Entropic independence via sparse localization

Abstract: Entropic independence is a structural property of measures that underlies modern proofs of functional inequalities, notably (modified) log-Sobolev inequalities, via ``annealing'' or local-to-global schemes. Existing sufficient criteria for entropic independence typically require spectral independence and/or uniform bounds on marginals under \emph{all} pinnings, which can fail in natural canonical-ensemble models even when strong mixing properties are expected. We introduce \emph{sparse localization}: a restricted localization framework, in the spirit of Chen--Eldan, in which one assumes $\ell_2$-independence only for a sparse family of pinnings (those fixing at most $cn$ coordinates for any $c > 0$), yet still deduces quadratic entropic stability and entropic independence with an explicit multiplicative loss of order $c^{-1}$. As an application, we give a rigorous proof of approximate conservation of entropy for the uniform distribution on independent sets of a given size in bounded degree graphs.

• The paper's main contribution is establishing that sparse localization suffices to guarantee entropic independence under targeted pinnings with bounded marginals.
• It introduces quadratic entropic stability and a sparse convex duality argument to derive explicit KL divergence bounds linking probability distributions.
• The methodology overcomes previous limitations by enabling rapid mixing and concentration results in conservative combinatorial sampling problems.

Entropic Independence via Sparse Localization

Introduction and Context

The paper "Entropic independence via sparse localization" (2604.10902) addresses a central issue in the study of high-dimensional discrete probability measures: obtaining entropic independence, a structural property that underlies proofs of functional inequalities such as modified log-Sobolev inequalities (mLSIs). mLSIs are widely used to establish concentration phenomena, mixing times, and stability properties for Markov chains over combinatorial structures, particularly in settings where state spaces are inherently constrained (such as the $k$-slice $\binom{[n]}{k}$ representing subsets of $k$ elements).

A prevailing approach for proving mLSIs involves an annealing scheme—connecting a complex measure to a tractable one via a sequence of interpolations and establishing that entropy contracts controllably along this sequence. The key technical step (step (i)) is a reduction to a so-called "easy regime" by proving an entropy contraction or factorization property, which is most succinctly captured by the notion of entropic independence. Existing paradigms for proving entropic independence critically rely on control of measures under all possible pinnings or conditionalizations; for many models of practical and theoretical interest, such as canonical-ensemble models with conservative dynamics, this uniformity is unattainable or at least technically prohibitive, even when global strong mixing or expansion properties are present.

Sparse Localization: Concept and Formalization

This work introduces and operationalizes the notion of sparse localization as a restricted alternative to full localization. Instead of requiring operator norm bounds (i.e., $\ell_2$-independence) for the influence matrix under all pinnings, the paper posits that it suffices to control this norm only for pinnings that fix at most a $c$-fraction of the coordinates, for any $c>0$. This constraint aligns with the subset of pinnings visited by the annealing path in applications, sidestepping pathological or degenerate conditionalizations that obviate global geometric structure.

The core formal result states:

If a measure $\nu$ on $\binom{[n]}{k}$ satisfies a marginal boundedness condition (marginals $\geq b > 0$) and is $\alpha$-$\binom{[n]}{k}$0-independent under all sparse pinnings (those that fix $\binom{[n]}{k}$1 coordinates), then $\binom{[n]}{k}$2 is entropically independent, with a multiplicative loss of order $\binom{[n]}{k}$3 in the constant.

That is, for all probability measures $\binom{[n]}{k}$4 on $\binom{[n]}{k}$5,

$\binom{[n]}{k}$6

where $\binom{[n]}{k}$7 are the normalized 1-site marginals.

This statement, Theorem 1 in the paper, generalizes the Chen–Eldan framework and makes the method viable in cases where strong mixing is accessible only along specific, restricted localization paths.

