Full text of DOI:10.1112/S0024610701002332

f
C
J. London Math. Soc. (2) 64 (2001) 501–512
2001 London Mathematical Society
DOI: 10.1112/S0024610701002332
ON SIMPLEX SHAPE SPACES
HUILING LE and DENNIS BARDEN
Abstract
The right-invariant Riemannian metric on simplex shape spaces in fact makes them particular Riemannian
symmetric spaces of non-compact type. In the paper, the general properties of such symmetric spaces
are made explicit for simplex shape spaces. In particular, a global matrix coordinate representation is
suggested, with respect to which several geometric features, important for shape analysis, have simple
´
and easily computable expressions. As a typical application, it is shown how to locate the Frechet means
of a class of probability measures on the simplex shape spaces, a result analogous to that for Kendall’s
shape spaces.
1. Introduction
In [1] and [2], Bookstein proposed a method for the representation of the shapes of
´
labelled triangles as points in the Poincare half plane – a space of constant negative
curvature, with the collinear triangles situated on the infinite horizon formed by
the x-axis. This representation has a geometrically appealing property: the geodesic
distance between any two triangle shapes is equal to a multiple of the logarithm of
the strain ratio of the affine transformation mapping one triangle to the other. In [13],
Small extended the Bookstein model by representing the shapes of labelled simplexes
in Rn on a Riemannian manifold in such a way that the Riemannian distance
between two simplex shapes is invariant under simultaneous transformation of the
simplexes by a common affine transformation and is a function of the principal
strains of the affine transformation mapping one simplex onto the other. These
manifolds are quite distinct from those proposed by D. G. Kendall (cf. [5]) based
upon procrustean arguments, and are designed mainly for applications, such as the
biological ones, where it is reasonable to assume that there is some geometrical
interdependence among the vertices which prevents the simplex they span being
degenerate. In such cases, it would be more appropriate to use Bookstein’s model
than Kendall’s model to measure the difference between shapes, since, in the former,
the shapes of degenerate simplexes are infinitely far from each non-degenerate one.
In [11], using the theory of Riemannian submersions, Le and Small obtained an
explicit formula for the Riemannian metric on simplex shape space and, hence, also
for the Riemannian distance between simplex shapes. In the case of Kendall shape
spaces, much more geometric information is known (cf. [6]), such as a formula
for the geodesic between two given shapes and a formula for the full curvature
tensor at each shape giving, in particular, bounds on the sectional curvature. All of
this information has practical applications in the statistical analysis of Shape. For
example, the geodesics have been used to model change of shape over time (cf. [9]);
the knowledge of the curvature enables one to make a quantitative assessment of
the validity of the common approximation for concentrated data of working in a
Received 21 July 1999; revised 28 September 2000.
2000 Mathematics Subject Classification 60D05.

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huiling le and dennis barden
tangent space to the shape space and it also plays an important role in the study
´
of existence and location of Frechet mean shapes (cf. [8]). In this paper, we take
a different approach to that of [11] in order to obtain further explicit geometric
information for simplex shape spaces and we also illustrate its practical application.
In the next section, we describe our model of simplex shape space. We explain
how the requirement that the distance between simplex shapes is invariant under
simultaneous affine transformation implies that this metric be invariant under a
natural action of the special linear group on the shape space. If this invariant metric
is induced by a Riemannian metric, this further implies that our simplex shape space
is in fact an irreducible Riemannian globally symmetric space of non-compact type
(cf. [3, p. 518]). This means that it has a number of geometric properties which
are of significance in the study of simplex shapes. For example, the right-invariant
Riemannian metric that we require is unique up to a homothety and, hence, the
choice that we shall make must agree up to a global scalar multiple with those
described rather differently in [11] and [13]. In fact, it agrees with that in [11]
√
and is n/2 times that in [13]. We describe this and other geometric properties of
symmetric spaces that we shall use and also, since they are not always easy to locate
explicitly in the literature, give readers some idea of how they arise.
In Section 3 we use this basic information to obtain a choice of matrix coordinates
for the simplex shape space in terms of which the various geometric features have
particularly simple expressions. In particular, we obtain explicit expressions for the
induced distance function, the geodesics and the global isometric involutions. All of
these formulae are elementary and easily susceptible to computation.
This lays the foundation for practical results analogous to those mentioned above
for Kendall shape spaces and, in the final section of the paper, we give an example
´
of such an application: namely, the proof of the existence and uniqueness of Frechet
means, as well as their identification, for appropriate probability measures on simplex
shape spaces.
