Nonlinear Perron–Frobenius Theory

• Nonlinear Perron–Frobenius theory is a framework that generalizes classical eigenvalue problems for positive matrices to order-preserving, homogeneous nonlinear maps on cones.
• It leverages tools such as Hilbert’s projective metric and extended Collatz–Wielandt numbers to establish necessary and sufficient conditions for the existence, uniqueness, and convergence of positive eigenvectors.
• The theory underpins practical applications in tensor eigenproblems, stochastic games, and population models while offering algorithmic strategies like power iteration for R-linear convergence.

Nonlinear Perron–Frobenius theory generalizes the classical Perron–Frobenius theorem on positive matrices to the setting of order-preserving, homogeneous nonlinear maps on cones, notably the positive orthant $\mathbb{R}^n_{>0}$. This theory provides powerful necessary and sufficient conditions for the existence, uniqueness, and computability of positive eigenvectors for nonlinear, possibly infinite-dimensional, operators. Central tools are extensions of the Collatz–Wielandt formula to the nonlinear context, the use of Hilbert and Thompson projective metrics, and combinatorial and hypergraph-based characterizations. The theory subsumes classical results, tensor eigenproblems, population models, game-theoretic Shapley operators, and generalizes to real-analytic, stochastic, or multi-homogeneous mappings.

1. Foundational Notions and Projective Metrics

Let $$\mathbb{R}^n_{++} = \{ x \in \mathbb{R}^n: x_i > 0\ \forall i \}$$, and define $x \leq y$ if $y - x \in \mathbb{R}^n_{+}$. The archetypal objects are maps

$f: \mathbb{R}^n_{++} \to \mathbb{R}^n_{++}$

that are

• Order-preserving: $x \leq y \implies f(x) \leq f(y)$,
• Homogeneous: $f(t x) = t f(x)$ for all $t > 0$.

On cones, Hilbert's projective metric is

$$d_H(x, y) = \log \left( \frac{ \max_i (y_i/x_i) }{ \min_j (y_j/x_j)} \right),$$

which is symmetric, scale-invariant, and makes each projective ray a metric space. Fundamental is nonexpansiveness: $d_H(f(x), f(y)) \leq d_H(x, y).$

A function $f$ may also be subhomogeneous—$f(t x) \leq t f(x)$ for $t \geq 1$—and type K order-preserving, meaning $x \leq y$ and $x_i < y_i$ imply $f(x)_i < f(y)_i$ coordinatewise.

2. Nonlinear Spectral Theory and Collatz–Wielandt Numbers

For $f$ as above, an eigenpair is a tuple $(x, \lambda)$ with $x \in \mathbb{R}^n_{++}$, $f(x) = \lambda x$, and $\lambda > 0$. The spectrum is generally nonlinear and parameterized by the so-called cone spectral radius

$$r(f) = \lim_{k \to \infty} \| f^k(x) \|^{1/k} = \inf_{x > 0} M(f(x)/x) = \sup_{x > 0} m(f(x)/x),$$

where $M(f(x)/x) = \inf\{ \beta > 0: f(x) \leq \beta x \}$ and $$m(f(x)/x) = \sup\{ \alpha > 0: \alpha x \leq f(x) \}$$ are the Collatz–Wielandt numbers, which quantify one-step expansion and contraction.

In the nonlinear setting, existence and boundedness of the set of positive eigenvectors $E(f) = \{ x > 0 : f(x) = \lambda(x) x \}$ are equivalent to a family of inequalities on these Collatz–Wielandt numbers for certain boundary restrictions of $f$: $$E(f) \neq \emptyset \ \text{and} \ d_H\text{-bounded} \iff \forall J \subsetneq \{1,\ldots,n\}, \ r(f^J_0) < \lambda(f^{[n] \setminus J}_\infty),$$ where $f^J_0$ and $f^{[n]\setminus J}_\infty$ are boundary extensions of $f$ defined by freezing coordinates outside $J$ to 0 or $\infty$.

3. Combinatorial and Hypergraph Criteria

Directed hypergraphs capture the behavior of $f$ at the boundary:

• $H^-$ with hyperarc $(I \to \{j\})$ if $\lim_{t\to\infty} f(e^{-t e_I})_j = 0$
• $H^+$ with hyperarc $(I \to \{j\})$ if $\lim_{t\to\infty} f(e^{t e_I})_j = \infty$

A subset $S \subseteq [n]$ is invariant under a hypergraph if no hyperarc leaves $S$. The Akian–Gaubert–Hochart criterion states that $E(f)$ is d$_H$–bounded if and only if, for all disjoint $I, J \subseteq [n]$, at least one of $I^c$ is invariant in $H^-$ or $J^c$ is invariant in $H^+$. These combinatorial conditions can be efficiently checked in many cases and cover tensor eigenproblems, population models, and stochastic games (Lins, 2021, Akian et al., 2018).

