Hilbert Metric Contraction

• Hilbert metric contraction is the property whereby positive, order-preserving operators on cones strictly contract the Hilbert projective metric, ensuring robust convergence.
• Birkhoff’s contraction theorem quantifies the contraction factor as tanh(Δ/4), providing a foundation for spectral gap estimates and stability analysis.
• This concept is pivotal in nonlinear Perron–Frobenius theory, supporting the convergence of iterative methods like Sinkhorn’s algorithm in various applied settings.

Hilbert metric contraction refers to the phenomenon that linear (or, more generally, homogeneous order-preserving) operators on cones strictly contract the Hilbert projective metric, with the contraction factor quantitatively expressed in terms of a projective diameter. This property underpins structural results in nonlinear Perron–Frobenius theory, spectral gap estimates for positive matrices and operators, the exponential convergence of scalable algorithms such as Sinkhorn’s, and ergodic properties of dynamical systems on cones. The Hilbert metric and its contraction properties are now central to operator theory, the study of Markov processes, convex and discrete geometry, and entropic optimal transport.

1. Definition and Fundamental Properties of the Hilbert Metric

The Hilbert projective metric is defined on the interior of a convex cone $C$ in a real Banach space or, equivalently, on the interior of a bounded convex domain $\Omega \subset \mathbb{R}^n$. For positive vectors $x, y$ in $C$, the real Hilbert metric reads

$$d_H(x, y) = \ln\left(\max_i \frac{x_i}{y_i}\right) - \ln\left(\min_i \frac{x_i}{y_i}\right)$$

or, in terms of the so-called domination constants for cones ($x, y > 0$),

$d_H(x, y) = \log\left(\frac{M(x/y)}{m(x/y)}\right)$

where $M(x/y)$ and $m(x/y)$ are the infimal and supremal scaling factors so that $m(x/y) y \le_C x \le_C M(x/y) y$ (Lemmens et al., 2013). On bounded convex domains $\Omega$, the metric can be equivalently represented by the logarithm of the cross-ratio of four points determined by $x, y$, and their intersections with $\partial\Omega$, or, via support functionals,

$$d_H(x, y) = \sup_{\ell \in \mathcal{F}} \left| \ln\frac{\ell(x)}{\ell(y)} \right|,$$

where $\mathcal{F}$ is the family of positive affine functionals vanishing on $\partial \Omega$ (Karlsson, 2013).

The Hilbert metric is:

• Projectively invariant (unchanged under scalar multiplication)
• Complete on the interior of $C$
• Monotone under cone-preserving maps

2. Contraction Principle and Birkhoff’s Theorem

Birkhoff’s contraction theorem is the foundational result quantifying Hilbert metric contraction under positive linear operators. For a positive linear map $T$ on a cone $C$, with projective diameter

$\Delta(T) = \sup_{x, y \in C} d_H(Tx, Ty),$

the optimal contraction constant is

$\kappa(T) = \tanh\left(\frac{\Delta(T)}{4}\right).$

The contraction property states

$d_H(Tx, Ty) \leq \kappa(T) d_H(x, y)$

for all $x, y \in C$ (Lemmens et al., 2013, Cohen et al., 2023, Karlsson, 2013).

In matrix settings, for positive matrices $A$, there is an explicit formula for the contraction coefficient (Birkhoff or Hopf): $$\phi(A) := \min_{i,j,k,l} \frac{a_{jk} a_{il}}{a_{ik} a_{jl}}, \qquad c_H(A) = \frac{1 - \phi(A)}{1 + \phi(A)}.$$ This forms the backbone of spectral gap estimates via Hopf’s inequality (Han et al., 2019).