Technical Contributions

Quadratic Entropic Stability via Sparse Pinning

A central element is a quadratic stability result (Theorem 2), which provides a bound on the squared $\binom{[n]}{k}$8-distance between the mean vectors $\binom{[n]}{k}$9 and $k$0 of two probability distributions on the hypercube: $k$1

The proof follows the Chen–Eldan localization martingale construction, but with two substantial modifications:

1. Restricted Localization Martingale: The path-wise evolution of the annealing process is constrained to only allow sparse pinning directions, controlled by the downward closure of the set of sign vectors corresponding to sparse vectors. This ensures that only measures with preserved independence and regularity are encountered.
2. Sparse Convex Duality Argument: The variational characterization of KL divergence is only over sparse linear test functions, which geometrically leads to a comparison using the Ky Fan $k$2-norm of the difference in means (with $k$3), and introduces a factor of $k$4. The precise technical handling, including the computation of the sparse quadratic conjugate and use of matrix norm inequalities, is detailed and extends beyond full support versions of the localization argument.

The paper also clarifies the relationship between spectral independence (correlation matrix operator norm) and $k$5-independence (influence matrix operator norm), demonstrating comparability under lower bounds on coordinate variances.

Application to Conservative Combinatorial Sampling

The practical utility of the sparse localization theory is demonstrated in the context of the "down-up walk" on independent sets of fixed size in bounded-degree graphs—a crucial model for Markov chain Monte Carlo under hard constraints (such as the uniform measure on $k$6-element independent sets, important for sampling and approximate counting). Prior approaches, dependent on full spectral independence under all pinnings, fail for this setting due to the emergence of pinnings with extremal marginals.

The newly established theory fills this gap: independence and marginal bounds are verified only for sparse pinnings, and the entropic independence—and consequently, entropy conservation under annealing—is rigorously proved in this regime. This completes and corrects an open step in earlier literature claims and yields sharp bounds for mixing and stability in conservative dynamics, with explicit dependence on parameters such as graph degree and independence set density.

Strong Numerical and Structural Claims

• The entropic independence constant, $k$7, has explicit dependence on the sparsity parameter $k$8, operator norm bound $k$9, and marginal lower bound $\ell_2$0 as $\ell_2$1. This allows for fine-grained quantitative estimates in applications.
• The factor $\ell_2$2 is shown to be the geometric price of restricting to sparse localizations, through explicit comparison between Ky Fan and $\ell_2$3 norms.
• The method is robust under higher-degree graphs and for independent set sizes in the "algorithmic tractability" regime, with constants depending explicitly on graph parameters.

Theoretical Implications

The principal theoretical implication is that the property of entropic independence can be certified under much weaker, application-driven hypotheses than previously considered. This decouples much of the complexity inherent in the structure of high-dimensional discrete measures from the technical machinery previously used to analyze them. It also refines the understanding of when log-Sobolev-type inequalities can be expected to hold, and opens new directions for analyzing mixing times and concentration without requiring uniformity over all pinnings or adversarial conditionings.

Moreover, the method suggests that in constrained systems, where annealing or other local-to-global methods are the only tractable analytic routes, one need only study independence properties along a realistic sample path, rather than the much larger universe of all possible conditionalizations.

Practical Implications and Future Directions

From an algorithmic perspective, the results validate and reinforce subset-localization-based analyses for rapid mixing and efficient sampling protocols in constrained combinatorial systems—particularly relevant for algorithms for approximate counting in the presence of conservation laws or fixed-size constraints.

Potential future developments include:

• Extension of sparse localization to models with additional forms of symmetry, such as graphical models with more intricate dependency structures.
• Development of refined sparse-convex-analytic tools for other functional inequalities (e.g., transportation or concentration inequalities beyond quadratic stability).
• Empirical investigation and numerical simulation to confirm that the theoretical constants yield practical improvements in the analysis or design of MCMC algorithms in high-dimensional, constrained regimes.
• Application of the theory to other ensembles in statistical physics and discrete probability where full localization is infeasible.

Conclusion

"Entropic independence via sparse localization" (2604.10902) advances the functional analysis of high-dimensional combinatorial probability measures by developing a localization-based framework that only requires local (sparse) independence properties. By replacing the stringent "all-pinnings" requirement with sparse localization, the paper provides both theoretical innovation and practical applicability, opening new possibilities for studying mixing, concentration, and stability in settings where full operator control is unattainable. The application to conservative Markov chains on fixed-size independent sets serves as a case study for the impact of this localization principle on algorithmic and probabilistic analysis.