ARI solves a structured prediction problem over a bipartite graph between source and target field paths. The objective is to infer a globally consistent mapping via maximum a posteriori (MAP) inference in a constrained graphical model.

Problem setup

Let the sets of source and target field paths be:

$$ S = \{s_i\} \qquad T = \{t_j\} $$

A mapping is a binary relation $R$:

$$ R \subseteq S \times T $$

Define indicator variables:

$$ x_{s,t} \in \{0,1\}, \quad (s,t) \in S \times T $$

where $x_{s,t}=1$ means that the source $s$ and target $t$ fields are compatible.

Then, equivalently, we can define $R$ as:

$$ R = \{(s,t) \mid x_{s,t} = 1\} $$

MAP objective

ARI defines a Gibbs distribution over mappings:

$$ P(R) \propto \exp\left(-E(R)\right) $$

and seeks the MAP solution:

$$ R^* = \arg\min_R E(R) $$

Energy decomposition

The energy decomposes into unary and pairwise terms:

$$ E(x) = \sum_{(s,t)\in S\times T} \psi_u(s,t)\,x_{s,t}

• \sum_{(s,t)\neq(s',t')} \psi_p\big((s,t),(s',t')\big)\,x_{s,t}x_{s',t'} $$