Generalized Logarithmic Sobolev Inequality

• Generalized Logarithmic Sobolev Inequality is an extension of classical LSI that replaces strict convexity with relaxed conditions, such as isotropy and diameter bounds, to control entropy via energy functionals.
• Techniques like stochastic localization and barrier functions enable precise control over moment dynamics and spectral norms, yielding improved convergence rates and mixing time estimates.
• The GLSI framework has practical implications for high-dimensional probability, providing sharp bounds for mixing times, concentration inequalities, and small-ball probabilities in complex settings.

A generalized logarithmic Sobolev inequality (GLSI) extends the classical logarithmic Sobolev inequality (LSI), which provides control of the entropy of a function by an appropriate "energy" (e.g., Dirichlet form, Fisher information, or difference operators) under a reference probability measure. The generalized form is tailored for measures or generators that lack strong convexity, permit weaker regularity, or arise from non-Euclidean geometry, subelliptic operators, or discrete/product structures. The GLSI framework is pivotal in quantifying convergence rates for Markov processes, deriving sharp concentration inequalities, and providing mixing and small-ball estimates for high-dimensional distributions beyond the reach of the classical (Gross) LSI.

1. Core Inequality and Main Theorem

A GLSI takes the general form

$\Ent_\mu(f^2) \leq C \int Q(f) \, d\mu,$

where $\Ent_\mu(f^2) := \int f^2 \log f^2\, d\mu - (\int f^2\, d\mu) \log \int f^2\, d\mu$ is the entropy, and $Q(f)$ is an energy-type functional encoding the regularity of $f$ under $\mu$. The constant $C$ depends explicitly on geometric, convexity, or structural features.

For isotropic log-concave densities $p$ on $\mathbb{R}^n$ with support of diameter $D$, the optimal GLSI is

$\Ent_p(f^2) \leq \frac{D}{c} \,\mathbb{E}_p[\|\nabla f\|^2]$

where $c$ is an absolute constant. Equivalently, the log-Sobolev constant $\rho_p$ satisfies

$\rho_p \geq \frac{c}{D}~,$

with no dependence on the ambient dimension $n$ in the leading order. This improves the previous estimate $\rho_p \gtrsim 1/D^2$ and is asymptotically tight (Lee et al., 2017).

2. Methodological Framework: Stochastic Localization and Barrier Techniques

The tight bound on the log-Sobolev constant for isotropic log-concave measures is achieved via the combination of:

• Stochastic Localization (Eldan SDE): Defines a diffusion-like stochastic flow $p_t$, formed by continuously tilting the measure with a Brownian driver, which preserves log-concavity and allows one to interpolate between the original measure and a limiting Gaussian-like regime.
• Stieltjes-type Barrier Functions: Sharp control of the moment/covariance structure of the flow is implemented using a Stieltjes barrier, ensuring the spectral norm of the covariance $\|A_t\|_{\text{op}}$ stays controlled up to a critical time scale.
• Log-Cheeger Constant: Links isoperimetric properties with Sobolev constants by bounding the log-Cheeger constant

$$\kappa_p = \inf_{S \subset \mathbb{R}^n} \frac{p(\partial S)}{ \min \{ p(S), p(\mathbb{R}^n \setminus S) \} \sqrt{ \log (1/p(S)) } }$$

and showing $\kappa_p \gtrsim 1/\sqrt{D}$, with $\rho_p = \Theta(\kappa_p^2) \gtrsim 1/D$ (Lee et al., 2017).

3. Generalizations of the Logarithmic Sobolev Inequality

GLSI extends the classical framework in multiple directions:

• Relaxed Convexity: Instead of global strong convexity, only isotropy and a diameter bound (no Hessian lower bound) are required.
• Weighted or Block-Structured Inequalities: For densities with weak interactions, one can derive weighted entropic inequalities (e.g., Marton's weighted relative-entropy, where local log-Sobolev constants of block conditionals are aggregated), leading to exponential mixing for weighted Gibbs samplers and recovery of the global log-Sobolev constant under suitable mixing conditions (Marton, 2012).
• Curvature-Dimension Criteria: GLSIs can be set under $CD(\rho,\infty)$ or more general Bakry–Émery-type curvature conditions, which permit nonuniform weights or growth conditions (as in generalized Cauchy measures) and yield entropy bounds with explicit, sometimes optimal, constants (Huguet, 2024).
• Subelliptic and Geometric Settings: On submanifolds and non-Euclidean spaces (e.g., Lie groups, subelliptic Laplacians, or submanifolds with mean curvature), GLSIs quantify entropy production through horizontal and vertical (curvature) components, often capturing geometric invariants in the constants (1908.10360).

