The concept of a mean is one of the most basic ideas in mathematics. The arithmetic mean gives us the center of a collection of numbers, but when the underlying space is equipped with a different distance or geometry, the notion of a center can change.
The paper “Scales of Fréchet means and Karcher quasi-arithmetic means” by Frank Nielsen develops a geometric theory connecting means, metric distances, Fréchet means, Karcher means, quasi-arithmetic means, convexity, Hessian geometry, and Bregman centroids.
The central idea is that a point lying between two numbers can be interpreted as their midpoint under a suitably chosen metric. More generally, different families of means can be understood as geometric centers expressed in different coordinate systems.
1. Means
For two numbers (a) and (b), the arithmetic mean is
[
m(a,b)=\frac{a+b}{2}.
]
A general mean is a function that satisfies three fundamental properties:
Idempotence
[
m(x,x)=x
]
Internality
[
\min(a,b)\leq m(a,b)\leq\max(a,b)
]
Symmetry
[
m(a,b)=m(b,a).
]
Thus, a mean produces a value lying between its inputs and does not depend on their ordering.
2. Metric Distance
A distance function (d(x,y)) measures how far two points are from each other.
A mathematical distance must satisfy four metric axioms:
[
d(x,y)\geq0
]
[
d(x,y)=0\iff x=y
]
[
d(x,y)=d(y,x)
]
and
[
d(x,z)+d(z,y)\geq d(x,y).
]
These correspond to non-negativity, identity of indiscernibles, symmetry, and the triangle inequality.
The important observation in the paper is that changing the distance can change the meaning of a midpoint.
3. Midpoint With Respect to a Distance
Normally, the midpoint of (a) and (b) is
[
\frac{a+b}{2}.
]
But mathematically, a point (c\in(a,b)) can be called a midpoint with respect to a distance (d) whenever
[
d(a,c)=d(c,b).
]
Therefore, the midpoint depends not only on (a) and (b), but also on the geometry used to measure distance.
This observation is fundamental to the paper.
4. Fréchet Mean
The Fréchet mean generalizes the ordinary mean to metric spaces.
For two points (a) and (b), it is defined as the point minimizing the sum of squared distances:
[
c=
\operatorname*{arg,min}_{x\in[a,b]}
\left[
d^2(a,x)+d^2(x,b)
\right].
]
In ordinary Euclidean space, this gives the arithmetic midpoint.
In a general metric space, however, the Fréchet mean can be different.
It can also fail to be unique. For example, on a sphere, two antipodal points can have an entire great circle of Fréchet means.
5. Quasi-Arithmetic Means
A central concept in the paper is the quasi-arithmetic mean.
Let (h) be a continuous, strictly monotone function. The two-variable quasi-arithmetic mean is
[
m_h(a,b)
h^{-1}
\left(
\frac{h(a)+h(b)}{2}
\right).
]
The corresponding (n)-variable form is
[
m_h(x_1,\ldots,x_n)
h^{-1}
\left(
\frac{1}{n}
\sum_{i=1}^{n}h(x_i)
\right).
]
The function (h) is called the generator of the mean.
The arithmetic mean is obtained by choosing
[
h(x)=x.
]
The geometric mean is obtained using
[
h(x)=\log x.
]
Thus, quasi-arithmetic means provide a general framework for constructing different types of averages.
6. Distance Generated by a Function
The connection between quasi-arithmetic means and geometry becomes clear by defining
[
d_f(x,y)=|f(x)-f(y)|.
]
If (f) is strictly monotone, then the midpoint with respect to this distance is
[
c=
f^{-1}
\left(
\frac{f(a)+f(b)}{2}
\right).
]
Therefore,
[
c=m_f(a,b).
]
This means:
Every quasi-arithmetic mean can be interpreted as a midpoint under an appropriate metric.
This is one of the fundamental mathematical connections developed in the paper.
7. Power Means
Power means are an important example of quasi-arithmetic means.