2. Simplex shape space as a symmetric space
Our model of simplex shape space starts from the observation that, since the
vertices of a simplex are ordered, or equivalently labelled, a non-degenerate simplex
in Rn can be represented by an n × (n + 1) matrix whose jth column is the location
of the jth vertex of the simplex. By moving the first vertex to the origin and then
omitting it, the quotient space of labelled non-degenerate simplexes in Rn modulo
translations is then identified with
GL(n) = {Y | Y is a (n × n)-matrix such that det(Y ) = 0}.
The absolute value of the determinant of a matrix in GL(n) is, up to a constant
multiple, equal to the volume of the corresponding simplex. Thus, by making
the natural choice of the volume as a measure of the ‘size’ of the simplex, the
quotient space of simplexes in Rn modulo translations, reflections and scaling may
be identified with
SL(n) = {Y ∈ GL(n) | det(Y ) = 1}.
Thus, since the shape of a simplex is also invariant under rotations, the space of sim-
plex shapes in Rn is homeomorphic with the quotient M(n) = SL(n)/SO(n) of SL(n)
by the left action of SO(n). In the following, we shall write π for the quotient map

on simplex shape spaces
503
from SL(n) to M(n) so that π(Y ) is the left coset SO(n)Y = {Y 0 ∈ SL(n) | Y 0 = TY
for some T ∈ SO(n)}. This is the element of M(n) representing the shape of all sim-
plexes that, after translation, reflection and re-scaling, are identified with Y ∈ SL(n).
Recall that the basic requirement in Bookstein’s and Small’s models is that
the distance between the shapes of two simplexes should depend only on the affine
transformation that takes one onto the other. Since shape is unaffected by translation,
reflection and scalar multiplication, this means that for each T in SL(n), regarded
as the linear part of an affine transformation that preserves volume and orientation,
we require the distance d(π(Y ), π(TY )) between the shapes π(Y ) and π(TY ) to be
independent of the choice of Y , and so equal to the distance d(π(I), π(T)) between
the shapes of two simplexes that, after standardising with respect to translation,
reflection and scaling, are identified with I and T respectively. Then, varying Y will
show that we need this metric to be invariant under the right action of SL(n) on
M(n).
In correlating the results we claim with those in the literature, two technical points
need to be observed. Both are consequences of the facts that we represent the shape
of a simplex in Rn by a left coset of SO(n) in SL(n) and that we are seeking a
right-invariant metric on M(n). Firstly, in order to have a group action of SL(n) on
itself, for T ∈ SL(n), we should define the corresponding right multiplication ρ˜T by
ρ˜T : Y → Y T−1. Similarly, for M(n), we define ρT : π(Y ) → π(Y T−1). Throughout
this paper, the term right invariance will refer to the appropriate one of these right
actions. Secondly, it will be convenient to identify the tangent space to SL(n) at the
identity I with the Lie algebra of right-invariant vector fields with the Lie product
corresponding to the Poisson bracket of vector fields. These are the opposites of the
conventions usually adopted, however all the proofs remain valid mutatis mutandis.
If we further require that the metric on M(n), invariant under the right action
of SL(n), be induced by a Riemannian metric, as was indeed the case in [11] and
[13], then M(n) becomes a Riemannian symmetric space of non-compact type. Such
spaces are classified (cf. [3] and [14]) and their main geometric properties are well
understood. In the remainder of this section, we explain how two basic properties
of symmetric spaces, the essentially unique right-invariant metric and the resulting
geodesics, arise in our specific context.
The group GL(n) has a natural differential structure as an open subset of Euclidean
space and SL(n) is a smooth submanifold of GL(n) for which the right translations
ρ˜T are diffeomorphisms. If we give M(n) the quotient topology, then the induced right
translations ρT will be homeomorphisms. However, M(n) is an example of a coset
manifold, and general theory (cf. [12, 307ff]) tells us that it has a unique differential
structure that makes the natural projection, or quotient map, π : SL(n) → M(n) a
submersion: that is, at each point of SL(n), the derivative of π has rank equal to
the dimension of M(n). In terms of this structure, the homeomorphisms ρT form
a group of diffeomorphisms which acts transitively on M(n). With respect to any
right-invariant metric, these diffeomorphisms will necessarily be isometries which
will be important for our calculations and applications.
Any right-invariant vector field, in the Lie algebra of SL(n), is of course determined
by its value in the tangent space at the identity I, its value at other points being
obtained from that by right translation. The tangent space to CL(n) at I is the space
of all n × n real-valued matrices, and the subspace tangent to SL(n), the kernel of
the derivative of the determinant map there, is the subspace of matrices of trace
zero. Thus, we may further identify the Lie algebra sl(n) of SL(n) with this subspace,

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huiling le and dennis barden
which we shall still denote by sl(n), and, similarly, the Lie algebra of SO(n) may be
identified with the subalgebra
so(n) = {X ∈ sl(n) | Xt = −X}.
