Differential-Escort Transformations

• Differential-escort transformations are operations on probability density functions that deform distributions while maintaining key information-theoretic metrics.
• They form a one-parameter abelian group with multiplicative inverses, linking classical divergences to generalized entropies through rescaling relations.
• Applications include variational inference, statistical complexity reduction, and geometric analysis in escort coordinate space for efficient centroid computation.

Differential-escort transformations are probability-preserving operations on probability density functions (pdfs) that provide a systematic method for deforming distributions while preserving key information-theoretic, entropic, and divergence-related properties. These transformations interpolate between densities and exhibit monotonicity, group structure, and strong algebraic relations with classical divergences and generalized entropies. They underpin results ranging from variational inference to statistical complexity and the geometric analysis of probability spaces.

1. Formal Definition of Differential-Escort Transformations

Let $\rho(x)$ be a probability density on a connected support $\Lambda\subseteq\mathbb{R}$, with $\int_\Lambda \rho(x)\,dx =1$. The differential-escort transform $\mathfrak{E}_\alpha[\rho]$ of order $\alpha \in \mathbb{R}$ is defined by

$\rho_\alpha(y) := [\rho(x(y))]^\alpha$

where the variable change $y=y(x)$ is determined by the requirement of local probability preservation:

$$\frac{dy}{dx} = [\rho(x)]^{1-\alpha}, \qquad y(x_0)=x_0$$

for any base point $x_0$ in $\Lambda$ (Puertas-Centeno, 2018). The transformation $\rho \mapsto \rho_\alpha$ guarantees

$$\int_{\Lambda_\alpha} \rho_\alpha(y)\,dy = \int_\Lambda \rho(x)\,dx = 1$$

where $\Lambda_\alpha = y(\Lambda)$.

For relative differential-escort with reference density $h$ and target $f$, both strictly positive on $\Omega = (x_i, x_f)$, define

$$y'(x) = f(x)^{1-\alpha} h(x)^\alpha, \qquad y(x_i)=x_i$$

and

$$f^{[h]}_\alpha(y) = \left(\frac{f(x(y))}{h(x(y))}\right)^\alpha$$

This recovers the standard escort density when $h \equiv 1$ (Iagar et al., 12 Dec 2025).

2. Algebraic Structure and Group Properties

The family of standard and relative differential-escort transformations forms a one-parameter abelian group under composition (Iagar et al., 12 Dec 2025):

• Multiplicativity: $$\mathfrak{E}_\beta \circ \mathfrak{E}_\alpha = \mathfrak{E}_{\alpha\beta}$$ and $\mathfrak{E}_1$ is the identity.
• Inverse: For $\alpha \neq 0$, the inverse of $\mathfrak{E}_\alpha$ is $\mathfrak{E}_{1/\alpha}$.
• Divergence Transformations: By algebraic conjugation, one derives "divergence" transforms

$$\mathcal{A}^{(h)}_\alpha[f] := \mathfrak{E}^{-1,[h]}_1 \circ \mathfrak{E}^{[h]}_\alpha[f]$$

which themselves form a multiplicative group: $$\mathcal{A}^{(h)}_\beta \circ \mathcal{A}^{(h)}_\alpha = \mathcal{A}^{(h)}_{\beta\alpha}$$ (Iagar et al., 12 Dec 2025).

3. Entropic and Divergence Properties

Differential-escort transformations induce simple and exact rescaling relations for Shannon, Rényi, and Tsallis entropies, as well as their entropic moments:

• Entropic moments: $W_q[\rho_\alpha] = W_{q_\alpha}[\rho]$ with $q_\alpha = 1 + \alpha(q-1)$.
• Shannon entropy: $S[\rho_\alpha] = \alpha S[\rho]$.
• Rényi entropy: $R_q[\rho_\alpha] = \alpha R_{q_\alpha}[\rho]$.
• Tsallis entropy: $T_q[\rho_\alpha] = \alpha T_{q_\alpha}[\rho]$ (Puertas-Centeno, 2018).

For Rényi divergences between transformed and reference densities:

$$D_\xi[\mathfrak{E}_\alpha[f] \,\|\, h] = \alpha D_{1+\alpha(\xi-1)}[f\,\|\, h] + (\alpha - 1) D_\alpha[h\,\|\, f]$$

Monotonicity is established:

• $\partial_\alpha D_\xi[g_\alpha \| h] \geq 0$ for $\alpha > 0$ and $\xi \geq 1$,
• $\partial_\alpha D_\xi[g_\alpha \| h] \leq 0$ for $\alpha < 0$
• $D_\xi[g_0 \| h]=0$ (Iagar et al., 12 Dec 2025).

