Log-Supermodular Functions: Theory and Applications

• Log‐supermodular functions are strictly positive functions on product lattices that satisfy the inequality f(x)f(y) ≤ f(x∧y)f(x∨y), linking them to submodularity.
• They underpin attractive graphical models, ensuring that variational methods like belief propagation provide rigorous lower bounds for partition functions.
• Recent advances explore their structural complexity and convolution properties, broadening applications in image denoising, combinatorial optimization, and robust probabilistic modeling.

A log-supermodular (LSM) function is a strictly positive function $f$ on a product lattice (e.g., $\{0,1\}^n$ or $\mathbb{R}^d$) that satisfies the fundamental inequality $f(x)\, f(y) \leq f(x \wedge y)\,f(x \vee y)$ for all arguments $x, y$, where the infimum/supremum is taken coordinatewise. This property is equivalent to the submodularity of $V = -\log f$, which connects the theory of LSM functions directly to the combinatorics of submodular set functions, graphical models, and important families of correlations in probability and information theory. LSM functions are central to the analysis of attractive graphical models, statistical physics, variational inference, and robust probabilistic modeling.

1. Formal Definitions and Equivalent Characterizations

For a discrete lattice $\{0,1\}^n$, a function $\varphi:\{0,1\}^n \to \mathbb{R}_+$ is log-supermodular if

$$\varphi(x)\, \varphi(y) \leq \varphi(x \wedge y)\, \varphi(x \vee y).$$

Equivalently, in logarithmic coordinates,

$$\log \varphi(x) + \log \varphi(y) \leq \log \varphi(x\wedge y) + \log \varphi(x\vee y).$$

A global function $$f(x) = \prod_i D_i(x_i) \prod_\alpha \psi_\alpha(x_\alpha)$$ is said to admit a log-supermodular factorization if every local factor $\psi_\alpha$ is log-supermodular. For real lattices $\mathbb{R}^d$, the definition carries over verbatim for $f: \mathbb{R}^d \to (0,\infty)$.

A submodular function $f_\theta$ on $\{0,1\}^n$ satisfies

$$f_\theta(x) + f_\theta(y) \geq f_\theta(x \wedge y) + f_\theta(x \vee y).$$

Log-supermodular distributions are then defined via $p_\theta(x) = \exp(-f_\theta(x)) / Z(\theta)$, where $f_\theta$ is submodular and $Z(\theta)$ ensures normalization (Shpakova et al., 2016).

These properties align LSM functions with the multivariate total positivity (MTP$_2$) property in continuous domains, and with “attractive” or ferromagnetic pairwise interactions in binary models (Ruozzi, 2012, Madiman et al., 22 Dec 2025).

2. Fundamental Inequalities and Closure Properties

A central technical tool for LSM functions is the Ahlswede–Daykin “four functions” theorem and its log-supermodular extensions. For $f_i \geq 0$,

$$f_1(x)\, f_2(y) \leq f_3(x\wedge y)\, f_4(x\vee y) \implies \int f_1\, \int f_2 \leq \int f_3\, \int f_4.$$

Ruozzi introduced the “log-supermodular $2k$-functions” inequality: for $f_1, ..., f_k : \{0,1\}^n \to \mathbb{R}_+$ and log-supermodular $g(\cdot)$,

$$g(x^1,\dots,x^k) \leq \prod_{i=1}^k f_i(z^i(x^1,\dots,x^k)) \implies \sum_{x^1,\dots,x^k} g(x^1,\dots,x^k) \leq \prod_{i=1}^k \sum_x f_i(x).$$

A key supporting result is that marginals of LSM functions remain LSM, enabling coordinatewise induction (Ruozzi, 2012).

Closure under convolution is recently established for log-concave product densities: $$(f * g)(x) = \int_{\mathbb{R}^d} f(z)\, g(x-z)\, dz$$ is LSM if $g$ is a product of one-dimensional log-concave densities and $f$ is LSM. The proof reduces to verifying convexity-induced inequalities in one dimension and leverages a “four-function” type argument (Madiman et al., 22 Dec 2025).