For (p\neq0),
[
M_p(a,b)
\left(
\frac{a^p+b^p}{2}
\right)^{1/p}.
]
For (p=0), the limiting case is
[
M_0(a,b)=\sqrt{ab}.
]
The corresponding generator is
[
h_p(x)=x^p
]
for (p\neq0), while
[
h_0(x)=\log x.
]
Power means produce the familiar hierarchy
[
QM\geq AM\geq GM\geq HM.
]
Here:
(QM) = quadratic mean
(AM) = arithmetic mean
(GM) = geometric mean
(HM) = harmonic mean
The paper uses power means as the starting point for a more general theory.
8. Scales of Means
A scale of means is a one-parameter family
[
{m_r}_{r\in\mathbb R}.
]
The parameter (r) changes the resulting mean continuously.
For an increasing scale,
[
\lim_{r\to-\infty}m_r(a,b)=\min(a,b)
]
and
[
\lim_{r\to+\infty}m_r(a,b)=\max(a,b).
]
For a decreasing scale, the limits are reversed.
Thus, a scale provides a continuous transition between the minimum and maximum.
9. Main Theorem: Every Interior Point Can Be a Midpoint
The central theoretical result can be expressed as follows.
Suppose
[
a<c<b.
]
If a family of strictly monotone differentiable functions
[
{s_\alpha}_{\alpha\in\mathbb R}
]
generates a strictly monotone scale of quasi-arithmetic means, then there exists a parameter (\alpha) such that
[
c
s_\alpha^{-1}
\left(
\frac{s_\alpha(a)+s_\alpha(b)}{2}
\right).
]
Consequently,
[
d_{s_\alpha}(a,c)
d_{s_\alpha}(c,b),
]
where
[
d_{s_\alpha}(x,y)
|s_\alpha(x)-s_\alpha(y)|.
]
Therefore, any interior point of an interval can be realized as the midpoint of the endpoints under a suitable distance from the scale.
This generalizes earlier results that were restricted to positive intervals and power means.
10. Exponential Means
The paper gives the exponential means as an important example that works over the entire real line.
Define
[
e_\alpha(u)=e^{\alpha u}
]
for (\alpha\neq0), with the limiting case
[
e_0(u)=u.
]
The corresponding exponential mean is
[
m_{e_\alpha}(x,y)
\frac{1}{\alpha}
\log
\left(
\frac{e^{\alpha x}+e^{\alpha y}}{2}
\right)
]
for (\alpha\neq0).
At (\alpha=0),
[
m_{e_0}(x,y)
\frac{x+y}{2}.
]
Thus, the arithmetic mean occurs naturally as the zero-parameter limit.
11. Exponential Mean and Log-Sum-Exp
The exponential mean is closely related to the log-sum-exp function.
For large positive (\alpha),
[
m_{e_\alpha}(x,y)\rightarrow\max(x,y).
]
For large negative (\alpha),
[
m_{e_\alpha}(x,y)\rightarrow\min(x,y).
]
Therefore,
[
\min(x,y)
\longleftarrow
m_{e_\alpha}(x,y)
\longrightarrow
\max(x,y)
]
as the parameter varies.
This provides a smooth approximation to the maximum and minimum functions.
The associated distance is
[
d_{e_\alpha}(x,y)
|e^{\alpha x}-e^{\alpha y}|.
]
For (\alpha=0), it reduces to ordinary Euclidean distance:
[
d_{e_0}(x,y)=|x-y|.
]
12. Radical Means
The paper also studies a scale of radical means on the positive real numbers.
These means are generated by transformations involving reciprocal powers.
A special case occurs when the parameter equals (1), producing the harmonic mean:
[
HM(a,b)=\frac{2ab}{a+b}.
]
The corresponding distance is based on the reciprocal transformation:
[
d(x,y)
\left|
\frac1x-\frac1y
\right|.
]
Thus, the harmonic mean can also be interpreted geometrically as a midpoint under a transformed metric.