4. Monotonicity and Statistical Complexity

The transformation acts as a monotone operation for LMC–Rényi complexity $C_{p,q}[\rho] = \exp(R_p[\rho] - R_q[\rho])$, with $p<q$:

$$C_{p,q}[\rho_\alpha] = (C_{p_\alpha, q_\alpha}[\rho])^\alpha, \qquad p_\alpha=1+\alpha(p-1),\; q_\alpha=1+\alpha(q-1)$$

Monotonicity:

• For $0 \leq \alpha \leq 1$, $C_{p,q}[\rho_\alpha] \leq C_{p,q}[\rho]$ with strict inequality unless $\rho$ is uniform.
• The family $\{ \mathfrak{E}_\alpha \}_{\alpha\in[0,1]}$ reduces complexity, with minimum at $\alpha=0$ (Puertas-Centeno, 2018).
• Extremal behavior: As $\alpha \to 0^+$, $C_{p,q}[\rho_\alpha] \to 1$ (uniformity). As $\alpha \to \infty$, $C_{p,q}[\rho_\alpha] \to \infty$ (highly peaked/heavy-tailed).

5. Connections to Generalized Entropies and Escort Distributions

Differential-escort transformations connect with classical escort distributions and generalized entropies/divergences:

• Escort distribution: For densities $p_0$, $p_1$, intermediate density minimizes divergence to $p_1$ with fixed KL distance to $p_0$:

$$p_q(x) = \frac{p_1(x)^q \, p_0(x)^{1-q}}{\int p_1^q \, p_0^{1-q}\,d\mu}$$

Varying $q$ traces the "escort-path" in density space (Bercher, 2012).

• Fisher information dynamics: Fisher information along the path,

$$I(q) = \text{Var}_q\left(\log\frac{p_1}{p_0}\right)$$

Integrating $I(q)$ yields thermodynamic divergence proportional to Jeffreys' divergence,

$\int_0^1 I(q)\,dq = D(p_1 \| p_0) + D(p_0 \| p_1)$

• Variational inference: Minimization problems under generalized-moment constraints (e.g., $q$-moments of observables) reduce to minimizing Rényi divergence subject to constraints (Bercher, 2012).

6. Geometric Structure and Applications

Escort and differential-escort transformations induce geometric structures in statistical manifolds:

• Conformal flattening: Escort probabilities provide global affine coordinates for the flattening of $\alpha$-geometry on the simplex. The pull-back of dually flat structure yields ordinary Bregman divergences in escort coordinates (Ohara et al., 2010).
• Alpha-Voronoi diagrams: Algorithms for Voronoi tessellation with respect to $\alpha$-divergences become convex hull problems in escort coordinate space, with computational advantages.
• Centroid computation: The escort centroid is the weighted mean in escort space, facilitating efficient centroid calculation for non-additive statistical geometries (Ohara et al., 2010).

7. Special Cases and Example Transformations

Differential-escort transformations unify disparate distributional families:

• Exponential to Tsallis $q$-exponential: Applying the transformation to the standard exponential,

$\mathcal E(x) = e^{-x}$

yields,

$$\rho_\alpha(y) = \left[1 + (1-\alpha)y\right]^{-\frac{\alpha}{1-\alpha}}$$

recognized as Tsallis $q$-exponential with $q=2-1/\alpha$ for $q<2$ (Puertas-Centeno, 2018).

• Generalized $q$-Gaussian: Minimized Rényi divergence under moment constraint leads to solution:

$$p^*(x) = \frac{1}{Z_q(\beta)} [1-(1-q)\beta A(x)]_+^{1/(1-q)} p_0(x)$$

which produces $q$-Gaussian for $A(x)=|x|^\alpha$, $p_0 \equiv 1$ (Bercher, 2012).

• Divergence maximization/minimization: By tuning $\alpha$, one may interpolate between distributions minimizing or maximizing divergence relative to a reference ($\mathcal{A}^{(h)}_\alpha$) (Iagar et al., 12 Dec 2025).

• "Differential-escort transformations and the monotonicity of the LMC-Rényi complexity measure" (Puertas-Centeno, 2018)
• "A new group of transformations related to the Kullback-Leibler and Rényi divergences and universal classes of monotone measures of statistical complexity" (Iagar et al., 12 Dec 2025)
• "A simple probabilistic construction yielding generalized entropies and divergences, escort distributions and q-Gaussians" (Bercher, 2012)
• "Dually flat structure with escort probability and its application to alpha-Voronoi diagrams" (Ohara et al., 2010)