In the additive setup, define $T = \log \circ f \circ \exp$, which is monotone and additively homogeneous. The dominion game $\Gamma_\infty(T)$ gives an alternative: $f$ has positive eigenvectors iff two players (MIN and MAX) do not have disjoint dominions in the game defined by the behavior of $T$ at infinity (Akian et al., 2018).

4. Uniqueness and Real-Analyticity

For real-analytic, order-preserving, homogeneous $f: \mathbb{R}^n_{++} \to \mathbb{R}^n_{++}$, the set $E(f)$ is nonempty and $d_H$-bounded if and only if it consists of a single projective class. The proof leverages the reduction to a nonexpansive real-analytic map on a Banach space slice (via the log transformation), and the fixed-point set is bounded; horofunction arguments show uniqueness if bounded (Lins, 2021).

Uniqueness can also be characterized game-theoretically: $u$ is the unique eigenvector (up to scaling) if and only if, in the local dominion game at $u$, MIN and MAX do not have disjoint dominions. This gives a precise, computationally checkable uniqueness test (Akian et al., 2018).

5. Multi-Homogeneous and Tensor Extensions

The classical nonlinear theory is subsumed in the broader framework of order-preserving multi-homogeneous maps, $F: K_+ \to K_+$, where $K_+$ is a product cone and $F$ is homogeneous blockwise with a nonnegative matrix $A$ (the homogeneity matrix): $$F( t_1 x^{(1)}, \ldots, t_d x^{(d)} ) = ( t_1, \ldots, t_d )^A \otimes F(x^{(1)}, \ldots, x^{(d)} )$$ The spectral radius, contraction properties, and Collatz–Wielandt principle generalize accordingly. If $\rho(A)<1$, $F$ is a strict contraction in a suitable projective metric, guaranteeing uniqueness and R-linear convergence; when $\rho(A)=1$ and $F$ is irreducible, there exists a unique positive eigenvector (Gautier et al., 2018, Gautier et al., 2017).

This paradigm unifies:

• Irreducibility and primitivity for nonnegative matrices/tensors,
• Tensor and multiplicative mean eigenproblems,
• General spectral problems for coupled block nonlinearities,
• The computation via generalized nonlinear power iterations, with explicit convergence guarantees.

6. Algorithmic Aspects and Applications

Power iteration algorithms for nonlinear PF problems iterate $x^{k+1} = f(x^k)$ (possibly with normalization) and, under appropriate conditions (type K property, irreducibility, contraction via Hilbert or Thompson metric), guarantee global R-linear convergence to the unique positive eigenvector. The rates depend on contractivity parameters or spectral gaps in the linearization at the fixed point (Lins, 2022, Gautier et al., 2018).

Applications include:

• Positive eigenvectors for nonnegative tensors, with precise combinatorial irreducibility conditions,
• Zero-sum stochastic games (game-theoretic dynamic programming operators/Shapley operators), with slice-boundedness ensuring existence/uniqueness of mean payoff equilibria,
• Population dynamics models, as in Schoen’s 4-state model or SIS epidemics,
• Consensus and stability in nonlinear multiagent systems (order-preserving, subhomogeneous maps, type K property assure global convergence) (Deplano et al., 2019),
• Growth maximization and optimal control in piecewise-linear or switched positive linear systems, where the nonlinear eigenvalue links to the ergodic constant of an associated Hamilton–Jacobi PDE (Calvez et al., 2014).

7. Extensions and Concluding Perspectives

Recent work extends nonlinear PF theory in several directions:

• The stochastic nonlinear Perron–Frobenius theorem establishes measurable random eigenpairs for cocycles of order-preserving, homogeneous operators on random cones under strict contractivity on average (Babaei et al., 2016).
• The theory connects to metrics for dynamical systems via RKHS Perron–Frobenius operators, leading to positive-definite pseudo-distances between discrete-time dynamical systems in data-driven applications (Ishikawa et al., 2018).
• The unified framework subsumes classical, tensor, mean, and coupled nonlinear spectral problems, with explicit algorithmic strategies, combinatorial and analytic criteria for existence and uniqueness, and a robust projective-metric geometric foundation.

The subject continues to attract significant research interest for its blend of deep analysis, combinatorics, and direct applications in applied mathematics, data science, game theory, and mathematical biology.