Quasi-arithmetic centers, quasi-arithmetic
mixtures, and the Jensen-Shannon∇-divergences

1[0000−0001−5728−0726]

Frank Nielsen

Sony Computer Science Laboratories Inc, Tokyo, Japan.

We rst explain how the information geometry of Bregman
manifolds brings a natural generalization of scalar quasi-arithmetic meansthat we term quasi-arithmetic centers. We study the invariance andequivariance properties of quasi-arithmetic centers from the viewpointof the Fenchel-Young canonical divergences. Second, we consider statistical quasi-arithmetic mixtures and dene generalizations of the JensenShannon divergence according to geodesics induced by ane connections.
Abstract.

Legendre-type function · quasi-arithmetic means · co-monotonicity
information geometry · statistical mixtures · Jensen-Shannon divergence.
Keywords:·

1

Introductiondi wi = 1} ⊂ R denotes the closedn(n − 1)-dimensional standard simplex sitting in R , ∂ be the set boundary op◦
erator, and ∆n−1 = ∆n−1 \∂∆n−1 the open standard simplex. Weighted quasiarithmetic means [12] (QAMs) generalize the ordinary weighted arithmetic mean
Let ∆n−1= {(w1 , . . . , wn ) : wi ≥ 0,

A(x1 , . . . , xn ; w) =PPi wi xi as follows:Denition 1 (Weighted quasi-arithmetic mean (1930's)). Let f : I ⊂R → R be a strictly monotone and dierentiable real-valued function. The
weighted quasi-arithmetic mean (QAM) Mf (x1 , . . . , xn ; w) between n scalars
x1 , . . . , xn ∈ I ⊂ R with respect to a normalized weight vector w ∈ ∆n−1 , isdened by!
Mf (x1 , . . . , xn ; w) := f −1nX

wi f (xi ) .

i=1

Mf (x1 , . . . , xn ) := Mf (x1 , . . . , xn ; n1 , . . . , n1 ), andMf,α (x, y) := Mf (x, y; α, 1 − α) for α ∈ [0, 1], the weighted bivariate QAM.
Let

us

write

for

short

A QAM satises the in-betweenness property:

min{x1 , . . . , xn } ≤ Mf (x1 , . . . , xn ; w) ≤ max{x1 , . . . , xn },and we have [16] Mg (x, y) = Mf (x, y) if and only if g(t) = λf (t)+c for λ ∈ R\{0}and c ∈ R. The power means Mp (x, y) := Mfp (x, y) are obtained for the following

2Frank Nielsen

continuous family of QAM generators indexed by p ∈ R:

 tp −1
fp (t) =, p ∈ R\{0},,log(t), p = 0.
pfp−1 (t) =1(1 + tp) p , p ∈ R\{0},,exp(t),
p = 0.

Special cases of the power means are the harmonic mean (H = M−1 ), the geometric mean (G = M0 ), the arithmetic mean (A = M1 ), and the quadratic meanalso called root mean square (Q = M2 ). A QAM is said positively homogeneous
if and only if Mf (λx, λy) = λ Mf (x, y) for all λ > 0. The power means Mp are
the only positively homogeneous QAMs [12].
In Section 2, we dene a generalization of quasi-arithmetic means called
quasi-arithmetic centers (Denition 3) induced by a Legendre-type function. We
show that the gradient maps of convex conjugate functions are co-monotone(Proposition 1). We then study their invariance and equivariance properties(Proposition 2). In Section 4, we dene quasi-arithmetic mixtures (Denition 4),
show their connections to geodesics, and dene a generalization of the JensenShannon divergence with respect to ane connections (Denition 5).

2

Quasi-arithmetic centers and information geometry

2.1 Quasi-arithmetic centers
To generalize scalar QAMs to other non-scalar types such as vectors or matrices,
we face two diculties:
1. we need to ensure that the generator G : X → R admits a global inverse

−1

G

1

, and

2. we would like the smooth function G to bear a generalization of monotonicity
of univariate functions.
We consider a well-behaved class F of non-scalar functions G (i.e., vector or−1

matrix functions) which admits global inverse functions G

belonging to thesame class F : Namely, we consider the gradient maps of Legendre-type functions
where Legendre-type functions are dened as follows:Denition 2 (Legendre type function [24]). (Θ, F ) is of Legendre type if
the function F : Θ ⊂ X → R is strictly convex and dierentiable with Θ 6= ∅ anopen convex set anddF (λθ + (1 − λ)θ̄) = −∞,λ→0 dλlim∀θ ∈ Θ, ∀θ̄ ∈ ∂Θ.