In terms of these matrices, the Lie product is given by [X1, X2] = X1X2 − X2X1.
For any element T of SL(n), the conjugation isomorphism CT : Y → TY T−1
determines a linear isomorphism Ad(T) = d(CT )I of the tangent space sl(n) at the
identity. Since, in terms of the standard coordinates on GL(n), both left and right
multiplication are linear, it follows that Ad(T) is the linear map X → TXT−1 of
1
sl(n) into itself. Using the trivial decomposition X = {(X − Xt) + (X + Xt)} for any
2
X ∈ sl(n), we have
sl(n) = so(n) ⊕ p(n),
where p(n) is the subspace {X ∈ sl(n) | Xt = X}, which is clearly Ad(SO(n))-
invariant. This means that M(n) will be what is termed a reductive coset manifold.
When equipped with an SL(n)-invariant metric, it becomes a reductive homogeneous
space and the possible SL(n)-invariant metrics may be determined as in the following
lemma, which is a special case of a result which holds for all reductive coset manifolds
(cf. [12]).
Lemma 1. There is a bijection between Ad(SO(n))-invariant scalar products on
p(n) and SL(n)-invariant Riemannian metrics on M(n), determined by the requirement
that dπI : p(n) → Tπ(I)(M(n)) be an isometry.
One particular choice of Ad(SO(n))-invariant inner product on p(n) is hX1, X2i=
tr(X1X2t) for which Ad(SO(n))-invariance follows from the fact that, for R in SO(n),
Ad(R) acts by conjugation. This inner product is in fact a constant multiple of
the restriction to p(n) of the Killing form on sl(n) and is essentially unique. This
uniqueness arises from the fact that the decomposition of sl(n) as so(n)⊕p(n), together
with an Ad(SO(n))-invariant inner product on p(n) and the involution of sl(n) that
is +id on so(n) and −id on p(n), satisfies a couple of other technical conditions that
make it an irreducible orthogonal involutive Lie algebra. The classification of such
algebras implicitly involves the following lemma. See, for example, [14], where the
result we require falls under class 3 of Theorem 8.2.9. However, given our explicit
knowledge of Ad(SO(n))-action on p(n), it is possible, by choosing an obvious basis
of p(n) and generators of SO(n), to give a direct elementary proof for our case.
Lemma 2. The only Ad(SO(n))-invariant inner products on p(n) are positive
scalar multiples of the restriction of the Killing form.
From Lemma 2, together with Lemma 1, we see that the SL(n)-invariant Rieman-
nian metric on M(n) is unique up to homothety, that is, multiplication by a constant.
The choice we make is that corresponding to the multiple 1/2n of the Killing form.
We now use it to characterise the geodesics on simplex shape space and, in the next
section, we shall use it to obtain a simple formula for the distance between two
shapes as well as for the geodesics themselves.
In order to describe the geodesics, we consider the Lie exponential map from sl(n)
to SL(n) given, in terms of matrices, by exp : X → n=0 Xn/n!. This series converges
P∞
and so exp (X) is defined for all X. Since exp (Y XY −1) = Y exp (X)Y −1, conjugation

on simplex shape spaces
505
of X to the Jordan normal form shows that det(exp (X)) = exp (tr(X)). Thus,
exp (X) is indeed in SL(n) for all X in sl(n). Moreover, when X1 and X2 commute,
exp (X1 + X2) = exp (X1) exp (X2) so that, for real s and t, exp (sX) exp (tX) =
exp ((s+t)X). This means that αX : t → exp (tX) is a 1-parameter subgroup of SL(n).
It is also the integral curve starting at I of the right-invariant vector field X. On the
other hand, since M(n) is a Riemannian manifold, for each v ∈ Tπ(Y )(M(n)), there is
a geodesic γv, a curve in M(n) starting at π(Y ) determined by the requirement that
γv0 (0) = v and Dv(t)v(t) = 0, where D is the Levi–Civita connection determined by
´
the Riemannian metric and v(t) = γv0 (t) is the tangent vector field along γv. We then
have the following, which is, again, a special case of a more general result. See, for
example, [12], Proposition 3.1 on p. 317.
Lemma 3. For X ∈ p(n), the geodesic γdπ (X), with respect to the above SL(n)-
I
invariant metric on M(n), is the image π ◦ αX of the 1-parameter subgroup of SL(n)
generated by X.
Since exp (tX) is defined for all t, this lemma confirms the completeness of M(n),
another property common to all Riemannian symmetric spaces.