3. Variational Inference and Partition Function Bounds in LSM Models

In graphical models with binary variables and LSM potential functions, every fixed point of (sum–product) belief propagation yields a Bethe partition function $Z_B(G)$ which is rigorously a lower bound on the true partition function $Z(G)$: $Z_B(G) \leq Z(G).$ This bound is established by combining a graph cover perspective (lifting the problem to $k$-covers and bounding $Z(H) \leq [Z(G)]^k$) with the log-supermodular $2k$-functions inequality. The approach guarantees that variational approximations based on the Bethe free energy do not overestimate $Z(G)$ in attractive (log-supermodular) binary models (Ruozzi, 2012).

In log-supermodular distributions $p_\theta(x) \propto \exp(-f_\theta(x))$, the log-partition function $A(\theta) = \log Z(\theta)$ is generally intractable. Two key convex upper bounds are available:

• The base-polytope (Lovász extension) bound
• The “perturb-and-MAP” (logistic) bound, exploiting random noisy perturbations and MAP inference, which always dominates the L-field method in tightness (Shpakova et al., 2016).

Stochastic subgradient methods based on these surrogates enable efficient parameter learning and inference even with missing data.

4. Structural Complexity and Functional Clones

A significant structural result demonstrates that the algebraically generated clone of LSM functions using only binary implication and unary functions is a strict subset of all LSM functions. Let $$LSM = \{ F: \{0,1\}^k \to \mathbb{R}_{\ge 0} \mid F\ \text{is LSM}\}$$. It is shown that

$$\left\langle \{IMP\}\cup_1 \right\rangle_\omega \neq LSM$$

where $IMP$ is the binary implication and $\_1$ the set of all unary functions. This is established by constructing an explicit 4-ary LSM counterexample $S$ not contained in this clone, using a bilinear form $B(F;w)$ and closure properties under tensor and contraction (McQuillan, 2011).

This negative result implies that, contrary to previous conjecture, higher-arity primitives are necessary for full LSM expressive power, impacting the complexity analysis of #CSPs and the structure of counting dichotomies.

5. Information-Theoretic and Probabilistic Implications

Log-supermodularity yields new classes of entropy power inequalities (EPIs) beyond the regime of independence. In particular, if $(X,Y)$ is an $\mathbb{R}^2$-valued random vector with joint LSM density, then the “conditional” EPI

$N(X+Y) \geq N(X|Y) + N(Y|X)$

holds, where $N(\cdot)$ denotes entropy power. The proof leverages the preservation of LSM property along Ornstein–Uhlenbeck semigroups and shows that associated cross-terms are nonpositive due to componentwise submodularity (Madiman et al., 22 Dec 2025).

The convolution stability result ensures that log-concave product convolution maintains LSM property, encompassing standard Gaussian kernels and their discrete analogues. This consolidates the connection between mass transport, correlation inequalities, and information geometry in the study of dependency structures and inference under LSM constraints.

6. Applications and Model Learning

LSM functions underlie probabilistic models for image denoising, combinatorial optimization, and certain constraint satisfaction problems. In practical terms:

• LSM-based denoising in binary images achieves error rates of $0.4$–$4.2\%$ at moderate noise levels, outperforming structured-SVM baselines.
• The LSM framework enables efficient learning even in the presence of missing or noisy data by maximizing lower bounds on marginal likelihoods using perturb-and-MAP and stochastic subgradient descent (Shpakova et al., 2016).
• Canonical inference and counting tasks such as the enumeration of independent sets or vertex covers in bipartite graphs can be reframed into LSM models, with reparametrizations transforming originally log-submodular functions into log-supermodular form for guaranteeable lower bounds on partition functions (Ruozzi, 2012).

These advances illustrate the robustness and tractability benefits conferred by LSM structure in high-dimensional probabilistic modeling and statistical inference.

7. Open Problems and Future Directions

Despite deep connections to submodularity, total positivity, and variational approximations, numerous fundamental questions remain:

• Full structural characterization of LSM functional clones, including the minimal higher-arity generators required.
• Tight classification of tractable versus intractable counting problems within LSM-weighted CSPs, in light of the failure of generation by binary primitives (McQuillan, 2011).
• Extensions of LSM-based information inequalities to broader classes of dependent random variables, including beyond log-concave convolution kernels (Madiman et al., 22 Dec 2025).
• Deeper exploration of links between LSM preservation under convolution, optimal transport, and functional inequalities interpolating Prékopa–Leindler type theorems.

A plausible implication is that further progress in LSM theory will inform tractability frontiers in both approximate counting and inference, and yield novel inequalities for dependent structures in information theory and geometry.