Legendre-type functions F (Θ) admits a convex conjugate F∗(1)(η) of Legendre

type via the Legendre transform (Theorem 1 [24]):F ∗ (η) = ∇F −1 (η), η − F (∇F −1 (η)),
1

The inverse function theorem [10, 11] in multivariable calculus states only the local
existence of an inverse continuously dierentiable function G−1 for a multivariate
function G provided that the Jacobian matrix of G is not singularJensen-Shannon ∇-divergences3hθ, ηi denotes the inner product in X (e.g., Euclidean inner producthθ, ηi = θ> η for X = Rd , the Hilbert-Schmidt inner product hA, Bi := tr(AB > )where tr(·) denotes the matrix trace for X = Matd,d (R), etc.), and η ∈ H with H∗−1
the image of the gradient map ∇F : Θ → H . Moreover, we have ∇F = (∇F )∗ −1and ∇F = (∇F ), i.e., gradient maps of conjugate functions are reciprocal towhereeach others.
The gradient of a strictly convex function of Legendre type exhibit a generalization of the notion of monotonicity of univariate functions: A function

G : X → R is said strictly increasing co-monotone if∀θ1 , θ2 ∈ X, θ1 6= θ2 ,hθ1 − θ2 , G(θ1 ) − G(θ2 )i > 0.

and strictly decreasing co-monotone if −G is strictly increasing co-monotone.

Proposition 1 (Gradient co-monotonicity [25]). The gradient functions∇F (θ) and ∇F ∗ (η) of the Legendre-type convex conjugates F and F ∗ in F arestrictly increasing co-monotone functions.
Proof. We have to prove thathθ2 − θ1 , ∇F (θ2 ) − ∇F (θ1 )i > 0,∗∗hη2 − η1 , ∇F (η2 ) − ∇F (η1 )i > 0,

∀θ1 6= θ2 ∈ Θ(2)∀η1 6= η2 ∈ H(3)The inequalities follow by interpreting the terms of the left-hand-side of Eq. 2
and Eq. 3 as Jereys-symmetrization [17] of the dual Bregman divergences [9]BF and BF ∗ :BF (θ1 : θ2 ) = F (θ1 ) − F (θ2 ) − hθ1 − θ2 , ∇F (θ2 )i ≥ 0,
BF ∗ (η1 : η2 ) = F ∗ (η1 ) − F ∗ (η2 ) − hη1 − η2 , ∇F ∗ (η2 )i ≥ 0,
where the rst equality holds if and only if θ1 = θ2 and the second inequality
holds i η1 = η2 . Indeed, we have the following Jereys-symmetrization of thedual Bregman divergences:

BF (θ1 : θ2 ) + BF (θ2 : θ1 ) = hθ2 − θ1 , ∇F (θ2 ) − ∇F (θ1 )i > 0,∗∗

BF ∗ (η1 : η2 ) + BF ∗ (η2 : η1 ) = hη2 − η1 , ∇F (η2 ) − ∇F (η1 )i > 0,

∀θ1 6= θ2∀η1 6= η2Denition 3 (Quasi-arithmetic centers, QACs)). Let F : Θ → R bea strictly convex and smooth real-valued function of Legendre-type in F . The
weighted quasi-arithmetic average of θ1 , . . . , θn and w ∈ ∆n−1 is dened by thegradient map ∇F as follows:!
M∇F (θ1 , . . . , θn ; w) := ∇F −1

X

wi ∇F (θi ) ,

(4)

i

!
= ∇F

∗

X

wi ∇F (θi ) ,(5)

iwhere ∇F ∗ = (∇F )−1 is the gradient map of the Legendre transform F ∗ of F .