3. Some explicit geometry of simplex shape spaces
The results of the previous section enable us to give an elementary account in our
context of the features of symmetric spaces which interest us. Firstly, we describe
a convenient set of matrix coordinates for simplex shape space, in terms of which
we then give explicit formulae for the distances and geodesics between shapes and
for the global isometric involutions which characterise it as a symmetric space. We
close with some comments on the sectional curvature of our spaces.
Our coordinates will lie in the submanifold
P(n) = {P ∈ SL(n) | Pt = P and P is positive definite},
a relatively open subset of a linear subspace, of SL(n). From the unique polar
factorisation of Y ∈ SL(n) as Y = RY PY , where RY ∈ SO(n) and PY ∈ P(n), it
follows that PY is the unique element of P(n) in the SO(n)-coset π(Y ). Thus, the
restriction of the quotient map π to P(n), mapping PY to π(PY ) = π(Y ), is bijective
and is easily checked to be a diffeomorphism.
This diffeomorphism π¯ of P(n) onto M(n) has particularly convenient properties,
which we now elucidate. Note first that exp maps the subspace p(n) into the
submanifold P(n). In fact, if P is in P(n), then all the eigenvalues of P are positive,
so that P may be written as P = TΛTt with T orthogonal and Λ a diagonal
matrix with positive entries and det(Λ) = 1. Thus, Λ = exp (X) with X ∈ p(n)
and so P = exp (TXTt) with TXTt also in p(n). This shows that the restricted
map exp : p(n) −→ P(n) is surjective and it may similarly be seen to be injective
and so a diffeomorphism between the two spaces. Thus, the composite π¯ ◦ exp is a
diffeomorphism of p(n) onto M(n). However, the Riemannian exponential Exp of
M(n) at π(I) is given by
Exp(tdπI (X)) = γtdπ (X)(1) = γdπ (X)(t),
I
I
so Lemma 3 says that π¯ ◦ exp = Exp ◦ dπ¯I . Since, by choice, in order to apply
Lemma 1, dπI is an isometry of p(n) with Tπ(I)M((n)), it follows that Exp is also
a diffeomorphism between Tπ(I)(M(n)) and M(n) and, by homogeneity, a similar

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huiling le and dennis barden
property holds at all points of M(n). This fact will be used in Section 4 when
applying the result of Lemma 4 to simplex shape spaces.
In view of the above diffeomorphism π¯, we shall, when it is convenient, identify
M(n) with P(n) and so denote π(Y ) just by PY or P. This gives another system of
local coordinates on the simplex shape space, which is different from those arising
from the upper-triangular representation given in [13] or from the representation
based on the singular-values decomposition used in [11]. Our present coordinates
have the advantage that by Lemma 3, when M(n) has the SL(n)-invariant metric
induced by the inner product hX1, X2i = tr(X1X2t) on p(n), the Riemannian geodesics
on M(n) starting at π(I) correspond in P(n) to the 1-parameter groups
γX(t) = exp (tX),
t ∈ R,
(1)
for X ∈ p(n). Since exp : p(n) → P(n) is bijective, this shows in particular that the
Riemannian distance from P = exp(X) to I, corresponding to that in M(n), is given
by d(I, P)2 = tr(XXt). However, if λ1, . . . , λn are the eigenvalues of P, then the
eigenvalues of X are log λ1, . . . , log λn, and so we have
n
X
d(I, P)2 = tr(XXt) =
(log λi)2.
i=l
Q
n
Note that P ∈ P(n) implies that all the λi are positive and i=1 λi = 1. The right
invariance of the metric on M(n), together with the fact that the eigenvalues of the
positive definite symmetric factor in the polar decomposition of Y ∈ SL(n) are the
positive square roots of the eigenvalues of Y Y t, gives us the following result.
Proposition 1. Let the general points P1 and P2 of P(n) represent the
simplex shapes π¯(P1) and π¯(P2) in M(n). Then, the squared distance d(π¯(P1), π¯(P2))2
between these two shapes, induced by the right-invariant Riemannian metric on M(n),
P
n
is
i=1(log λi)2, where the λi are the positive square roots of the eigenvalues of
P1P2−1(P1P2−1)t.
˜
Now, to correlate this distance function with that given in [11], for any Y ∈ GL(n),
we define
˜
Y
˜
Y = diag{1, . . . , 1, sign(det(Y ))}
.