3. Spectral Gap Estimates and Hopf’s Inequality

In Perron–Frobenius theory, the spectral gap of a positive matrix $A$ with maximal eigenvalue $\rho_1$ and second-largest modulus eigenvalue $\rho_2$ is crucial for quantifying convergence and stability. Using Hilbert-metric contraction, Hopf’s inequality yields

$$\frac{\rho_2}{\rho_1} \leq c_H(A) \implies \rho_1 - \rho_2 \geq \rho_1(1 - c_H(A)).$$

This spectral gap estimate is a direct application of the contraction property for the projective map $f_A(w) = N(Aw)$ acting on the positive cone, where $N(w)$ denotes normalization to the simplex (Han et al., 2019). The argument generalizes via the complex Hilbert metric, allowing treatment of non-real eigenvectors and yielding sharp quantitative control of the gap.

4. Generalizations: Pseudo- and Generalized Hilbert Metrics

Variants of the Hilbert metric—bounded or “pseudo” metrics—have been formulated on extended cones or infinite-dimensional spaces. For example, the pseudo-Hilbert metric $$d(\bar{f}, \bar{g}) = \frac{1 - m(f, g)}{1 + m(f, g)}$$ is a strictly bounded analogue controlling contraction for positive linear operators, where $m(f, g) = \aleph(f, g)\, \aleph(g, f)$, and $\aleph(f, g)$ is the maximal scalar such that $b f \leq g$ (Ligonnière, 2023).

Further, generalizations admit cones of functions with bounded growth or include “tail-mass” constraints. In these settings, kernel integral operators are shown to contract a suitably tailored Hilbert metric, as in entropic optimal transport for measures with light tails and unbounded costs. The contraction factor is again given by $\tanh(\Delta/4)$ for an appropriate projective diameter $\Delta$ (Eckstein, 2023).

5. Infinite-Dimensional and Geometric Extensions

The Hilbert metric contraction principle extends uniformly to arbitrary real Hilbert spaces and function spaces. A uniform contraction theorem—generalizing Birkhoff’s original result—states that for nested proper subsets $U \subset V$ of the one-point compactification $\widehat{H}$, the associated metrics satisfy

$$d_V(x, y) \leq \tanh\left(\frac{\Delta}{4}\right) d_U(x, y), \quad \Delta = \sup_{u_1, u_2 \in U} d_V(u_1, u_2).$$

This principle is known as Apollonian contraction in infinite dimensions and is conformally invariant, unifying Birkhoff's and Dubois's results for real, complex, and Hilbert-space geometries (Dubois et al., 2011).

6. Dynamical Systems and Nonlinear Perron–Frobenius Theory

Hilbert-metric contraction provides the foundation for fixed-point, ergodicity, and convergence results in dynamical systems on cones and bounded convex domains. For Hilbert-nonexpansive maps $T$, the Banach fixed-point theorem applies if the contraction ratio is strictly less than one, guaranteeing unique fixed points and exponential convergence (Karlsson, 2013). In more general settings, the theory accounts for orbit behaviors and limit sets in terms of faces of the domain boundary.

Applications include:

• Sinkhorn iteration: Matrix scaling algorithms and entropic optimal transport, with exponential convergence in Hilbert metric and total variation (Eckstein, 2023).
• Transfer and Perron-Frobenius operators: Spectral gaps and long-term statistical properties in ergodic theory (Lemmens et al., 2013).
• Population dynamics models: Geometric stability analysis via contraction (Karlsson, 2013).

7. Applications and Comparative Metrics

The contraction property of the Hilbert metric ensures robust convergence properties for positive operators and provides quantitative bounds expressible in projective diameter or kernel structure. In probability theory, contraction in the Hilbert metric implies convergence in stronger metrics such as total variation and Wasserstein distances, often with explicit sharp constants: $$\|\mu - \nu\|_{\mathrm{TV}} \leq 2\, \tanh\left(\frac{\mathcal{H}(\mu, \nu)}{4}\right),$$ for probability measures $\mu, \nu$ (Cohen et al., 2023). Additionally, for $f$-divergences, one has

$$D_f(\mu \| \nu) \leq \max\{ f(e^{\mathcal{H}(\mu, \nu)}), f(e^{-\mathcal{H}(\mu, \nu)}) \} \to 0 \text{ as } \mathcal{H}(\mu, \nu) \to 0.$$

These implications position Hilbert metric contraction as a unifying tool across analysis, geometry, dynamics, and applied probability.