Non-Monotonic Metric Functions

• Non-monotonic metric functions are mathematical constructs where increasing the input does not guarantee an increase in the output, often exhibiting multiple extrema and inflection points.
• They are rigorously quantified using measures such as the L1-distance to monotone rearrangement, weighted indices, and derivative-based methods to capture deviations from classical monotonicity.
• These functions underpin advanced generalizations in quasi-metrics, φ-metrics, and non-monotonic activation functions in neural networks, influencing system stability, learning performance, and evaluation metrics.

Non-monotonic metric functions are mathematical, statistical, or algorithmic constructs in which the functional relationship between variables does not preserve a single direction of change—i.e., increasing the argument does not always yield an increase (or decrease) in the function value. These functions arise across diverse areas including dynamical systems, functional analysis, statistical inference, machine learning, optimization, and computational geometry. Their paper encompasses theoretical characterization, quantification of non-monotonicity, impact on system stability, fixed point theory, practical evaluation metrics, and implications for learning and control.

1. Foundational Concepts of Non-Monotonic Metric Functions

A function $f$ defined on an ordered set or within a structured domain is monotonic if it preserves order, for instance, $x \leq y$ implies $f(x) \leq f(y)$ (or the reverse for monotonic decreasing). In the context of metric functions, the classical requirement is that the "distance" between points satisfies properties including monotonicity—either directly (as in monotone functions) or indirectly (as in triangle inequality for metrics). Non-monotonic metric functions violate these requirements in a controlled or structural fashion, either globally or on specific "slices" of the domain.

Several archetypes illustrate their occurrence:

• In dynamical systems (bioreactors, chemical networks), functional response rates (e.g., microbial growth rates as functions of substrate concentration) can be non-monotonic, exhibiting initial growth followed by inhibition (cf. Haldane kinetics in chemostats (Rapaport et al., 2012)).
• In statistical metrics and functionals, non-monotonicity may be intrinsic due to complex data structures or non-linear geometric relationships (e.g., metric distribution functions in non-Euclidean spaces (Wang et al., 2021)).
• In learning theory, particularly neural networks, non-monotonic activation functions and loss surfaces present nontrivial optimization landscapes with implications for convergence and expressivity (Wu, 2022, Chen et al., 2023, Biswas et al., 2023).
• In metric generalizations, such as φ-metrics or point pair quasi-metrics, the triangle inequality or related axioms are relaxed by non-monotonic error terms or parameters (Dautova et al., 2022, Das et al., 2022).

Non-monotonic metric functions are characterized by the presence of multiple extrema, inflection points, or alternating behaviors, and consequently, they may lead to multistability, complex equilibrium structures, or require generalized notions of order and distance.

2. Quantification and Measurement of Non-Monotonicity

Quantitative assessment of non-monotonicity is essential for analysis, validation, and comparison in both theoretical and applied settings. Rigorous indices have been proposed to measure how much a function deviates from being monotonic:

• $L_1$-distance to monotone rearrangement: For a real function $h$ on $[0,1]$, one defines its monotone rearrangement $I_h$ via its quantile function and measures

$\mathcal{I}_h = \int_0^1 |h(t) - I_h(t)|\,dt,$

which captures total deviation from monotonicity ($\mathcal{I}_h = 0$ iff $h$ is non-decreasing) (Qoyyimi et al., 2014).

• Weighted (linearized) indices: Using

$\mathcal{L}_h = \int_0^1 (h(t) - I_h(t))(1-t)dt,$

with analogous invariance and homogeneity properties, and $\mathcal{L}_h \leq \mathcal{I}_h$ (Qoyyimi et al., 2014).

• Indices based on derivatives: For an absolutely continuous $g$,

$$\text{LOI}_y(g) = \int_0^y (g'(x))^- dx, \quad \text{LOD}_y(g) = \int_0^y (g'(x))^+ dx,$$

and the overall lack of monotonicity,

$$\text{LOM}_y(g) = 2 \cdot \min\{ \text{LOI}_y(g), \text{LOD}_y(g) \},$$

normalized variants are also introduced for unit-free comparison (Davydov et al., 2017).