1/n
˜
(| det(Y |)
so that Y is in SL(n). If ˜γ1 > . . . > ˜γn > 0 and γ1 > . . . > γn > 0 are the
˜
singular-values of Y and Y respectively, then we have
˜γi
γI =
,


1/n
n
Y


˜γj
j=1
and the point PY in P(n) that corresponds to π(Y ) ∈ M(n) has eigenvalues γi,
i = 1, . . . , n. Thus, the squared Riemannian distance from the simplex shape π(Y ) to
the shape π(I) is
(
)
2
n
n
n
X
which is equivalent to the formula given in [11].
X
X
1
d(π(I), π(Y ))2 =
(log γi)2 =
log ˜γi −
log ˜γj
,
n
i=1
i=1
j=1

on simplex shape spaces
507
We have already given the simple formula (1) for geodesics on M(n) starting at
π(I) in terms of their coordinates in P(n). For geodesics starting from other points,
a little care is needed. Firstly, we note that, if we define an inner product on sl(n) by
the same formula, hX1, X2i = tr(X1X2t), that we are using on p(n) and then, regarding
sl(n) as the tangent space to SL(n) at the identity, extend this to a right-invariant
Riemannian metric on SL(n) using the derivatives of right translations, then the
projection π of SL(n) onto M(n) will be a Riemannian submersion with respect to
the SL(n)-invariant metric on M(n). This follows from the fact that so(n) and p(n)
are orthogonal subspaces of sl(n) so that, as so(n) is the tangent space to the fibre
SO(n), p(n) is the horizontal subspace at I ∈ SL(n). Then, since right translations
preserve fibres and are isometric, HY = dρ˜Y −1 (p(n)) is the horizontal subspace
at Y ∈ SL(n). However, the projection dπY : HY → Tπ(Y )(M(n)) is the composite
dρY −1 ◦ dπ1 ◦ dρ˜Y of maps all of which are isometries by the definitions.
Given points P1 and P2 in P(n), representing the simplex shapes π¯(P1) and
π¯(P2) in M(n), let P2P1−1 = RPP P−1 , where R ∈ SO(n) and PP P−1 ∈ P(n). If
2
2
1
1
P2P1−1
X ∈ T (P(n)) = p(n) is such that P
= exp (X), then P(t) = exp (tX) is a,
I
necessary horizontal, geodesic in P(n) from I to PP P−1 and so
2
1
ρ˜P−1 (P(t)) = P(t)P1 = exp (tX)P1
1
is a horizontal geodesic in SL(n) from P1 to RtP2. Thus, π(ρ˜ (P(t))) is the geodesic
in M(n) from π¯(P1) to π¯(P2). Since
P1−1
d
(ρ˜P−1 (P(t)))|t=0 = XP1,
1
dt
we have
g
ρ˜P−1 (P(t)) = exp (tX)P1 = Exp (tXP1),
P1
1
g
where ExpP is the Riemannian exponential map on SL(n) at P1. Hence, the required
1
geodesic π(ρ˜P−1 (P(t)) is Expπ¯(P )(tdπ(XP1)), where, as before, Exp
nian exponential map on M(n) at π¯(P1). Thus, we have established the following
is the Rieman-
π¯(P1)
1
1
result.
Proposition 2. Writing the general tangent vector V at π¯(P1) ∈ M(n), where P1 ∈
P(n), as the image V = dπ(XP1) of the horizontal vector XP1 at P1 in SL(n), where
X
∈
p(n), the geodesic Expπ¯(P )(tV) in M(n) is given by π¯(P(t)), where
I
exp (tX)P1 = R(t)P(t) with R(t) ∈ SO(n) and P(t) ∈ P(n). This geodesic joins π¯(P1)
and π¯(P2) when X is chosen such that P2P1−1 = R exp (X) with R in SO(n).
The defining property of a Riemannian symmetric space is the existence at each
point π(Y ) of an isometric involution fixing π(Y ) whose derivation is −id on the
tangent space at π(Y ) (cf. [4] and [12]). These involutions will play an important role
in Theorem 1 below. They are easily identified in our context: if σ = exp ◦ ι ◦ exp−1
is the ‘polar’ diffeomorphism that the involution ι = −id of p(n) induces on P(n),
then the involution ζ = π¯ ◦ σ ◦ π¯−1 that σ induces on M(n) via π¯ is the required
isometric involution at π(I). That dζπ(I) = −id follows immediately from the fact
that d exp (0) = id and dπ¯I is an isomorphism. Writing P = exp (X) for X in p(n),
we see that σ(P) = P−1 = (Pt)−1, since P ∈ P(n), so that σ is the restriction of the
involutive automorphism σ˜ : Y → (Y −1)t of SL(n). From the polar decomposition

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huiling le and dennis barden
Y = RY PY of Y , we see that
π ◦ σ˜(Y ) = π(RY (PY )−1) = π¯ ◦ σ(PY ) = ζ ◦ π(Y ).