13. Same Center, Different Coordinates
A deeper interpretation appears when the real line is viewed through different coordinate systems.
Suppose two points have coordinates
[
a=x(A),\qquad b=x(B).
]
Their Euclidean center of mass is
[
C=\frac{A+B}{2}.
]
Now introduce another coordinate system related by
[
x=h(x').
]
Then
[
c'=h^{-1}
\left(
\frac{h(a')+h(b')}{2}
\right).
]
Therefore,
[
c'=m_h(a',b').
]
So a quasi-arithmetic mean can be interpreted as the same Euclidean center of mass represented in a transformed coordinate system.
This provides an important geometric interpretation of generalized means.
14. Riemannian Geometry
The paper then interprets these ideas using Riemannian geometry.
A Riemannian manifold is a space equipped with a metric that allows lengths, angles, and distances to be defined locally.
For a one-dimensional manifold with coordinate (\theta), consider a metric
[
g(\theta)>0.
]
The infinitesimal length is
[
ds=\sqrt{g(\theta)},d\theta.
]
The distance between two points is
[
\rho(\theta_1,\theta_2)
\int_{\theta_1}^{\theta_2}
\sqrt{g(u)},du.
]
If
[
g(\theta)=f''(\theta)
]
for a strictly convex potential (f), then the metric is called a Hessian metric.
15. Coordinate Transformation of the Distance
Define
[
h(\theta)
\int^\theta
\sqrt{f''(u)},du.
]
Then
[
h'(\theta)=\sqrt{f''(\theta)}>0.
]
Therefore, (h) is strictly increasing.
The Riemannian distance becomes
[
\rho(\theta_1,\theta_2)
|h(\theta_2)-h(\theta_1)|.
]
Thus, the one-dimensional Riemannian geometry can be transformed into ordinary Euclidean geometry through the coordinate (h).
16. Karcher Mean
The Karcher mean is the Riemannian version of a center of mass.
For two points (a) and (b), the Karcher mean minimizes
[
\rho^2(a,x)+\rho^2(x,b).
]
Because the Riemannian distance can be expressed through (h), the center is
[
c
h^{-1}
\left(
\frac{h(a)+h(b)}{2}
\right).
]
Therefore,
[
c=m_h(a,b).
]
This establishes the connection:
[
\boxed{
\text{Karcher mean}
\text{quasi-arithmetic mean}
}
]
under the appropriate Hessian geometry.
17. Convex Potentials
The Hessian metric is generated by a strictly convex function (f):
[
g(\theta)=f''(\theta)>0.
]
Convexity guarantees that the metric remains positive.
This creates a connection between:
Convex analysis
Riemannian geometry
Means
Optimization
The potential (f) therefore determines the geometry, while the geometry determines the corresponding notion of distance and mean.
18. Dual Coordinates
Convex functions naturally produce dual coordinate systems.
If
[
\eta=f'(\theta),
]
then (\theta) and (\eta) form a pair of dual coordinates.
The relationship is governed by Legendre duality.
This leads to pairs of dual means.
One coordinate system may produce one type of quasi-arithmetic mean, while the dual coordinate system produces another related mean.
This is particularly important in information geometry.
19. Dual Scales of Means
The paper introduces the idea that a convex potential can generate two related families of means.
These are associated with:
A primal coordinate system
A dual coordinate system
A primal potential
A convex conjugate potential
Thus, the geometry naturally produces dual scales of means.
The two means are not independent; they are connected through convex duality.
20. Higher-Dimensional Extension
The theory is extended beyond the one-dimensional real line.
Consider a strictly convex function
[
F(\theta)
]
defined on a multidimensional domain.
Its Hessian is
[
\nabla^2F(\theta).
]
This Hessian defines a multidimensional Hessian metric.
The corresponding geometry can be transformed into Euclidean geometry using an appropriate coordinate representation.
