Local Dependence of Coefficients (LDC)

• Local Dependence of Coefficients (LDC) is a framework that constrains coefficient behavior through localized data such as initial segments, neighborhoods, or local conditions.
• In commutative algebra, LDC governs the bounded dependence of Hilbert coefficients in Hilbert–Samuel polynomials based on depth and initial coefficients, ensuring finiteness in function behavior.
• In high-dimensional statistics and algorithm design, LDC informs sparse covariance estimation, pointwise dependence measures, and partition-function approximations by enforcing local constraints for improved consistency and efficiency.

Searching arXiv for recent and foundational papers on Local Dependence of Coefficients and related uses of local dependence. Local Dependence of Coefficients (LDC) is used in several distinct senses in current research. In commutative algebra it denotes explicit bounded dependence of the last Hilbert coefficients on an initial segment of lower coefficients and the dimension; in ordered high-dimensional Gaussian models it denotes the constraint that each variable depends only on nearby predecessors, with row-specific bandwidth; and in recent zero-freeness-based algorithms it denotes divisibility statements showing that low-order coefficients of suitable ratio functions are insensitive to distant modifications (Dung et al., 2017, Yu et al., 2016, Shao et al., 8 Sep 2025). Closely related statistical literature studies pointwise local dependence coefficients for densities, copulas, lift functions, and quantile regions, so the expression names a family of locality principles rather than a single universal formalism (Sukeda et al., 2024, Dastbaravarde et al., 2024, Marcondes et al., 2019).

1. Recurrent locality patterns

Across these uses, the common structure is that a coefficient, coefficient sequence, or coefficient-like function is not treated as freely varying. Instead, its admissible behavior is constrained by a local datum: an initial segment of a Hilbert polynomial, a neighborhood in an ordered index set, a graph-theoretic distance, or a local conditioning region in probability.

Context	Object	Local statement
Commutative algebra	$e_i(I,M)$	$e_{d-t+1},\dots,e_d$ are bounded by $e_0,\dots,e_{d-t}$
Ordered Gaussian models	$a_{jl}$ in a Cholesky factor	$a_{jl}=0$ for $l<j-k_j$
Partition-function algorithms	$P_{G,m}^{m_1}(z)$	low-order coefficients are unchanged by distant modifications

The locality parameter changes with the subject. In Hilbert–Samuel theory it is depth $t$ and the initial block $e_0,\dots,e_{d-t}$; in ordered covariance estimation it is the row-wise bandwidth $k_j$; in zero-freeness-based algorithms it is graph distance, encoded through divisibility statements such as $e_{d-t+1},\dots,e_d$0. The same general pattern also appears in local dependence coefficients for distributions: one studies how pointwise dependence at $e_{d-t+1},\dots,e_d$1 or $e_{d-t+1},\dots,e_d$2 is determined by behavior in an infinitesimal neighborhood rather than by a single global scalar (Dung et al., 2017, Lee et al., 2021, Shao et al., 8 Sep 2025).

2. Hilbert coefficients and depth-sensitive dependence

In commutative algebra, LDC concerns the Hilbert coefficients of the Hilbert–Samuel polynomial. For a finitely generated $e_{d-t+1},\dots,e_d$3-module $e_{d-t+1},\dots,e_d$4 of dimension $e_{d-t+1},\dots,e_d$5 and depth $e_{d-t+1},\dots,e_d$6 over a Noetherian local ring $e_{d-t+1},\dots,e_d$7, with $e_{d-t+1},\dots,e_d$8 an $e_{d-t+1},\dots,e_d$9-primary ideal, the Hilbert–Samuel polynomial is written

$e_0,\dots,e_{d-t}$0

The coefficients $e_0,\dots,e_{d-t}$1 are the Hilbert coefficients. Writing

$e_0,\dots,e_{d-t}$2

Dung and Hoa prove that the last $e_0,\dots,e_{d-t}$3 coefficients are bounded in terms of the first $e_0,\dots,e_{d-t}$4 coefficients and the dimension. In the $e_0,\dots,e_{d-t}$5-adic case, for every $e_0,\dots,e_{d-t}$6 with $e_0,\dots,e_{d-t}$7,

$e_0,\dots,e_{d-t}$8

More generally, for a good $e_0,\dots,e_{d-t}$9-filtration $a_{jl}$0 with reduction number $a_{jl}$1,

$a_{jl}$2

This is a depth-sensitive locality statement: once $a_{jl}$3, $a_{jl}$4, and, in the filtered setting, $a_{jl}$5 are fixed, the remaining coefficients cannot vary arbitrarily. The paper also proves a finiteness theorem: fixing $a_{jl}$6 and positive integers $a_{jl}$7, there are only finitely many Hilbert–Samuel functions $a_{jl}$8 occurring for pairs $a_{jl}$9 with $a_{jl}=0$0, $a_{jl}=0$1, and $a_{jl}=0$2 for $a_{jl}=0$3. The proof runs through bounds on $a_{jl}=0$4, superficial elements, local cohomology estimates, and bounds on lengths of Artinian quotients. The threshold $a_{jl}=0$5 is also shown to be essentially sharp: examples demonstrate that one cannot generally reduce the number of independent controlling coefficients below $a_{jl}=0$6 (Dung et al., 2017).

