Multimarginal Interpolants: Theory & Applications

• Multimarginal interpolants are mathematical constructs that form joint structures satisfying several prescribed marginal constraints, bridging probability distributions and logical formulas.
• They are implemented through efficient algorithms in both continuous domains—via optimal transport and generative modeling—and discrete domains like tree-structured SMT interpolation.
• This framework enables explicit solutions, flexible multi-way correspondences, and diverse applications including style transfer, data decorruption, and algorithmic fairness.

A multimarginal interpolant is a construct that interpolates among multiple prescribed marginals (probability distributions or logical formulas), yielding a joint structure that simultaneously respects each marginal constraint while exposing the internal multi-way correspondences. These interpolants arise in several fields, including optimal transport, generative modeling, and satisfiability modulo theories (SMT). The concept generalizes classical two-marginal interpolation to the multimarginal context, enabling explicit solutions, efficient algorithms, and novel forms of correspondence extraction. Recent work formalizes multimarginal interpolants both in continuous domains—via mass transportation and generative modeling—and in the discrete/logical regime through tree-structured interpolation algorithms for quantified SMT formulas.

1. Multimarginal Interpolants in Optimal Mass Transport

The classical multimarginal optimal transport problem seeks a joint measure $\mu$ on $X_1 \times \cdots \times X_N$ that attains given marginals and minimizes a cost functional. For example, in the explicit construction on $[0,1]^3$ with cost $c(x,y,z) = xyz$ and uniform marginals, the solution is supported on a closed set $S = M_x \cup M_y \cup M_z \cup M_2$, where the components $M_x$, $M_y$, $M_z$ are one-dimensional spikes and $M_2$ is a two-dimensional curved face defined by $xyz = c_0$ with $c_0 = \ell r^2$ and explicit $\ell, r$ values. Any (3,1)-stochastic measure $\mu$ supported on $S$ achieves the optimal value, while the corresponding dual potentials (denoted $\varphi, \psi, \theta$) are unique up to constants and constructed via an explicit switching function $\lambda(x)$ (Gladkov et al., 2018).

This construction exhibits the following salient features:

• Non-unique Primal Solution: Multiple distinct optimal measures $\mu$ exist, all supported on $S$ and maintaining the uniform projection property. Mass can be arbitrarily redistributed on $M_2$ subject to marginal constraints.
• Unique Dual Interpolant: The dual potentials, when fixed, serve as multimarginal interpolants in the sense that any compatible primal measure is algebraically optimal, but the internal interpolation structure—encoded in the distribution on $M_2$—is non-unique.
• Support with Non-constant Local Dimension: The optimal coupling is concentrated both on one-dimensional spikes and a two-dimensional sheet, reflecting highly nontrivial geometry.

2. Dynamical and Statistical Multimarginal Interpolation

In generative modeling, multimarginal stochastic interpolants generalize classical two-way interpolation to $K+1$ marginals. Points in the $K$-simplex $\Delta^K$ are used as interpolation parameters $\alpha$, and the barycentric stochastic interpolant is defined by $x(\alpha) = \sum_{k=0}^K \alpha_k x_k$, where $x_k$ are drawn from prescribed distributions $\rho_k$. Multi-way correspondences among marginals arise by coupling the full collection $(x_1, ..., x_K)$ and then extracting conditional mean fields $g_k(\alpha, x) = E[x_k|x(\alpha) = x]$ (Albergo et al., 2023).

Key aspects of this framework include:

• Continuity Equations: For each $k$, the joint density $\rho(\alpha, x)$ satisfies continuity equations linking derivatives in $\alpha_k$ to divergence operators in $x$, realized by conditional means.
• Transport PDEs on the Simplex: Interpolating along paths within the simplex leads to ODEs/SDEs with velocity fields $b(t, x)$ built from the $g_k$ fields, which admit Fokker–Planck/score-based diffusion generalizations.
• Variational Quadratic Regression: Fields $g_k$ (or parameterized neural nets) are learned as minimizers of quadratic losses over samples, ensuring the compatibility of multi-way correspondences.
• Extraction of Multi-way Correspondences: In cases where the coupling is of Monge form ($x_k = T_k(x_0)$), the learned field $g_k(\alpha, x)$ gives explicit multi-way mappings, and more generally nearest-neighbor matching yields soft correspondences for arbitrary couplings.

3. Tree-Structured Multimarginal Interpolation in Logic

In the SMT context, multimarginal interpolation is implemented via tree interpolation algorithms that compute interpolants from unsatisfiable proofs composed of resolution steps and quantifier instantiations (Henkel et al., 2023). Here, the tree structure generalizes traditional binary interpolation to arbitrary acyclic graphs with possibly many leaves, each corresponding to a formula partition.