• Non-monotonicity in measures: Measures of lack of sign-constancy are analogously defined for signed measures via Hahn/Jordan decomposition, e.g., $\text{LOS}(\nu) = 2 \min(\|\nu^+\|, \|\nu^-\|)$ (Davydov et al., 2017).

Computational approaches for these indices involve discretization, rearrangement, and integration techniques, with proven convergence and bounded approximation errors for discretized versions (Qoyyimi et al., 2014).

3. Structural and Theoretical Generalizations

Non-monotonicity challenges classical properties such as the triangle inequality, or global order-preservation, leading to advanced generalizations:

• Quasi-metrics and Generalized Metrics: The point pair function

$$p_G(x, y) = \frac{|x-y|}{\sqrt{ |x-y|^2 + 4\,d_G(x)d_G(y) }},$$

where $d_G(x)$ is the distance to the boundary, may not always satisfy the triangle inequality but can be shown to be a quasi-metric with sharp constants—becoming a metric in punctured space, and only for specific parameters in generalized versions $p^\alpha_G(x,y)$ (with, e.g., $\alpha\leq 12$ in certain domains) (Dautova et al., 2022).

• φ-metrics: Functions $d_\phi(x, y)$ satisfying modified triangle inequalities,

$$d_\phi(x,y) \leq d_\phi(x,z) + d_\phi(z,y) + \phi(x,y,z),$$

where the correction $\phi$ term decays for nearby points, allow generalizations encompassing S-metrics and b-metrics. These structures admit metrization, well-behaved convergence criteria, and fixed point properties—even when the underlying metric is non-monotonic (Das et al., 2022).

• Non-monotonic Functions in Lattice Theory: In abstract fixpoint theory, functions may not be monotonic over the full lattice, but retain monotonicity with respect to stratified levelwise preorders. The fixed point theorem for a-monotonic functions over stratified complete lattices ensures existence and uniqueness of least fixed points and generalizes Tarski and Kleene theorems for standard monotonic (or continuous) maps (Ésik et al., 2014). This structure accommodates operators arising in logic programming with negation, weighted automata, or layered systems exhibiting non-monotonic transitions.

4. Dynamical Systems, Stability, and Control

Non-monotonic metric functions play a pivotal role in the stability and equilibrium analysis of complex dynamical systems:

• Buffered chemostat models: For growth kinetics $\mu(S)$ with substrate inhibition (non-monotonic in $S$; e.g., Haldane kinetics), classical single-vessel chemostats exhibit multiple equilibria depending on initial substrate levels. The introduction of a buffer compartment—with parameters $(\alpha, r)$—renders the equilibrium condition

$D\,\phi_{\alpha, r}(s) = \mu(s)$

typically uniquely solvable by suitable $(\alpha, r)$ choices, thereby restoring global asymptotic stability while suppressing bistability—a property unobtainable with monotonic kinetic responses (Rapaport et al., 2012).

• Non-monotonic Lyapunov functions in MPC: Economic model predictive control (MPC) for nonlinear systems often employs non-monotonic Lyapunov functions $V^{(\delta)}$ enforced to decrease only every $m\geq1$ steps. The resulting relaxed Lyapunov-based constraints permit improved cost performance (e.g., in HVAC control) while retaining recursive feasibility and asymptotic convergence (Wang et al., 2017).
• Chemical reaction networks and non-normality: Linear master equations with non-normal rate matrices exhibit non-monotonic transient observables and Rényí entropy evolution. Non-normality amplifies non-monotonicity but is not strictly required; geometric positioning of initial states and observables can generate transient departures from equilibrium. The phenomenon is accentuated in structured networks (e.g., linear SCC chains in hydrogen combustion vs. giant component graphs in unstructured networks) (Nicolaou et al., 2020).

5. Non-Monotonic Metric Functions in Inference and Machine Learning

Statistical inference and machine learning leverage or encounter non-monotonic metric functions in distributional characterizations and learning algorithms:

• Metric Distribution Functions (MDF): In non-Euclidean metric spaces, the MDF

$$F_\mu^m(\mathbf{u}, \mathbf{v}) = \mu\big(\prod_{k=1}^K \bar{B}(u_k, d_k(u_k, v_k))\big)$$

is constructed via concentric metric balls, circumventing the absence of a natural monotone order. MDFs uniquely determine the measure under mild conditions and enable robust, nonparametric inference—such as homogeneity and mutual independence tests—on general spaces (Wang et al., 2021).

