A Probabilistic Approach to Pose Synchronization for Multi-Reference Alignment with Applications to MIMO Wireless Communication Systems

TL;DR

The paper tackles multi-reference alignment under unknown transformations by reframing alignment from absolute poses to relative poses $\mathbf{R}_{ij} = \mathbf{P}_i^{-1}\mathbf{P}_j$ and enforcing cycle-consistency, enabling a probabilistic graphical model that marginalizes over nuisance poses. It introduces both a direct triplet-based estimation and an iterative EM-like refinement that denoises the channel estimates $\mathbf{H}'$ while jointly refining the relative poses; the Orthogonal Procrustes step provides closed-form, non-decreasing updates, and a Matrix-Normal prior supports correlation across blocks. The key contributions are (i) the triplet-based, decentralized synchronization framework that scales as $\mathcal{O}(D^3)$, (ii) the iterative refinement algorithm that yields denoised channels and improved synchronization, and (iii) theoretical justification of cycle-consistency linking relative poses to absolute pose equivalence classes. Empirical results on synthetic 5G-like MIMO data show consistently lower reconstruction error than pairwise methods, confirming the approach’s potential for robust, scalable MRA in communications and sensing contexts.

Abstract

From molecular imaging to wireless communications, the ability to align and reconstruct signals from multiple misaligned observations is crucial for system performance. We study the problem of multi-reference alignment (MRA), which arises in many real-world problems, such as cryo-EM, computer vision, and, in particular, wireless communication systems. Using a probabilistic approach to model MRA, we find a new algorithm that uses relative poses as nuisance variables to marginalize out -- thereby removing the global symmetries of the problem and allowing for more direct solutions and improved convergence. The decentralization of this approach enables significant computational savings by avoiding the cubic scaling of centralized methods through cycle consistency. Both proposed algorithms achieve lower reconstruction error across experimental settings.

Figures (4)

• Figure 1: Iterative refinement achieves the lowest reconstruction error compared to signal denoising only. This is a synthetic 6-by-6 grid of 12 measurements each. Error bars indicate standard error among 25 random seeds. Additionally, the 'synchronization base' approach, by itself, achieves lower reconstruction error than prior art for reasonable signal-to-noise ratios.
• Figure 2: The reconstruction error for varying number of iterative refinement steps. Most improvement over the direct approach (blue shaded region) is achieved in two to five steps.
• Figure 3: Numerical results along varying length scales, which determine the correlation properties. Our synchronization improves the channel reconstruction across these settings.
• Figure 4: Structured Probabilistic Graphical Model to illustrate the joint distribution in Equation \ref{['eqn:joint_distr']}