Defining characteristics:

• Tree Interpolation Problem: For a tree-shaped DAG $(V,E)$ with leaf formulas $F(p)$ (for $p \in P \subset V$) and global unsatisfiability $\bigwedge_{p \in P} F(p) \vdash \bot$, a tree interpolant $I : V \to \text{Form}$ is assigned to every node so that root is $\bot$, leaves satisfy local entailment, and inductivity/symbol-conditions propagate up the tree.
• Coloring of Proof Literals: Literals are virtually assigned (colored) to interpolation partitions, enabling flexibility and procedural control in constructing interpolants.
• Quantifier Management via Flattening: Compound ground terms are flattened to fresh variables with auxiliary equations, facilitating partition-specific quantification (existential, universal, or inlined according to locality).
• Algorithmic Construction: The TreeInterpolate algorithm initializes leaf interpolants, performs internal resolution using McMillan’s rules, and systematically eliminates unsupported flattening variables with quantifier rules.
• Theory-specific Back-ends: Separate handling for equality with uninterpreted functions (EUF) and linear arithmetic (LA), including congruence, transitivity, Farkas, trichotomy rules, and Nelson–Oppen combination strategies.

4. Uniqueness and Non-uniqueness in Multimarginal Interpolants

Multimarginal interpolants exhibit a fundamental dichotomy between uniqueness in the dual potentials and non-uniqueness in the primal coupling:

• Dual Uniqueness: In transport, the dual potentials $(\varphi, \psi, \theta)$ are unique up to additive constants, as dictated by convexity/Legendre-transform analysis, and subject only to the constraint $\varphi + \psi + \theta \le c$ with tightness on the support $S$ (Gladkov et al., 2018).
• Multiple Optimal Primal Structures: The failure of the “twist” condition in $c(x, y, z) = xyz$ admits infinitely many optimal primal couplings $\mu$, permitting redistribution of mass on faces such as $M_2$ without violating marginal constraints.
• Encoding of Multi-way Interpolation: In practical applications, the choice of primal measure among admissible couplings reflects different internal multi-way matchings and can be exploited, e.g., for style transfer or fairness correction.

5. Algorithmic and Computational Aspects

Algorithmic frameworks for multimarginal interpolation share several strategies:

• Regression-based Learning: In generative modeling, vector-valued fields $g_k(\alpha, x)$ are learned by minimizing empirical quadratic losses, with the complexity scaling linearly in the number of marginals: learning $O(K)$ outputs vs.\ $O(K^2)$ for exhaustive pairwise flows (Albergo et al., 2023).
• Path Optimization: Once the multimarginal field is learned, optimal transport paths $\alpha(t)$ within the simplex can be optimized, reducing mean-squared path length by 20% over naive linear paths in experiments.
• Tree Traversal and Quantifier Handling: In SMT algorithms, proof nodes are processed for each interpolation partition, with complexity $O(|\text{proof}| \cdot |V|)$; quantifiers are inserted as necessary based on variable locality and partition sharing (Henkel et al., 2023).
• Theory Combination: Integration of EUF and LA theories is achieved by Nelson–Oppen style combination, inline handling of mixed equalities, and quantifier-minimizing techniques to avoid spurious bindings.

6. Applications and Experimental Evaluation

The multimarginal interpolant framework enables a broad range of practical and theoretical applications:

• All-to-All Style Transfer: Learned generative maps facilitate translation among multiple image domains (e.g., MNIST, AFHQ, Oxford-Flowers, CelebA); conditioning on simplex variables yields smooth morphing and semantic correspondence preservation (Albergo et al., 2023).
• Data Decorruption and Algorithmic Fairness: Multimarginal couplings of clean, corrupted, and de-biased datasets allow construction of joint samples for artifact correction and debiasing.
• SMT Interpolation for Quantified Formulas: Tree-structured interpolants handle arbitrary partitionings, support quantifiers, and are implemented in tools such as SMTInterpol for resolution-based proofs (Henkel et al., 2023).
• Quantitative Metrics: Functional improvements are documented via FID score reductions, enhanced LPIPS perceptual similarity, and reduced path integration costs, with visualization via simplex polygons and Petrie polygon schemes.

7. Theoretical Properties and Completeness

Underlying all multimarginal interpolation methods are detailed theoretical invariants:

• Complementary Slackness: In transport, the equivalence $f(x) + f(y) + f(z) = x y z$ on the support set ensures primal and dual optimality (Gladkov et al., 2018).
• Tree-Inductivity and Symbol Conditions: In logical interpolants, tree-inductive properties and symbol-matching constraints guarantee correctness and minimality for each interpolant.
• Completeness of Construction: The coloring-based tree interpolation algorithm is complete: for every unsatisfiability proof there exists a method to construct a valid multimarginal (tree) interpolant.
• Quantifier Control: The appearance of existential/universal quantifiers is explicitly governed by partition-locality of terms, with all quantifiers ranging over shared bridge variables.

This synthesis demonstrates how multimarginal interpolants provide explicit solutions, efficient algorithms, and flexible correspondences for multi-way interpolation problems across mathematical, algorithmic, and logical domains.