Dual Fréchet Quasi-Arithmetic Means

• Dual Fréchet quasi-arithmetic means are a class of barycentric methods that generalize scalar means through convex duality and Legendre-type functions.
• They leverage primal and dual constructions via Bregman divergences to achieve unique Fréchet characterizations in both coordinate systems.
• This framework forms a complete lattice with invariance properties, effectively bridging convex analysis, information geometry, and classical mean theory.

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 (Nielsen, 2023).

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 $d_{(\varphi^*)}$ (Nielsen, 26 Nov 2025).

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 (Nielsen, 26 Nov 2025).

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 (Pasteczka, 2018).

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 (Nielsen, 2023).

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 (Nielsen, 2023).

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 (Nielsen, 26 Nov 2025).

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 (Nielsen, 2023).

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 (Nielsen, 2023, Pasteczka, 2018, Nielsen, 26 Nov 2025).