4Frank Nielsen

Mf (Denition 1) whenf(u)dufor a strictly increasing or strictly decreasing and continuousa−1function f : Mf = MF 0 (with f= (F 0 )−1 ). Notice that we only need to consider
F to be strictly convex or strictly concave and smooth to dene a multivariate
QAM since M∇F = M−∇F .
We recover the usual denition of scalar QAMs

F (t) =

Rt

Example 1 (Matrix example). Consider the strictly convex function [8] F :Sym++ (d) → R with F (θ) = − log det(θ), where det(·) denotes the matrix determinant. Function F (θ) is strictly convex and dierentiable [8] on the domainof d-dimensional symmetric positive-denite matrices Sym++ (d) (open convexcone). We have

F (θ) = − log det(θ),∇F (θ) = −θ−1 =: η(θ),∇F −1 (η) = −η −1 =: θ(η)
F ∗ (η) = hθ(η), ηi − F (θ(η)) = −d − log det(−η),
where the dual parameter η belongs to the d-dimensional negative-denite matrix
domain, and the inner matrix product is the Hilbert-Schmidt inner producthA, Bi := tr(AB > ), where tr(·) denotes the matrix trace. It follows thatM∇F (θ1 , θ2 ) = 2(θ1−1 + θ2−1 )−1 ,is the matrix harmonic mean [1] generalizing the scalar harmonic mean H(a, b) =2aba+b for a, b > 0. Other examples of matrix means are reported in [7].

2.2 Quasi-arithmetic barycenters and dual geodesics
A Bregman generator F : Θ → R induces a dually at space [4](Θ, g(θ) = ∇2θ F (θ), ∇, ∇∗ )that we call a Bregman manifold (Hessian manifold with a global chart), where∇ is the at connection with Christoel symbols Γijk (θ) = 0 and ∇∗ is the dual
∗ ijkconnection with respect to g such that Γ(η) = 0.
In a Bregman manifold, the primal geodesics γ∇ (P, Q; t) are obtained asline segments in the θ -coordinate system (because the Christoel symbols of
the connection ∇ vanishes in the θ -coordinate system) while the dual geodesicsγ∇∗ (P, Q; t) are line segments in the η -coordinate system (because the Christof∗
fel symbols of the dual connection ∇ vanishes in the η -coordinate system).
∇
The dual geodesics dene interpolation schemes (P Q) (t) = γ∇ (P, Q; t) and∇∗(P Q) (t) = γ∇∗ (P, Q; t) between input points P and Q with P = γ∇ (P, Q; 0) =γ∇∗ (P, Q; 0) and Q = γ∇ (P, Q; 1) = γ∇∗ (P, Q; 1) when t ranges in [0, 1]. We
express the coordinates of the interpolated points on γ∇ and γ∇∗ using quasiarithmetic averages as follows:Jensen-Shannon ∇-divergences

5∇-geodesic γ∇ (P, Q; t) = (P Q)∇ (t)
Mid (θ(P ), θ(Q); 1 − t, t)(P Q)∇ (t) =M∇F ∗ (η(P ), η(Q); 1 − t, t)
QP∗∇∗ -geodesic γ∇∗ (P, Q; t) = (P Q)∇ (t)∗M∇F (θ(P ), θ(Q); 1 − t, t)(P Q)∇ (t) =Mid (η(P ), η(Q); 1 − t, t)∗(M, g, ∇, ∇ )The points on dual geodesics in a dually at spaces have dual coordinates
expressed with quasi-arithmetic averages.
Fig. 1.

Mid (θ(P ), θ(Q); 1 − t, t)(P Q) (t) = γ∇ (P, Q; t) =,M∇F ∗ (η(P ), η(Q); 1 − t, t)∗M∇F (θ(P ), θ(Q); 1 − t, t)(P Q)∇ (t) = γ∇∗ (P, Q; t) =,Mid (η(P ), η(Q); 1 − t, t)
∇

(6)

(7)

where id denotes the identity mapping. See Figure 1.
Quasi-arithmetic centers were also used by a geodesic bisection algorithm
to approximate the circumcenter of the minimum enclosing balls with respect
to the canonical divergence in Bregman manifolds in [21], and for dening the
Riemannian center of mass between two symmetric positive-denite matrices
with respect to the trace metric in [15]. See also [22, 23].