4. Applications and Consequences

The improved GLSI for isotropic log-concave measures entails:

• Mixing Time Bounds: The ball walk with step size $\delta = O(1/\sqrt{n})$ mixes in $O(n^2 D)$ steps from any starting point, a significant improvement over the former $O(n^2 D^2)$ bound and shown to be optimal (Lee et al., 2017).
• Large Deviation and Concentration Inequalities: For $L$-Lipschitz functions $g$,

$$\Pr(|g(x) - \bar{g}| \geq c L t) \leq \exp\left( -\Omega\left( \frac{t^2}{t + \sqrt{n}} \right) \right),$$

and specifically for $\|x\|$, the probability tails decay as $\exp(-\Omega(\min\{t,t^2\}\sqrt{n}))$ (Lee et al., 2017).

• Small-Ball Probability Bounds: Using Cheeger-type inequalities, one derives dimension-free exponential decay for small-ball probabilities, matching the best known bounds from Paouris (Lee et al., 2017).
• Tensorization and Weak Interactions: Weighted inequalities imply fast convergence and concentration in systems with dependencies, such as lattice models or Gibbsian random fields, provided the inter-block Hessian is controlled in supremum norm (Marton, 2012).

5. Relationship to Classical and Modified Logarithmic Sobolev Inequalities

GLSI bridges and interpolates between classical Gross-style LSI and various modified or weighted forms:

• Sharpness: For the standard Gaussian, the optimal constant is recovered ($\rho_p=2$), attesting to the tightness of the result in the limit $D\to O(1)$.
• Classical Log-Sobolev for Strongly Log-Concave Measures: If $p$ is strongly log-concave (e.g., Hessian $\geq \Lambda I$), the classical Bakry–Émery approach yields the dimension-free constant $\frac{1}{2\Lambda}$, but fails for measures not satisfying strict convexity.
• Block-Tensorization and Weak Dependence: The methodology of weighted local log-Sobolev inequalities enables extension to high-dimensional dependent structures, with contraction in entropy and convergence to equilibrium for Markov generators with suitable block decomposition (Marton, 2012).
• Connection to Cheeger and Isoperimetry: The quantitative equivalence between log-Sobolev and isoperimetric inequalities (via the log-Cheeger constant) ensures that concentration and mixing phenomena are tightly linked to isoperimetric profiles in the underlying measure (Lee et al., 2017).

6. Analytical and Probabilistic Proof Ingredients

The proof structure includes:

• Stochastic Processes: The localization via stochastic differential equations (SDEs) coupled to the measure, tracking moments and covariance.
• Barrier Potentials and Spectral Control: Analysis of evolution and control of spectral norms of covariance operators through carefully constructed barrier functions (e.g., Stieltjes-type functionals).
• Log-Isoperimetric Profile Tracking: Showing that key isoperimetric quantities do not deteriorate under the stochastic localization up to crucial time scales, thereby preserving strong inequalities back at $t=0$.
• Time Reversal and Gaussian Interpolation: At each point in the localization process, the measure exhibits a Gaussian component, allowing direct invocation of optimal Gaussian inequalities and recursive interpolation.

7. Broader Impact and Extensions

The GLSI for isotropic log-concave measures and associated methodology mark a culmination of advances connecting stochastic process representation, geometric functional inequalities, and high-dimensional analysis:

• Provides essentially optimal bounds for a broad class of convex bodies and log-concave distributions, with immediate implications for algorithmic sampling, randomized numerical computation, and random matrix theory.
• The barrier approach and stochastic localization have been influential for later analytical works on concentration, mixing, and isoperimetry in nonstrongly convex scenarios.
• The invariance with respect to dimension in the leading constant and minimal geometric or regularity assumptions make this framework robust for diverse applications in convex geometry, Markov chain Monte Carlo, and high-dimensional probability.
• The techniques and resulting inequalities also serve as a prototype for more general non-Euclidean, weighted, subelliptic, and geometrically nontrivial spaces, leveraging curvature-dimension hypotheses, measure perturbations, and geometric flows.

References:

• "Stochastic Localization + Stieltjes Barrier = Tight Bound for Log-Sobolev" (Lee et al., 2017)
• "An inequality for relative entropy and logarithmic Sobolev inequalities in Euclidean spaces" (Marton, 2012)
• "Logarithmic Sobolev inequalities for generalised Cauchy measures" (Huguet, 2024)