We then find that, for all P ∈ P(n), ρπ¯(P−1 ◦ζ = ζ ◦ρπ¯(P) on M(n). This result follows
)
from the bijectivity of π¯ and the fact that, for all P0 in P(n),
ζ ◦ ρπ¯(P)(π¯(P0)) = ζ ◦ π(P0P−1) = π ◦ σ˜(P0P−1
)
= π((P0)−1P) = ρπ¯(P−1) ◦ π¯((P0)−1) = ρπ¯(P−1) ◦ ζ(π¯(P0)).
Thus, for any v ∈ Tπ¯(P)(M(n)), writing v0 = dρπ¯(P)(v) ∈ Tπ(I)(M(n)), we get
hdζ(v), dζ(v)i = hdζ ◦ dρπ¯(P−1 (v0), dζ ◦ dρπ¯(P−1)(v0)i
)
= hdρπ¯(P) ◦ dζ(v0), dρπ¯(P) ◦ dζ(v0)i
= hdζ(v0), dζ(v0)i = h−v0, −v0i = hv, vi.
Thus, ζ is the induced global isometric involution of M(n) fixing π(I).
In terms of our chosen coordinates, ζ is just represented by the polar
diffeomorphism σ of P(n). As right translations are isometries and the composite
ρπ¯(P−1 ◦ζ ◦ρπ¯(P ) has derivative at π¯(P0) equal to −id, it must be the global isometric
)
0
0
involution at π¯(P0). Since the coordinate representation of the right translation ρπ¯(P )
0
P1P0−1
is P1 → P
, we have the following result.
Proposition 3. The global isometric involution M(n) at the shape π¯(P0) has coor-
dinate representation
ꢀ
ꢁ−1
P1 → P
,
(2)
PP1
P0
−1
P
0
where PY is the positive definite symmetric factor in the polar decomposition RY PY
of Y .
Finally in this section we briefly mention the sectional curvature of simplex shape
spaces, referring to [3], for example, for details. Since it is a symmetric space of
non-compact type of rank n − 1, we know that M(n) has non-positive sectional
curvature with a totally geodesic flat submanifold of dimension n − 1 through each
point. There is also, through each point, a totally geodesic submanifold of maximally
negative curvature. When calculated with respect to the metric induced by the Killing
form, this minimal value is −(2 − 3/n). Since the metric we are using is induced by
1/2n times the Killing form, it follows that, for our choice, the minimal curvature is
6 − 4n. The curvature will influence the range over which it is reasonable to make
linear approximations, projecting for example onto the tangent space at a convenient
point. However, for any desired level of accuracy, the different metrics will give the
√
same result since any two given shape points will be deemed to be 2n times further
apart with respect to the ‘Killing metric’ than with respect to ours; the domains on
which the approximation is valid will have a different measure, but will contain the
same shape points.
4. Fr´echet means of certain probability measures on simplex shape spaces
We now use the geometric properties of the simplex shape space consequent on its
structure as a symmetric space, when it is equipped with its unique SL(n)-invariant

on simplex shape spaces
509
Riemannian metric, to study a probabilistic issue that arises in the statistical analysis
of shape.
´
Recall that a Frechet mean, a generalisation of the mean or the expectation of a
probability distribution on a Euclidean space to that of a probability measure µ on
a general metric space (M, dist), is defined to be any point that achieves the global
minimum of the function
Z
dist(p, q)2dµ(q).
1
2
F(p) =
M
´
Frechet means of probability measures on shape spaces have been used in studying
the shape of the means, although such Frechet means are in general not unique (cf.
´
[6] and [7]). Nevertheless, when it is equipped with its SL(n)-invariant Riemannian
structure, M(n) has properties from which it will follow that any probability measure
´
on it has unique Frechet mean with respect to this Riemannian metric and that there
is a simple criterion to identify this mean. The properties of M(n) that we require
are its non-positive curvature and the fact that the Riemannian exponential map
at each point is a diffeomorphism of the tangent space there onto M(n). A point
with the latter property is called a pole. As for M(n), though not necessarily by
homogeneity, it suffices that one point of a Riemannian manifold be a pole for all
´
its points to be poles. The following, then, are the implications for Frechet means
of these two properties of a manifold.
Lemma 4. Let M be a Riemannian manifold of non-positive sectional curvature
for which every point is a pole. Then, p ∈ M is a Fr´echet mean, with respect to the
Riemannian distance d on M, of a given probability measure µ on M if and only if
´echet mean of the induced probability measure µ ◦ Exp on T (M)
0 ∈ T (M) is a Fr
p
p
p
˜
with respect to the distance d on T (M) determined by the Riemannian metric gp at
p
p. Furthermore, such Fr´echet means are unique.