3

Invariance and equivariance properties∗

A dually at manifold [4] (M, g, ∇, ∇ ) has a canonical divergence [2] D∇,∇∗
which can be expressed either as a primal Bregman divergence in the ∇-anecoordinate system θ (using the convex potential function F (θ)) or as a dual∗

Bregman divergence in the ∇ -ane coordinate system η (using the convex conjugate potential function F∗(η)), or as dual Fenchel-Young divergences [18] using∗

the mixed coordinate systems θ and η . The dually at manifold (M, g, ∇, ∇ )(a particular case of Hessian manifolds [26] which admit a global coordinate
system) is thus characterized by (θ, F (θ); η, F∗(η)) which we shall denote by(M, g, ∇, ∇ ) ← DFS(θ, F (θ); η, F (η)) (or in short (M, g, ∇, ∇∗ ) ← (Θ, F (θ))).
However, the choices of parameters θ and η and potential functions F (θ) andF ∗ (η) are not unique since they can be chosen up to ane reparameterizations∗∗
and additive ane terms [4]: (M, g, ∇, ∇ ) ← DFS([θ, F (θ); η, F (η)]) where [·]∗∗denotes the equivalence class that has been called purposely the ane Legendre
invariance in [14]:6Frank Nielsen

First, consider changing the potential function F (θ) by adding an ane term:F̄ (θ) = F (θ) + hc, θi + d. We have ∇F̄ (θ) = ∇F (θ) + c = η̄ . Inverting ∇F̄ (x) =∇F (x) + c = y , we get ∇F̄ −1 (y) = ∇F (y − c). We check that BF (θ1 : θ2 ) =BF̄ (θ1 : θ2 ) = D∇,∇∗ (P1 : P2 ) with θ(P1 ) =: θ1 and θ(P2 ) =: θ2 . It is indeed wellknown that Bregman divergences modulo ane terms coincide [5]. For the quasiarithmetic averages M∇F̄ and M∇F , we thus obtain the following invarianceproperty:

M∇F̄ (θ1 , . . . ; θn ; w) = M∇F (θ1 , . . . ; θn ; w).
Second, consider an ane change of coordinates θ̄ = Aθ + b for A ∈ GL(d)dand b ∈ R , and dene the potential function F̄ (θ̄) such that F̄ (θ̄) = F (θ). We
have θ = A−1(θ̄ − b) and F̄ (x) = F (A−1 (x − b)). It follows that∇F̄ (x) = (A−1 )> ∇F (A−1 (x − b)),and we check that BF̄ (θ :θ ) = BF (θ1 : θ2 ):1

2

BF̄ (F̄ (θ1 :θ2 ) = F̄ (θ1 ) − F̄ (θ2 ) − θ1 − θ2 , ∇F̄ (θ2 ) ,= F (θ1 ) − F (θ2 ) − (A(θ1 − θ2 ))> (A−1 )> ∇F (θ2 ),= F (θ1 ) − F (θ2 ) − (θ1 − θ2 )> A> (A−1 )> ∇F (θ2 ) = BF (θ1 : θ2 ).
| {z }(A−1 A)> =I

This highlights the invariance that D∇,∇∗ (P1 : P2 ) = BF (θ1 : θ2 ) = BF̄ (θ̄ :θ̄ ) ,1

2i.e., the canonical divergence does not change under a reparameterization of the∇-ane coordinate system. For the induced quasi-arithmetic averages M∇F̄ andM∇F , we have ∇F̄ (x) = (A−1 )> ∇F (A−1 (x − b)) = y , we calculate
x = ∇F̄ (x)−1 (y) = A ∇F̄ −1 (((A−1 )> )−1 y) + b,
and we have

X
M∇F̄ (θ̄1 , . . . , θ̄n ; w) := ∇F̄ −1 (wi ∇F̄ (θ̄i )),i!
−1= (∇F̄ )

(A

−1 >

)

X

wi ∇F (θi ) ,i= A ∇F −1 ((A−1 )> )−1 (A−1 )>
{z
}
|
=I

X

wi ∇F (θi ) + b,i

M∇F̄ (θ̄1 , . . . , θ̄n ; w) = A M∇F (θ1 , . . . , θn ; w) + b
More generally, we may dene F̄ (θ̄) = F (Aθ + b) + hc, θi + d and get via(η̄) = F ∗ (A∗ η + b∗ ) + hc∗ , ηi + d∗ (with A∗ , b∗ , c∗and d expressed using A, b, c and d since these parameters are linked by the
Legendre transformation F̄

∗Legendre transformation).

