂ Nxβα ⊂ ρα(Uα)⊂ M.

In other words, B_xβ

αis the metric ball centred at x

β

αin normal coordinates under the radial

distance function.

Since ραis a diffeomorphism for all α ∈ ΛM, this implies ρ−1α (Bxβα) =: Wxβα ⊂ Uαfor each

xβ α∈ ρα(Uα). Hence n ρα, W_xβ α o

β∈Λ0_M is the desired refinement of {(ρα

Observe that the proof did not use the fact that ραis orientation-preserving, as the trans-

ition map between elements of the open cover does not have to be orientation-preserving. Therefore it suffices to consider open cover {(ρα, Uα)}α∈ΛM of M such that ραare just diffeo-

morphisms with the following simple result:

Lemma 6.1.2. Let f : M → N be a local diffeomorphism between manifolds M, N. Then there exists local orientation-preserving diffeomorphism ˜f : M _{→ N.}

Proof. Since f : M → N is a local diffeomorphism, the pushforward f∗ : TpM → Tf (p)N is

a linear isomorphism for all x ∈ M. Since f∗ is a linear isomorphism, the determinant of the

matrix Df is non-zero: detDf 6= 0. If f is orientation-preserving, then there’s nothing left to prove. Hence we will now assume f is orientation reversing, in other words: detDf < 0.

Let (x1_{, . . . , x}n_{)denote local coordinates in M, let f denote coordinate representation of}

f, then we can write:

f x1(p), . . . , xn(p)
= f₁(x), . . . , f_n(x)
,

where x := (x1_{(p), . . . , x}n_(p)).

Choose a ∈ [1, . . . , n], and let ˜f : M _{→ N denote the diffeomorphism from M to N} defined by the following coordinate representation:

˜

f x1(p), . . . , xn(p)
= f₁(x), . . . , f_a+1(x), f_a(x), . . . , f_n(x)
.

In other words, we define ˜f by swap the ath _{and a + 1}th _{coordinates of f. The matrix}

representation of ˜f∗ in standard coordinates is thus given by:

where I0_{is the matrix given by:} I0 =          Ia−1 0 1 1 0 In−(a+1)          ,

where Ikis the identity matrix of the dimension k, the sub-matrix

   0 1 1 0    is located at the

(a, a)th to the (a + 1, a + 1)th position of I0, and the rest of the entries are all zero. Since detDf < 0, this means detD ˜f =detI0_{· detDf = −1 · detDf > 0.}

For the rest of the discussion we will consider the orientation-preserving open cover by metric ballsnρα, W_xβ α o β∈Λ0 M of Riemannian manifold M.