´
Proof. A Frechet mean of the induced probability measure on T (M) with respect
p
˜
to d is a global minimum of the function
Z
2
p
kX − X0k dµ(Expp(X0)), X ∈ T (M),
1
2
˜
Fp(X) =
p
T (M)
p
where k · kp is the norm determined by the inner product gp.
˜
Firstly, since Exp is a diffeomorphism between T (M) and M and, since d(0, X) =
p
p
d(p, Exp (X)) for all X in T (M), we have
p
p
˜
Fp(0) = F(p)
(3)
and, in particular, one is defined if and only if the other is. Secondly, it fol-
lows from Gauss’ lemma together with the Rauch comparison theorem (cf. [4])
that, since M has non-positive sectional curvature, Expp is distance-increasing:
˜
d(Exp (X1), Exp (X2)) > d(X1, X2) for all X1, X2 ∈ T (M). This implies that
p
p
p
˜
Fp(X) 6 F(Exp (X)), ∀ X ∈ T (M).
(4)
p
p
´
Thus, it follows from (3) and (4) that, if 0 is a Frechet mean of µ ◦ Exp on T (M)
p
p
˜
with respect to the distance d, then p is a Frechet mean of µ on M with respect to
´
the Riemannian distance d.
On the other hand, the global minimum of Fp can only occur at a critical point
˜

510
huiling le and dennis barden
R
0
˜
ˆ
and, since T (M) dµ(Expp(X)) = 1, we see that Fp(X) = 0 if and only if
p
Z
ˆ
X =
Xdµ(Expp(X)).
(5)
Tp(M)
00
˜
We also find that Fp (X) = gp so that this unique critical point is a local, and hence
global, minimum. However, since gradp(d(p, q)2) = −2Expp−1(q), we have
Z
Thus, when p is a critical point, in particular a global minimum, of F, then the
Z
gradF(p) = − Exp−p 1(q)dµ(q) = −
Xdµ(Expp(X)) = −X.
ˆ
M
T (M)
p
˜
unique global minimum of Fp occurs at the origin:
˜
Fp(0) < Fp(X),
˜
∀ X ∈ T (M)\{0}.
(6)
p
Then, (3), (4) and (6) imply that p is the unique global minimum of F.
q
ˆ
´
ˆ
It is clear from (5) that X is the Frechet mean of µ ◦ Expp on T (M) with
p
˜
respect to the distance d if and only if X is the Euclidean mean of µ ◦ Expp on
´
T (M). Note also that this lemma gives us a context in which Frechet means are
p
unique and, together with our proof of it, two further sufficient conditions for such
˜
´
a Frechet mean to exist: it will exist if, for any point p, the corresponding Fp
has its unique global minimum at the origin or if, on any domain on which the
integral defining it converges, F has any critical point p and, in each case, that
´
point p will be the unique Frechet mean. However, since it is usually difficult to
express the Riemannian exponential map explicitly, their identification still remains
a hard problem in general. On the other hand, in the case of simplex shape space,
our detailed knowledge of the geometry enables us to make further progress in
that context. Identifying M(n) with P(n) and T (P(n)) with the horizontal subspace
P
˜
H (SL(n)), the function FP defined in the proof of Lemma 4 can be expressed as
P
Z
0
2
0
1
2
˜
Fp(XP) =
g
kXP − X PkP dµ ◦ ExpP (X P)
HP
HP
p(n)
Z
Z
2
1
2
=
=
kXP − X0PkP dµ(exp (X0)P)
2
kX − X0kI dµ(exp (X0)P).
1
2
ˆ
Let Xp be the Euclidean mean of µ˜p on p(n) defined by
dµ˜P (X0) = dµ(exp (X0)P),
ˆ
which is clearly computable once µ is given. Then, XP P is the Euclidean mean of
g
´
µ ◦ ExpP on HP . Thus, the criterion of Lemma 4 for π¯(P) to be a Frechet mean of
µ is effectively calculatable, at least for simple µ.
We can go further than this to obtain an implementable algorithm for locating
´
´
the Frechet mean of µ, given that it exists. Suppose that π¯(P) is not the Frechet
ˆ
mean of µ, so that Xp = 0. Then,
π¯(P)(t) = F(Expπ¯(P)(tdπ(XP P)))
is a convex function, which is again directly computable using Proposition 2. From
ˆ
G

on simplex shape spaces
511
the proof of Lemma 4, we see that Gπ¯(p) is decreasing at zero. Thus, there is a unique
t0 in (0, l] such that Gπ¯(P)(t0) 6 Gπ¯(P)(t) for all t in [0, 1]. In particular,
ˆ
F(π¯(P)) = Gπ¯(P)(0) > Gπ¯(P)(t0) = F(Expπ¯(P)(t0dπ(XP P))).