∗Jensen-Shannon ∇-divergences

7

Third, the canonical divergences should be considered relative divergences(and not absolute divergences), and dened according to a prescribed arbitrary unit λ > 0. Thus we can scale the canonical divergence by λ > 0, i.e.,Dλ,∇,∇∗ := λD∇,∇∗ . We have Dλ,∇,∇∗ (P1 : P2 ) = λBF (θ1 : θ2 ) = λBF ∗ (η2 : η1 ),and λBF (θ1 : θ2 ) = BλF (θ1 : θ2 ) (and ∇λF = λ∇F ). We check the scale invariance of quasi-arithmetic averages: Mλ∇F = M∇F .

Proposition 2 (Invariance and equivariance of QACs). Let F (θ) be afunction of Legendre type. Then F̄ (θ̄) := λ(F (Aθ + b) + hc, θi + d) for A ∈ GL(d),b, c ∈ Rd , d ∈ Rd and λ ∈ R>0 is a Legendre-type function, and we have
M∇F̄ = A M∇F + b.
This proposition generalizes the invariance property of scalar QAMs, anduntangles the role of scale λ > 0 from the other invariance roles brought by the
Legendre transformation.
Consider the Mahalanobis divergence ∆2(i.e., the squared Mahalanobis dis-

tance ∆) as a Bregman divergence obtained for the quadratic form generator

FQ (θ) = 21 θ> Qθ + cθ + κ for a symmetric positive-denite d × d matrix Q,
c ∈ Rd and κ ∈ R. We have:∆2 (θ1 , θ2 ) = BFQ (θ1 : θ2 ) =1(θ2 − θ1 )> Q (θ2 − θ1 ).
2

When Q = I , the identity matrix, the Mahalanobis divergence coincides with

2 (i.e., the squared Euclidean distance). The Legendre

the Euclidean divergence
convex conjugate is

1 > −1
η Q η = FQ−1 (η),
2
−1∗
η . Thus we get theand we have η = ∇FQ (θ) = Qθ and θ = ∇FQ (η) = QF ∗ (η) =following dual quasi-arithmetic averages:−1

M∇FQ (θ1 , . . . , θn ; w) = QnX!
wi Qθi=

i=1

M∇FQ∗ (η1 , . . . , ηn ; w) = Q

n
X

n
X

wi θi = Mid (θ1 , . . . , θn ; w),i=1!
wi Q−1 ηi= Mid (η1 , . . . , ηn ; w).

i=1
The dual quasi-arithmetic centers M∇FQ and M∇F ∗ induced by a MahaQ

lanobis Bregman generator

FQ coincide since M∇FQ = M∇FQ∗ = Mid . Thismeans geometrically that the left-sided and right-sided centroids of the underlying canonical divergences match. The average M∇FQ (θ1 , . . . , θn ; w) expresses
the centroid C = C̄R = C̄L in the θ -coordinate system (θ(C) = θ ) and the average M∇F ∗ (η1 , . . . , ηn ; w) expresses the same centroid in the η -coordinate system
Q(η(C) = η ). In that case of self-dual at Euclidean geometry, there is an ane

2

The squared Euclidean/Mahalanobis divergence are not metric distances since they
fail the triangle inequality.

8Frank Nielsen−1transformation relating the θ - and η -coordinate systems:η = Qθ and θ = Q

η.

As we shall see this is because the underlying geometry is self-dual Euclidean∗

at space (M, gEuclidean , ∇Euclidean , ∇Euclidean = ∇Euclidean ) and that both dual
connections coincide with the Euclidean connection (i.e., the Levi-Civita connection of the Euclidean metric). In this particular case, the dual coordinate
systems are just related by ane transformations.

