---
title: Dual Fréchet Quasi-Arithmetic Means
url: https://www.emergentmind.com/topics/dual-frechet-quasi-arithmetic-means
type: topic
---

# Dual Fréchet Quasi-Arithmetic Means

Dual Fréchet quasi-arithmetic means arise at the confluence of convex analysis, information geometry, and the classical theory of means. They generalize scalar quasi-arithmetic means by leveraging convex (Legendre-type) potentials and their conjugates, leading to paired families of means related via convex duality. This duality allows every interior point of an interval, or more generally, every barycentric average in a dually flat manifold, to be realized simultaneously as a Fréchet mean in both the primal and the dual coordinate systems. The formalism encompasses the Bregman divergence setting, delivers unique characterization via minimization procedures, and extends to a lattice-like structure on the family of quasi-arithmetic means.

## 1. Primal and Dual Quasi-Arithmetic Means

Given an open convex set $\Theta \subset X$ (with $X$ a real inner product space), and a strictly convex, differentiable Legendre-type function $F: \Theta \rightarrow \mathbb{R}$, the classical quasi-arithmetic mean is extended as follows. For points $x_1,\dots,x_n \in \Theta$ and weight vector $w \in \Delta_{n-1}$ (the $(n-1)$-simplex), the **primal quasi-arithmetic mean** generated by $F$ is defined as

$$
M_{\nabla F}(x_1,\ldots,x_n; w) = \nabla F^{-1}\Big(\sum_{i=1}^n w_i \nabla F(x_i) \Big).
$$

Convex duality yields the Fenchel-Legendre conjugate $F^*: H \to \mathbb{R}$ on $H = \nabla F(\Theta)$, with $\nabla F^* = (\nabla F)^{-1}$. The **dual quasi-arithmetic mean** for dual parameters $\eta_1,\ldots,\eta_n \in H$ is then

$$
M_{\nabla F^*}(\eta_1,\ldots,\eta_n; w) = \nabla F^* \Big(\sum_{i=1}^n w_i \nabla F^*(\eta_i)\Big) = \nabla F^{-1} \Big(\sum_{i=1}^n w_i \eta_i\Big).
$$

In the scalar setting (with $F(t) = \int f(t) dt$), these reduce, via suitable choices, to standard means such as arithmetic, geometric, and harmonic [2301.10980].

## 2. Fréchet (Barycentric) Characterization and Metric Structure

Both primal and dual quasi-arithmetic means admit a Fréchet mean (or barycentric) characterization via Bregman divergences:

- **Primal mean as Bregman barycenter:**
  $$
  M_{\nabla F}(x_1,\ldots,x_n;w) = \arg\min_{x \in \Theta} \sum_{i=1}^n w_i B_F(x:x_i),
  $$
  where $B_F(x:y) = F(x) - F(y) - \langle x - y, \nabla F(y) \rangle$ is the Bregman divergence associated with $F$.

- **Dual mean as dual Bregman barycenter:**
  $$
  M_{\nabla F^*}(\eta_1,\ldots,\eta_n;w) = \arg\min_{\eta \in H} \sum_{i=1}^n w_i B_{F^*}(\eta : \eta_i).
  $$

In one dimension, for a strictly monotone $\varphi$, the mean $M_{(\varphi)}(x,y;w) = \varphi^{-1}(w \varphi(x) + (1-w)\varphi(y))$ realizes the Fréchet mean for the metric $d_{(\varphi)}(x,y)=|\varphi(x)-\varphi(y)|$. The dual mean $M_{(\varphi^*)}(x,y;w)$ does likewise for $d_{(\varphi^*)}$ [2511.21173].

## 3. Duality, Uniqueness, and Lattice Structure

The duality between $F$ and $F^*$ is central; both are Legendre-type and their gradient maps are (global) inverses. A remarkable result is that for any open interval $(x, y) \subset I$, *every* interior point $z$ can be realized *simultaneously* as both a Fréchet mean $M_{(\varphi)}(x, y; w)$ and a dual mean $M_{(\varphi^*)}(x, y; w)$ for a unique $w \in (0,1)$, with a consistent relationship between the primal and dual weights mediated by the Legendre transform [2511.21173].

Beyond the individual construction, the family of $\mathcal{C}^2$ quasi-arithmetic means (with nowhere vanishing derivative) forms a complete lattice under the pointwise ordering. Explicitly, for any finite set $\{f_1,\ldots,f_k\}$, there exist unique supremum and infimum means (up to affine equivalence), generated by solving the ODEs:
- $\frac{h''}{h'} = \sup_i \frac{f_i''}{f_i'}$ for the supremum,
- $\frac{\underline{h}''}{\underline{h}'} = \inf_i \frac{f_i''}{f_i'}$ for the infimum.

The dual (infimum) construction is obtained via a reflection symmetry on the generating functions [1811.04865].

## 4. Invariance, Equivariance, and Information Geometry

The mean $M_{\nabla F}$ exhibits natural invariance and equivariance properties with respect to affine and linear transformations of the generating function $F$. For $\bar F(\theta) = \lambda F(A\theta + b) + \langle c, \theta\rangle + d$ with $A \in \mathrm{GL}(X)$, $b, c \in X$, and $\lambda > 0$,
$$
M_{\nabla \bar F}(\theta_1, \dots, \theta_n; w) = A M_{\nabla F}(\theta_1, \dots, \theta_n; w) + b,
$$
while affine translations and rescalings in $F$ leave the mean invariant [2301.10980].

In the context of information geometry, these means and their duals naturally describe points (and barycenters) in dually flat manifolds: geodesics in primal and dual affine coordinates correspond to straight segments in $\theta$ and $\eta$ coordinates, with the respective means characterizing geodesic midpoints and barycenters [2301.10980].

## 5. Riemannian and Hessian Geometric Interpretation

For one-dimensional means, the setting of Hessian geometry illuminates the underlying structure. The line element $ds = \sqrt{\varphi''(t)} dt$ determines a Riemannian metric $g(t) = \varphi''(t)$, and the geodesic distance is expressed as $\rho(t_1, t_2) = |h(t_2) - h(t_1)|$ with $h'(t) = \sqrt{\varphi''(t)}$. In the $h$-coordinate system, all means become arithmetic means, and duality between charts ($\varphi$ and $\varphi^*$) establishes two distinct yet equivalent “scales of means,” with every interior point of an interval being a dual Fréchet mean in both coordinates [2511.21173].

## 6. Extensions and Examples

Notable instances of self-dual means occur when $\nabla F = \nabla F^*$; e.g., $F(\theta) = -\log \det \theta$ on the cone of symmetric positive definite (SPD) matrices yields the harmonic mean $M_{\nabla F}(\theta_1, \theta_2) = 2(\theta_1^{-1} + \theta_2^{-1})^{-1}$. More generally, by selecting $F(t) = \int f(t) dt$ with a univariate $f$, one recovers the full spectrum of classical scalar means [2301.10980].

In summary, dual Fréchet quasi-arithmetic means subsume an extensive class of barycentric constructions via convex duality, admit universal realization for interval interior points, and organize into a complete lattice with rich invariance, geometric, and analytical properties [2301.10980, 1811.04865, 2511.21173].

Source: https://www.emergentmind.com/topics/dual-frechet-quasi-arithmetic-means