Logarithmically Enhanced Quadratic Divergence

• Logarithmically Enhanced Quadratic Divergence is a divergence measure that combines a quadratic kernel with logarithmic amplification to capture complex statistical and geometric interactions.
• It underpins methods in robust statistical inference and information geometry by introducing logarithmic corrections that adjust curvature and metric properties.
• Applications span robust estimation, quantum spectral analysis, group theory thresholds, and quantum gravity, offering practical insights into nontrivial asymptotic behaviors.

Logarithmically enhanced quadratic divergence refers to divergences or asymptotic scaling behaviors in which a fundamentally quadratic structure acquires logarithmic amplification—either in the context of statistical divergences, metric entropy, or physical observables—due to underlying geometric, algebraic, or spectral features. This phenomenon appears across diverse domains including robust statistics, quantum statistical mechanics, geometric information theory, and random group theory, often signaling a nontrivial interplay between quadratic and logarithmic mechanisms.

1. Definitions and Fundamental Construction

The canonical instantiation of the logarithmically enhanced quadratic divergence (LEQD) in robust statistics arises as a special case of the Logarithmic Super Divergence (LSD) family with parameter choice $\beta=1$. For densities $g$ and $f$ on a common support, the LEQD is given by

$$D_{\rm LEQ}(g,f) = \log\left[\frac{\int f^2 \int g^2}{\bigl(\int f\,g\bigr)^2}\right],$$

which is symmetric, non-negative, and vanishes iff $f=g$ almost everywhere—a direct consequence of the Cauchy–Schwarz inequality. This divergence can be seen as the logarithm of a quadratic (Pearson $\chi^2$-type) kernel and thus encodes a "logarithmic enhancement" atop the usual quadratic divergence (Maji et al., 2014).

2. Information Geometry and Statistical Robustness

The geometric structure induced by LEQD forms a dual flat (Bregman-type) geometry on spaces of densities. The induced Riemannian metric at a model $f_\theta$ relates to the Hessian of $\log\int f_\theta^2$ minus twice the derivative of $\log\int f_\theta g$ at $g=f_\theta$, aligning with Fisher information under regularity conditions. Minimum-LEQD estimation procedures exhibit bounded, redescending influence functions, conferring robustness to outliers and contamination. For the minimum-divergence estimator $\hat\theta$ based on LEQD, asymptotic normality holds with asymptotic covariance determined by the information and variability of quadratic forms (Maji et al., 2014). Simulation studies confirm high robustness of LEQD-based estimators, with minimal loss in efficiency for clean data and strong resilience under contamination.

3. Logarithmic Corrections in Divergence Geometry

The family of $L^{(\pm\alpha)}$-divergences from optimal transport interpolates between quadratic (Bregman) divergence and pure logarithmic divergence through a one-parameter deformation. For a smooth potential $\varphi$,

$$\mathbf{D}^{(\alpha)}[\xi:\xi'] = \frac{1}{\alpha}\log(1+\alpha D\varphi(\xi')\cdot(\xi-\xi')) - (\varphi(\xi)-\varphi(\xi')),$$

with $\alpha\to 0$ recovering the Bregman divergence and $\alpha=1$ yielding the original $L$-divergence. The Taylor expansion for small $\alpha$ explicitly displays the logarithmic correction: $$\mathbf{D}^{(\alpha)}[\xi:\xi'] = D\varphi(\xi')\cdot(\xi-\xi') - (\varphi(\xi)-\varphi(\xi')) - \frac{\alpha}{2}(D\varphi(\xi')\cdot(\xi-\xi'))^2 + \cdots.$$ This reflects how logarithmic enhancement alters the curvature of the induced geometry, creating dually projective flatness with constant negative curvature $-\alpha$ (Wong, 2017).