Repeating the calculation at π¯(P0) = Expπ¯(P)(t0dπ(XpP)), we obtain a sequence of
ˆ
points at which F takes a strictly decreasing sequence of values. Since these values
are bounded below by the assumed global minimum value of F, they must converge
ˆ
and the above argument implies that they can only do so at a point π¯(P) such that
ˆ
´
XPˆ = 0, that is, by Lemma 4, they can only do so at the Frechet mean.
In certain cases, making use of the global isometric involutions of M(n), we can
´
also locate the Frechet mean immediately without any need for such an approxi-
mating algorithm, as in the following result which is analogous to one obtained for
Kendall shape spaces (cf. [7]) and is a generalisation of a result in [10].
Theorem 1. Suppose that M(n) is equipped with the unique SL(n)-invariant Rie-
mannian metric and that the Radon–Nikody´m derivative f, with respect to the volume
element ν, of a given probability measure µ on the simplex shape space is a function of
distance to a fixed point π(Y0). If any Fr´echet mean, with respect to the Riemannian
distance, of µ exists, then it is π(Y0) and is unique.
´
Proof. By Lemma 4, we know the Frechet mean of µ is unique. Suppose that
π(Y1) = π(Y0) is the Frechet mean of µ. Then, if ζ is the global isometric involution
´
of M(n) centred on π(Y0) and if ζ(π(Y1)) = π(Y2), we have π(Y1) = π(Y2), again since
M(n) is complete and has non-negative curvature. Then, since ζ(π(Y0)) = π(Y0),
Z
F(π(Y2)) =
d(π(Y2), π(Y ))2f(d(π(Y0), π(Y )))dν(π(Y ))
M(n)
M(n)
M(n)
Z
=
=
d(π(Y1), ζ−1(π(Y )))2f(d(π(Y0), ζ−1(π(Y ))))dν(ζ−1(π(Y )))
d(π(Y1), π(Y ))2f(d(π(Y0), π(Y )))dν(π(Y )) = F(π(Y1)).
Z
This contradicts the fact that F(π(Y2)) > F(π(Y1)).
q
Note that this proof only involves the single involution ζ and so we actually only
need to require that µ be invariant with respect to ζ. Such probability measures
are easily identifiable, since the explicit coordinate representation (2) that we have
given for this involution only requires the computation of the polar decomposition
of a matrix which is readily available in mathematical computer packages. Note
´
also that, if the Radon–Nikodym derivative f in the statement of Theorem 1 has
compact support, then the Frechet mean of the corresponding probability measure
´
does exist.
References
1. F. L. Bookstein, ‘Size and shape spaces for landmark data in two dimensions’ Statist. Sci. 1 (1986)
181–242 (with discussion).
2. F. L. Bookstein, Morphometric tools for landmark data: geometry and biology (Cambridge University
Press, Cambridge, 1991).
3. S. Helgason, Differential geometry, Lie groups and symmetric spaces (Academic Press, New York,
1978).

512
on simplex shape spaces
4. J. Jost, Riemannian geometry and geometric analysis (Springer, Berlin, 1998).
5. D. G. Kendall, ‘Shape manifolds, procrustean metrics, and complex projective spaces,’ Bull. London
Math. Soc. 16 (1984) 81–121.
6. D. G. Kendall, D. Barden, T. K. Carne, and H. Le, Shape and shape theory (John Wiley, Chichester,
1999).
7. H. Le, ‘On consistency of procrustean mean shapes’, Adv. Appl. Prob. 30 (1998) 53–63.
´
8. H. Le, ‘Locating Frechet means with application to shape spaces,’ Adv. Appl. Prob. (2001).
9. H. Le and A. Kume, ‘Detection of shape changes in biological features,’ J. Microscopy 200 (2000)
140–147.
10. H. Le and A. Kume, ‘The Frechet mean shape and the shape of the means,’ Adv. Appl. Prob. 32
´
(2000) 101–113.
11. H. Le and C. G. Small, ‘Multidimensional scaling of simplex shapes,’ Pattern Recognition 32 (1999)
1601–1613.
12. B. O’Neill, Semi-Riemannian geometry (Academic Press, Orlando, 1983).
13. C. G. Small, The statistical theory of shape (Springer, New York, 1996).
14. J. A. Wolf, Spaces of constant curvature (McGraw–Hill, New York, 1967).
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