3. Ordered high-dimensional Gaussian models

In high-dimensional statistics, LDC refers to ordered dependence encoded through the modified Cholesky decomposition. For $a_{jl}=0$7, one writes

$a_{jl}=0$8

where $a_{jl}=0$9 is strictly lower triangular and $l<j-k_j$0 is diagonal. Local dependence means that row $l<j-k_j$1 of $l<j-k_j$2 has a row-specific bandwidth $l<j-k_j$3, so that $l<j-k_j$4 for all $l<j-k_j$5; equivalently,

$l<j-k_j$6

Yu and Bien estimate this structure through a convex penalized Gaussian likelihood in the inverse Cholesky factor $l<j-k_j$7, using a hierarchical group lasso penalty that forces the nonzeros in each row to form a contiguous block near the diagonal. The optimization decomposes into independent row-wise subproblems, solved by ADMM, and yields a sparse, symmetric, positive definite estimator $l<j-k_j$8. Under irrepresentability and signal-strength conditions, the method achieves signed support recovery and estimation consistency rates in multiple norms that are described as being as mild as those in a regression problem (Yu et al., 2016).

A Bayesian formulation replaces penalized likelihood by priors on the row bandwidths $l<j-k_j$9, the local regression coefficients $P_{G,m}^{m_1}(z)$0, and the residual variances $P_{G,m}^{m_1}(z)$1. The prior

$P_{G,m}^{m_1}(z)$2

penalizes larger neighborhoods, while a Zellner $P_{G,m}^{m_1}(z)$3-prior centered at the OLS estimator gives closed-form posteriors under an $P_{G,m}^{m_1}(z)$4-fractional likelihood. Posterior inference is row-wise, fully parallelizable, and does not require MCMC. Under eigenvalue, beta-min, and bandwidth-growth conditions, the posterior consistently recovers the true vector of local neighborhood sizes $P_{G,m}^{m_1}(z)$5 and achieves posterior contraction rates for $P_{G,m}^{m_1}(z)$6 that are nearly or exactly minimax optimal under the corresponding norms (Lee et al., 2021).

4. Pointwise dependence coefficients in statistics

A related but distinct statistical use of local dependence studies pointwise dependence coefficients rather than sparse regression coefficients. For a copula density $P_{G,m}^{m_1}(z)$7, the classical local dependence function is

$P_{G,m}^{m_1}(z)$8

Because this is not invariant under marginal transformations at the level of a joint density, the relative local dependence is defined by

$P_{G,m}^{m_1}(z)$9

This quantity is invariant under strictly increasing transformations of the margins. It also yields PDE characterizations of copulas: the Frank copula satisfies

$t$0

so its relative local dependence is constant, while the FGM copula satisfies

$t$1

The paper also studies the general PDE class $t$2 and relates it to Frank and FGM copulas (Sukeda et al., 2024).

Quantile dependence coefficients localize dependence in quantile space rather than at density points. For continuous $t$3, the coefficient

$t$4

is defined as the limit of the conditional probability that $t$5 lies in a shrinking quantile band around $t$6, given that $t$7 lies in a shrinking quantile band around $t$8. Tail dependence appears as the boundary case: $t$9 These coefficients are local in the sense of shrinking quantile neighborhoods. The paper shows, for example, that the Gaussian copula has $e_0,\dots,e_{d-t}$0 for all $e_0,\dots,e_{d-t}$1, whereas the Student $e_0,\dots,e_{d-t}$2 copula has positive tail dependence but zero quantile dependence at every finite interior $e_0,\dots,e_{d-t}$3 (Dastbaravarde et al., 2024).

The Lift Function provides another local dependence scale. Under $e_0,\dots,e_{d-t}$4, where $e_0,\dots,e_{d-t}$5 is the joint law and $e_0,\dots,e_{d-t}$6 is the product of marginals, it is defined by

$e_0,\dots,e_{d-t}$7

Independence is equivalent to $e_0,\dots,e_{d-t}$8, and mutual information becomes

$e_0,\dots,e_{d-t}$9

The construction is extended to many singular distributions by replacing the ordinary density with a scaled local density $k_j$0 derived from small-ball asymptotics. In another line of work, the multivariate local dependence function

$k_j$1

is defined from normalized products of residuals after subtracting conditional expectations, and in the three-variate case it satisfies $k_j$2 and vanishes under joint independence (Marcondes et al., 2019, Bayramoglu et al., 2024).

