# Correct Convergence of Min-Sum Loopy Belief Propagation in a Block Interpolation Problem

## TL;DR

This paper proves that Min-Sum Loopy Belief Propagation correctly converges to local solutions in a grid interpolation problem within a number of iterations proportional to the grid size, similar to trees.

## Contribution

It establishes the correct convergence time of Min-Sum LBP for grid graphs with one-run boundary conditions, extending tree convergence results to loopy graphs.

## Key findings

- LBP converges in 2N iterations on N×N grids with one-run boundary.
- Local solutions can be accurately computed using message passing.
- Convergence time matches that of tree structures.

## Abstract

This work proves a new result on the correct convergence of Min-Sum Loopy Belief Propagation (LBP) in an interpolation problem on a square grid graph. The focus is on the notion of local solutions, a numerical quantity attached to each site of the graph that can be used for obtaining MAP estimates. The main result is that over an $N\times N$ grid graph with a one-run boundary configuration, the local solutions at each $i \in B$ can be calculated using Min-Sum LBP by passing difference messages in $2N$ iterations, which parallels the well-known convergence time in trees.