21. Multivariate Quasi-Arithmetic Means
For a vector-valued dataset
[
x_1,x_2,\ldots,x_n,
]
a multivariate quasi-arithmetic mean can be represented using a coordinate transformation.
Conceptually,
[
M_h(x_1,\ldots,x_n)
h^{-1}
\left(
\frac1n
\sum_{i=1}^{n}h(x_i)
\right).
]
In the higher-dimensional Hessian setting, the transformation is connected to the gradient of a convex potential.
Thus, generalized means become geometric centers in transformed coordinate systems.
22. Bregman Divergences
Convex functions also generate Bregman divergences.
For a differentiable strictly convex function (F),
[
D_F(x:y)
F(x)-F(y)
-\langle\nabla F(y),x-y\rangle.
]
Bregman divergence measures the discrepancy between two points using the geometry generated by (F).
Unlike a metric distance, it generally does not satisfy symmetry:
[
D_F(x:y)\neq D_F(y:x).
]
Nevertheless, Bregman divergences have a powerful connection with convex optimization and statistical geometry.
23. Bregman Centroids
Given points
[
x_1,\ldots,x_n,
]
a Bregman centroid minimizes an average Bregman divergence.
For the appropriate orientation, the centroid can be expressed through the gradient of the convex potential.
This creates the connection:
[
\text{Hessian geometry}
\rightarrow
\text{Karcher mean}
\rightarrow
\text{quasi-arithmetic mean}
\rightarrow
\text{Bregman centroid}.
]
The paper proves that, in its higher-dimensional squared Hessian setting, the Riemannian center of mass expressed in primal coordinates coincides with a multivariate quasi-arithmetic mean and with a left-sided Bregman centroid.
24. Geometric Interpretation
The complete theory can be summarized as a sequence:
[
\boxed{
\text{Mean}
\rightarrow
\text{Coordinate Transformation}
\rightarrow
\text{Distance}
\rightarrow
\text{Fréchet Mean}
}
]
and, in the Riemannian setting,
[
\boxed{
\text{Convex Potential}
\rightarrow
\text{Hessian Metric}
\rightarrow
\text{Geodesic Distance}
\rightarrow
\text{Karcher Mean}
}
]
while in the convex-analytic setting,
[
\boxed{
\text{Convex Potential}
\rightarrow
\text{Bregman Divergence}
\rightarrow
\text{Bregman Centroid}
}
]
These are not unrelated constructions. They are different mathematical descriptions of generalized centers.
25. Main Mathematical Insight
The main insight of the paper can be stated simply:
There is no single universal notion of a midpoint or average independent of geometry.
The arithmetic mean is the natural center under ordinary Euclidean geometry.
A transformed distance produces a quasi-arithmetic mean.
A Riemannian metric produces a Karcher mean.
A metric-space formulation produces a Fréchet mean.
A convex potential produces a Hessian geometry and associated Bregman divergence.
Therefore, the notion of an average is deeply connected to the geometry of the space in which the data is represented.
Download the PDF for free:
https://arxiv.org/pdf/2511.21173
Conclusion
The theory of Scales of Fréchet Means and Karcher Quasi-Arithmetic Means establishes a deep connection between seemingly different mathematical concepts.
A generalized mean can be viewed as a midpoint under an appropriately constructed distance. A family of quasi-arithmetic means can form a scale whose parameter moves the midpoint continuously from the minimum toward the maximum. The paper shows that this construction works for arbitrary intervals through suitable scales such as exponential and radical means.
The geometric interpretation becomes even richer when the real line is considered as a Hessian Riemannian manifold. In this setting, Karcher means arise naturally as Riemannian centers of mass, while convex duality generates corresponding dual coordinate systems and dual scales of means.
In higher dimensions, squared Hessian metrics connect Riemannian centers of mass with multivariate quasi-arithmetic means and Bregman centroids.paper ultimately demonstrates that averaging is not merely an arithmetic operation—it is a geometric operation whose form depends on the distance, coordinates, and structure of the underlying space.
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