5. Divisibility, zero-freeness, and spatial mixing

A recent and explicitly coefficient-level formulation of LDC appears in zero-freeness-based algorithms for the ferromagnetic Ising model. The key objects are ratios of partition functions under partial evaluations of edge activities. For suitable partial evaluations $k_j$3, the modified LDC statement is

$k_j$4

meaning that the Taylor expansions at $k_j$5 agree through degree $k_j$6. The proof is based on a new division relation, exemplified by

$k_j$7

obtained by a bijection on pairs of spin configurations. Low-order coefficients are therefore local: distant edge modifications cannot affect them.

This coefficient locality is separated from zero-freeness. Zero-freeness in the Lee–Yang region is enough to obtain a Weitz-type FPTAS for the ferromagnetic Ising model across the entire Lee–Yang zero-free region, without relying on strong spatial mixing. The algorithm writes $k_j$8 as a telescoping product of edge-deletion ratios, approximates each ratio by a truncated Taylor series, and computes the needed coefficients on a truncated self-avoiding walk tree. LDC is not needed for the FPTAS itself, but it becomes decisive for deriving strong spatial mixing: LDC gives exact agreement of low-order coefficients between the full graph ratio and its local truncation, while zero-freeness and Montel-type bounds convert that agreement into exponential decay with distance. The result is a generalized edge-SSM for Ising and, via the random-cluster representation, the first SSM result for the random cluster model on general graphs, beyond lattices. The same program is then carried out for hypergraph independence, binary symmetric Holant, and Potts models, showing that LDC is a combinatorial property independent of zero-freeness, while zero-freeness plus LDC yields correlation decay (Shao et al., 8 Sep 2025).

6. Adjacent formal frameworks

Outside the settings where the term LDC is explicit, several adjacent formalisms impose local dependence on coefficients, variables, or operators. In elliptic PDE, local regularity is quantified in terms of the local summability of the coefficient matrix $k_j$9 and the nonhomogeneous term $e_{d-t+1},\dots,e_d$00. For

$e_{d-t+1},\dots,e_d$01

with ellipticity controlled by measurable functions $e_{d-t+1},\dots,e_d$02, the paper proves local boundedness, Harnack inequality, and local Hölder continuity under

$e_{d-t+1},\dots,e_d$03

and shows by counterexamples that the threshold $e_{d-t+1},\dots,e_d$04 and the condition $e_{d-t+1},\dots,e_d$05 are essentially sharp. The resulting estimates are local in the precise sense that $e_{d-t+1},\dots,e_d$06, the Harnack constant, and the Hölder modulus depend only on $e_{d-t+1},\dots,e_d$07, $e_{d-t+1},\dots,e_d$08, and the local quantity $e_{d-t+1},\dots,e_d$09 built from the $e_{d-t+1},\dots,e_d$10-norm of $e_{d-t+1},\dots,e_d$11 and the $e_{d-t+1},\dots,e_d$12-norm of $e_{d-t+1},\dots,e_d$13 on $e_{d-t+1},\dots,e_d$14 (Guo, 2024).

Probability theory uses a different locality language. One line of work distinguishes local dependence (LD1), graphical dependence (GD), and hypergraph dependence (HD). The paper proves that the usual definition of local dependence does not imply concentration for general Hamming Lipschitz functions, while hypergraph dependence implies concentration if the maximal neighborhood size is small; under $e_{d-t+1},\dots,e_d$15, Hamming-Lipschitz and Talagrand-type inequalities acquire variance factors proportional to $e_{d-t+1},\dots,e_d$16 (Paulin, 2012). Another line establishes Cramér-type moderate deviations for sums $e_{d-t+1},\dots,e_d$17 under local dependence neighborhoods $e_{d-t+1},\dots,e_d$18, overlap complexity

$e_{d-t+1},\dots,e_d$19

and local exponential moments

$e_{d-t+1},\dots,e_d$20

The resulting theorem gives

$e_{d-t+1},\dots,e_d$21

on an explicit range of $e_{d-t+1},\dots,e_d$22, thereby extending optimal i.i.d.-type moderate deviation behavior to locally dependent fields and combinatorial CLTs (Liu et al., 2021).

Algebra and logic provide further formalizations. A vector space $e_{d-t+1},\dots,e_d$23 is locally linearly dependent if every vector $e_{d-t+1},\dots,e_d$24 is annihilated by some non-zero operator in $e_{d-t+1},\dots,e_d$25; by duality, classifying such spaces is equivalent to classifying matrix spaces of bounded rank. The paper develops this correspondence, identifies operator spaces of the alternating kind as the dominant large-range case, classifies all 4-dimensional LLD operator spaces over fields with more than 3 elements, and derives improved upper bounds on maximal rank in minimal LLD spaces (Pazzis, 2013). In logic, the base system LFD has dependence atoms $e_{d-t+1},\dots,e_d$26 and modalities $e_{d-t+1},\dots,e_d$27, expressing local dependence between variables and local dependence of statements on variables. Through translations between LFD and the guarded fragment GF, satisfiability in LFD is shown to be ExpTime-complete when the variable set is fixed and 2ExpTime-complete when it is part of the input, while Craig interpolation and finite model properties transfer between the two settings (Benthem et al., 2022).