Dual Scales of Means: Unified Averaging

• Dual scales of means are averaging processes that apply independent horizontal (domain) and vertical (codomain) transformations to generalize classical averaging methods.
• The framework leverages convex duality and Riemannian geometry to establish comparison theorems and interpolate among quasi-arithmetic, Stolarsky, and classical means.
• Matrix means, including permanental and scaling means, extend the dual scale approach by offering sharp bounds and ergodic convergence insights in high-dimensional analyses.

Dual scales of means generalize classical averaging processes by systematically incorporating two independent transformations: one acting on the domain ("horizontal scale") and one acting on the codomain ("vertical scale") of a function or data set. This dual structure—manifested in frameworks such as isomorphic means, dual Fréchet quasi-arithmetic means, and bivariate constructions—unifies and orders the rich taxonomy of modern means, encompassing classical means, quasi-arithmetic forms, and majorizations within convex and Riemannian formulations. The concept motivates comparison theorems, connections to ergodic theory, and dualities grounded in convex analysis and metric geometry (Liu, 2023, Nielsen, 26 Nov 2025, Bochi et al., 2015).

1. Foundational Concepts: Isomorphic Variables, Dual Functions, and Duality

The dual scale paradigm arises when two strictly monotone bijections $g: X \to U$ (domain transformation) and $h: Y \to V$ (codomain transformation) define the "isomorphic variable" $u = g(x)$ and the "dual-variable-isomorphic function" $\varphi(f:g,h) = h \circ f \circ g^{-1}$ (Liu, 2023). For the analysis of means, this approach extends averaging from classical definitions (arithmetic, geometric, harmonic, power) to generalized "isomorphic means", with both argument and value transformed before aggregation.

In convex analysis, "dual scales of means" are formulated via convex conjugation: a generator $f$ (strictly convex, differentiable) produces a dual generator $f^*$ by Legendre transform. Both induce associated metric distances ($d_f, d_{f^*}$), yielding Fréchet means in both primal and dual coordinates (Nielsen, 26 Nov 2025). This duality has a Riemannian geometric counterpart, with Hessian structures on both coordinate charts, and manifests particularly elegantly in the quasi-arithmetic mean context.

2. Classification and Structure of Dual Scale Means

Isomorphic means admit a seven-class taxonomy (Liu, 2023):

• Class I (PVI mean): Pure vertical transformation by $h$.
• Class II (IVI mean): Pure horizontal transformation by $g$.
• Class III (same-mapping DVI): Dual transformation with the same mapping.
• Class IV (general DVI): Fully dual transformation ($g,h$).
• Class V (one-variable DVI): Special case correlating with Cauchy mean, bivariate means (e.g., quasi-Stolarsky).
• Class VI: Ordinary function mean, no transformation.
• Class VII (inverse-pair DVI): Inverse-pair transformation.

This classification supports a framework where classical means and their generalizations become special cases (arithmetic, geometric, harmonic, power, logarithmic). The dual-variable approach generalizes mean value problems (e.g., recovery of Cauchy's mean value via Class V) and allows for the interpolation between diverse mean families (Stolarsky, quasi-Stolarsky).

The Fréchet mean formalism in metric spaces further refines the notion: any interior point of a finite interval becomes the mean for some dual pair of quasi-arithmetic functions, the two arising from convex-conjugate generators along primal and dual metrics (Nielsen, 26 Nov 2025).

3. Comparison Inequalities and Scale Interactions

Comparison theorems in the dual scale setting leverage the underlying structure of both domain and codomain transformations:

• Jensen-type inequality: Convexity of $g \circ h^{-1}$ on $h(f([a,b]))$ under PVI means yields $M_f|_g \ge M_f|_h$.
• Derivative-ratio criterion: For monotone $f$ and $C^1$ bijections $g,h$, increasing ratio $\rho(x) = g'(x)/h'(x)$ implies $M_f|_g^{II} \ge M_f|_h^{II}$.
• Losonczi’s criterion for Class V: Sharp necessary/sufficient conditions for ordering means $M_x|_{g,h} \leq M_x|_{G,H}$ are given by inequalities (second derivatives and mixed ratios), reflecting deep geometric interplay between transformations (Liu, 2023).

These theorems elucidate how horizontal and vertical scale transformations interact, with comparison orderings governed by convexity, monotonicity, or higher-order derivative relations.

4. Dual Scales, Convex Duality, and Riemannian Geometry

Dual scales of means are intrinsically connected to convex duality and differential geometry. Given a strictly convex generator $f$, its Legendre conjugate $f^*$ swaps the roles of coordinate charts, metrics, and mean computations (Nielsen, 26 Nov 2025). In the Riemannian setting, the real line is regarded as a Hessian manifold with metric $g(\theta) = f''(\theta)$; the associated geodesic mean is the quasi-arithmetic mean. On the dual chart ($\eta=f'(\theta)$), the dual Hessian metric and corresponding mean structure reflect the same phenomena with respect to $f^*$.

This duality extends to families of means: every one-parameter scale of generators $\{s_\alpha\}$ admits a dual scale $\{s_\alpha^*\}$, with algebraic correspondence

$$m_{s_\alpha}(\theta_1, \theta_2) = m_{(s_\alpha)^*}(s'_\alpha(\theta_1), s'_\alpha(\theta_2)).$$

Varying the generator thus systematically produces a rich hierarchy of means in both primal and dual representations, and every interior point in a finite interval can be realized as the Fréchet mean in both scales (Nielsen, 26 Nov 2025).

5. Matrix Means: Permanental and Scaling Approaches

Dual scale phenomena also arise in matrix theory through the permanental mean ($\pm(A)$) and scaling mean ($\sm(A)$) of nonnegative matrices (Bochi et al., 2015):

• Permanental mean: $\pm(A) = (\perm(A))^{1/n}$ aggregates products over permutations, connecting to symmetric and Muirhead means for vectors via block-matrix construction.
• Scaling mean: Defined by a convex-concave optimization problem under geometric mean normalization constraints, equivalently characterized by Sinkhorn decomposition $A = DSE$.
• Hierarchy: Concavity, monotonicity, and symmetry properties are rigorously established; the scaling mean always provides a lower bound: $\sm(A) \leq \pm(A)$.
• Law of Large Permanents: In dynamically generated matrices, the limit of permanental means coincides almost surely with the scaling mean, generalizing ergodic results (Birkhoff) for matrix-valued processes.

These constructions further realize classical means (arithmetic, symmetric, Muirhead) for data sets and random processes within the dual scale model.

6. Applications and Unified Perspective

The dual scale framework unifies classical and generalized means, supports robust comparison theorems, and provides sharp bounds in matrix and functional analysis. It models averaging via interacting geometric transformations, capturing a broad spectrum from arithmetic and harmonic to Stolarsky types and beyond. Ergonomic and convex duality arguments reveal that every interior point in a given interval is interpretable as a dual Fréchet mean for suitably chosen generator/dual pairs (Nielsen, 26 Nov 2025), and matrix means inherit matching dual scale structures with precise bounds (Bochi et al., 2015).

A plausible implication is that dual scales of means articulate the full geometric and convex-analytic structure underlying fundamental averaging processes, enabling systematic generalization, comparison, and application in high-dimensional analysis, metric geometry, and ergodic theory.