• Non-monotonic activation functions: Recent research has shifted towards leveraging non-monotonic neural activation functions (e.g., SiLU/Swish, GELU, Mish, SGELU, SSiLU, SMish, Sqish), motivated by both empirical performance and improved optimization landscapes. Non-monotonic functions typically take the form $f(x) = x \cdot s(\beta x)$ for $x < 0$ and $f(x) = x$ for $x \ge0$ (for “saturated” designs), preserving gradient flow for positive activations and capturing beneficial nonlinear features for negative regions (Wu, 2022, Chen et al., 2023, Biswas et al., 2023). The learnability of single neurons with such activations is ensured under a "dominating linear component" in the Hermite expansion, while the risk landscape remains benign. These designs result in improved classification accuracy, adversarial robustness, and generalization capacity—attributed to the interplay between non-monotonic flexibility and lossless positive propagation.

6. Non-Monotonicity in Evaluation Metrics and Applications

Classical monotonic metrics in evaluation (e.g., SNR, error rates) may not faithfully capture tradeoffs between artifact removal and signal preservation, especially in label-free scenarios:

• Area of the Continuous Contrast Curve (AOCC): For event camera denoising, AOCC is proposed as a non-monotonic, label-free evaluation metric. AOCC computes the area under the continuous contrast curve (CCC)—which plots contrast (computed from Sobel gradients on event frames) as the accumulation interval $\Delta t$ varies:

$A_c = \int_0^\tau C_{avg}(t)dt$

This metric captures the intrinsic trade-off—contrast rises to a peak and then falls as $\Delta t$ increases, penalizing both excessive noise removal and loss of edge-preserving events (Shi et al., 13 Jun 2024). This non-monotonic property allows AOCC to identify an optimal operational regime and provides a holistic, interpretable value not attainable by monotonic metrics.

7. Non-Monotonic Metric Functions in Complex Analysis and Functional Theory

Non-monotonicity manifests in mathematical analysis and functional spaces in both geometric and algebraic forms:

• Continuity in non-monotonic domains: Holomorphically contractible systems (e.g., Kobayashi, Carathéodory, Lempert, and Green pseudodistances) retain continuity under sequences of domains converging in the Hausdorff metric—even without inclusion monotonicity—provided all compacts are eventually included. The continuity of these intrinsic pseudodistances is crucial for domain approximation, complex geometry, and several complex variables (Lewandowski, 2015).
• Entire functions with non-monotonic quotients: In Laguerre–Pólya I class entire functions, coefficient sequences with non-monotonic second quotients (alternating between $a$ and $b$) still admit a sharp membership characterization: $f \in \mathcal{L}\mbox{-}PI$ iff there exists $z_0\in[-1,0]$ with $f(z_0)\leq 0$, generalizing classical results that presuppose monotonicity of coefficient ratios (Nguyen et al., 2021).

Summary Table: Classes and Properties of Non-Monotonic Metric Functions

Area	Manifestation of Non-Monotonicity	Core Reference(s)
Dynamics, chemostats	Non-monotonic kinetic response (Haldane)	(Rapaport et al., 2012)
Metrics, quasi-/φ-metrics	Triangle inequality loosened, φ-term	(Dautova et al., 2022Das et al., 2022)
Learning (neural nets)	Non-monotonic activations (GELU, SiLU, etc.)	(Wu, 2022Chen et al., 2023Biswas et al., 2023)
Statistical inference	Non-monotonic MDFs in non-Euclidean spaces	(Wang et al., 2021)
Quantification indexes	L1-/weighted distance, LOI/LOD/LOM, LOS	(1403.58411705.02742)
Evaluation metrics	AOCC (bell-shaped response curve)	(Shi et al., 13 Jun 2024)
Functional analysis	Non-monotonic coefficient ratios in entire fns	(Nguyen et al., 2021)

Non-monotonic metric functions represent a broad class of objects where the failure of monotonicity is either structural or exploited for superior performance, expressivity, or analytical reach. Their rigorous quantification, tailored generalizations, deep connections to convergence and stability theory, and their necessity in contemporary machine learning and statistical evaluation mark them as a crucial subject at the intersection of modern mathematics and data science.