Quadratic Bipartite Matching Loss

• Bipartite matching loss is a cost functional for optimal pairing of two independently sampled point clouds, defined by minimizing the average squared Euclidean displacement.
• The method employs a linearized electrostatic analogy where the optimal displacement field, given as the gradient of a potential, is computed using Green's functions.
• The framework quantifies finite-size effects and spatial correlations, revealing asymptotic scaling laws and logarithmic corrections across different dimensions.

The quadratic stochastic Euclidean bipartite matching loss is a cost functional arising in the optimal assignment between two independently sampled point clouds over a domain $\Omega \subset \mathbb{R}^d$. When each point cloud consists of $N$ points sampled according to a density $\rho(x)$ and $N \gg 1$, the objective is to determine a permutation mapping that minimizes the average squared Euclidean distance between matched pairs. This framework is closely related to stochastic optimal transport and extends Monge–Kantorovich theory by capturing finite-$N$ fluctuations through a linearized, electrostatic analogy—with explicit formulae for the expected optimal cost and the two-point correlation function of the matching field (Caracciolo et al., 2015).

1. Mathematical Formulation and Problem Structure

Consider a bounded domain $\Omega\subset\mathbb{R}^d$, or its d-dimensional flat torus version $\mathsf{T}^d = [0,1]^d$ with periodic boundary conditions. Two independent point sets, $R = \{r_i\}_{i=1}^N$ and $B = \{b_j\}_{j=1}^N$, are sampled i.i.d. from the density $\rho(x) > 0$ on $N$0. The assignment $N$1 solves: $N$2 The loss quantifies the minimal average squared displacement required to match sources to targets.

2. Empirical Measures and Continuum Limit

Define empirical measures: $N$3 and their difference $N$4. As $N$5, both $N$6 and $N$7 converge weakly to $N$8. This continuum perspective enables the reformulation of the matching loss using fields and measures, facilitating analytic derivations.

3. Electrostatic Analogy and Linearization

The optimal displacement field $N$9 transporting $\rho(x)$0 to $\rho(x)$1 is (in the weak sense) the gradient of a potential: $\rho(x)$2. The push-forward constraint linearizes, yielding the PDE: $\rho(x)$3 or periodic BC on $\rho(x)$4. Introducing the Green's function $\rho(x)$5 solving: $\rho(x)$6 the displacement field admits

$\rho(x)$7

4. Two-Point Correlation Function

Averaging over all random instantiations of $\rho(x)$8, the two-point correlation for the displacement field is

$\rho(x)$9

with

$N \gg 1$0

This yields (up to an $N \gg 1$1-dependent short-distance cutoff): $N \gg 1$2

5. Continuum Formula for Expected Optimal Cost

The average optimal matching loss in the continuum approximation uses the diagonal part of the correlation: $N \gg 1$3 which leads to the general formula: $N \gg 1$4 A short-distance cutoff proportional to the typical nearest-neighbor spacing $N \gg 1$5 in the integrals is required to regularize divergences.

6. Flat Hypertorus and Explicit Asymptotics

On $N \gg 1$6 with Poisson density ($N \gg 1$7), the Green function $N \gg 1$8 solves: $N \gg 1$9 The corresponding ansatz: $N$0 yields (mode-sum or zeta-function regularization):

• $N$1: $N$2, $N$3,
• $N$4: $N$5 for constant $N$6,
• $N$7: $N$8 as $N$9, with $\Omega\subset\mathbb{R}^d$0.

7. Generalizations and Theoretical Context

For nonuniform density $\Omega\subset\mathbb{R}^d$1 and general domains, correlation functions and expected loss follow the described Green-function prescription. Mean-field scaling $\Omega\subset\mathbb{R}^d$2 (with logarithmic correction for $\Omega\subset\mathbb{R}^d$3) is recovered. This approach extends Monge–Kantorovich theory to stochastic settings by modeling finite-$\Omega\subset\mathbb{R}^d$4 effects via a weakly linearized electrostatic analogy, capturing random fluctuations in discrete matching problems (Caracciolo et al., 2015).

A plausible implication is that this framework facilitates the systematic study of stochastic transport costs beyond mean-field, including spatial correlation structures and finite-size scaling in bipartite matching problems, with rigorous connection to probabilistic transport theory.