4. Spectral Theory and Logarithmic Boundary Laws

In quantum statistical mechanics, logarithmically enhanced quadratic functions govern entanglement area laws for fermionic systems in vanishing magnetic fields. The Hilbert-space trace of quadratic functions of Fermi projections,

$\mathrm{Tr}\,f(P_B(L\Lambda)),\quad f(t)=t(1-t),$

shows leading scaling

$$\mathrm{Tr}\,f(P_B(L\Lambda)) = a(f,\mu)|\partial\Lambda| L\ln L + o(L\ln L) \;\;\;\;\; (B=0),$$

and

$$\mathrm{Tr}\,f(P_B(L\Lambda)) = a(f,\mu)|\partial\Lambda| L\ln(\mu/B) + o(L\ln(\mu/B)) \;\;\;\;\; (L \gg 1/B).$$

Here $a(f,\mu)$ encodes geometric and spectral data, and the logarithmic enhancement over the quadratic area law is tied to the spectral asymptotics of Wiener–Hopf and Landau–Widom operators. The "enhancement" is thus a boundary-induced, geometry-dependent amplification fundamentally tied to the nonlocality of quadratic forms (Pfeiffer et al., 2023).

5. Logarithmically Enhanced Quadratic Divergence in Group Theory

A structurally novel instance occurs in random right-angled Coxeter groups and their associated square-graph percolation. For a random graph $\Gamma_{n,p}$, the emergence of a unique square-graph component spanning all vertices corresponds to a sharp threshold,

$p_c(n) = \frac{\sqrt{\sqrt{6} -2}}{\sqrt{n}},$

removing previous polylogarithmic uncertainty. The percolation of induced squares (i.e., 4-cycles) enables the group $W_{\Gamma_{n,p}}$ to become strongly algebraically thick of order 1, leading to exactly quadratic divergence of the associated metric spaces. The logarithmic "enhancement" here refers to the sharpness of the threshold—previous bounds had extra polylog$(n)$ factors, and their removal pinpoints the precise regime in which quadratic divergence sets in (Behrstock et al., 2020).

6. Physical Models: Logarithmic IR Enhancement in Quadratic Gravity

In four-derivative theories such as quadratic gravity, logarithmically enhanced infrared terms arise in loop corrections due to the $1/k^4$ propagator for the graviton. These IR log-enhanced quadratic corrections do not correspond to genuine RG running—they are process-dependent, basis-dependent, and gauge-dependent. Nevertheless, they represent true logarithmic amplification of quadratic asymptotics in physical observables (e.g., Green's functions and S-matrix elements). These IR log terms are distinct from universal UV $\beta$-functions, and their presence signals peculiar features of nonlocality and long-range propagation in higher-derivative field theories (Salvio et al., 11 Jul 2025).

7. Applications and Analytical Summary

Logarithmically enhanced quadratic divergences and associated area/divergence laws arise in:

• Robust statistical inference and hypothesis testing: LEQD provides a highly robust alternative to classical divergences and is instrumental in contamination-resistant estimation.
• Geometric information theory: $L^{(\pm\alpha)}$-divergences characterize statistical manifolds with nonzero curvature, connecting quadratic and logarithmic behaviors through an analytic deformation.
• Quantum many-body and spectral theory: Logarithmically enhanced scaling laws quantify leading asymptotics of entropy or energy in fermionic systems, especially near criticality or vanishing fields.
• Random group theory and geometric group theory: The precise percolation threshold for square-induced quadratic divergence demarcates phase transitions in group thickness and geometry.
• Quantum gravity and high-energy physics: IR log-enhanced corrections modify loop-integral behaviors in quadratic gravity, relevant for emergent macroscopic couplings.

In all settings, the essential feature is the interplay between quadratic structure and logarithmic enhancement, driven either by underlying geometry, spectral properties, algebraic connectivity, or field-theoretic propagator structure. As these mechanisms are illustrated in recent works (Maji et al., 2014, Wong, 2017, Pfeiffer et al., 2023, Behrstock et al., 2020, Salvio et al., 11 Jul 2025), logarithmically enhanced quadratic divergence constitutes a fundamental and unifying concept at the intersection of statistics, geometry, spectral analysis, and mathematical physics.