4

Quasi-arithmetic mixtures and Jensen-Shannon-typedivergences

Consider a quasi-arithmetic mean Mf and n probability distributions P1 , . . . , Pn

dPn
1
= dP
dµ , . . . , pn = dµ theirRadon-Nikodym derivatives. Let us dene statistical Mf -mixtures of p1 , . . . , pn :all dominated by a measure µ, and denote by p1

Denition 4. The Mf -mixture of n densities p1 , . . . , pn weighted by w ∈ ∆◦n isdened by(p1 , . . . , pn ; w)Mf (x) := R

Mf (p1 (x), . . . , pn (x); w).
Mf (p1 (x), . . . , pn (x); w)dµ(x)(p1 , . . . , pn ; w)Mf generalizes theordinary statistical mixture
i=1 wi pi (x) when f (t) = t and Mf = A isthe arithmetic mean. A statistical Mf -mixture can be interpreted as the Mf integration of its weighted component densities, the densities pi . The powerM
mixtures (p1 , . . . , pn ; w) p (x) (including the ordinary and geometric mixtures)
1−α
are called α-mixtures in [3] with α(p) = 1 − 2p (or equivalently p =2 ). A nicecharacterization of the α-mixtures is that these mixtures are the density centroids of the weighted mixture components with respect to the α-divergences [3]The quasi-arithmetic mixture (QAMIX)

Pd(proven by calculus of variation):(p1 , . . . , pn ; w)Mα = arg minpX

wi Dα (pi , p),iwhere Dα denotes the α−divergences [4, 20]. See also the entropic means dened
according to f -divergences [6]. Mf -mixtures can also been used to dene a generalization of the Jensen-Shannon divergence [17] between densities p and q as
follows:

M

DJSf (p, q) :=1
DKL (p : (pq)Mf ) + DKL (q : (pq)Mf ) ≥ 0,2

(8)p(x) log p(x)q(x) dµ(x) is the Kullback-Leibler divergence, and(pq)Mf := (p, q; 12 , 12 )Mf . The ordinary JSD is recovered when f (t) = t andM f = A:





1
p+q
p+q
DJS (p, q) =DKL p :+ DKL q :.
222
where DKL (p : q) =R

Jensen-Shannon ∇-divergences

9

In general, we may consider quasi-arithmetic paths between densities on thespace P of probability density functions with a common support all dominatedby a reference measure. On P , we can build a parametric statistical model calleda Mf -mixture family of order n as follows:Mf
Fp0 ,p(p0 , p1 , . . . , pn ; (θ, 1))Mf : θ ∈ ∆◦n .
1 ,...,pn :=In particular, power q -paths have been investigated in [13] with applications inannealing importance sampling and other Monte Carlo methods.
To conclude, let us give a geometric denition of a generalization of theJensen-Shannon divergence on P according to an arbitrary ane connection [4,27] ∇:

Denition 5 (Ane connection-based ∇-Jensen-Shannon divergence).
Let ∇ be an ane connection on the space of densities P , and γ∇ (p, q; t) the
geodesic linking density p = γ∇ (p, q; 0) to density q = γ∇ (p, q; 1). Then the ∇Jensen-Shannon divergence is dened by:
1JS
D∇(p, q) :=

2

When ∇ = ∇m


DKL p : γ∇



1
p, q;
2




+ DKL



1
.
q : γ∇ p, q;
2(9)

is chosen as the mixture connection [4], we end up with thep+q1e2 ) = 2 . When ∇ = ∇ ,the exponential connection, we get the geometric Jensen-Shannon divergence [17]1Gsince γ∇e (p, q; ) = (pq)is a statistical geometric mixture. We may consider
2α
the α-connections [4] ∇ of parametric or non-parametric statistical models, andαskew the geometric Jensen-Shannon divergence to dene the β -skewed ∇ -JSD:
ordinary Jensen-Shannon divergence since γ∇m (p, q;JSD∇
α ,β (p, q) = β DKL (p : γ∇α (p, q; β)) + (1 − β) DKL (q : γ∇α (p, q; β)).

(10)

A longer technical report of this work is available [19].

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