Various alignment problems arising in cryo-electron microscopy, community detection, time synchronization, computer vision, and other fields fall into a common framework of synchronization problems over compact groups such as Z/L, U (1), or SO(3). The goal of such problems is to estimate an unknown vector of group elements given noisy relative observations. We present an efficient iterative algorithm to solve a large class of these problems, allowing for any compact group, with measurements on multiple 'frequency channels' (Fourier modes, or more generally, irreducible representations of the group). Our algorithm is a highly efficient iterative method following the blueprint of approximate message passing (AMP), which has recently arisen as a central technique for inference problems such as structured low-rank estimation and compressed sensing. We augment the standard ideas of AMP with ideas from representation theory so that the algorithm can work with distributions over compact groups. Using standard but non-rigorous methods from statistical physics we analyze the behavior of our algorithm on a Gaussian noise model, identifying phases where the problem is easy, (computationally) hard, and (statistically) impossible. In particular, such evidence predicts that our algorithm is information-theoretically optimal in many cases, and that the remaining cases show evidence of statistical-to-computational gaps. Among the most common data problems in the sciences is that of recovering low-rank signal present in a noisy matrix. The standard tool for such problems is principal component analysis (PCA), which estimates the signal by the top eigenvectors. One example out of many is in macroeconomics, where large, noisy correlation matrices reveal useful volatility and yield predictions in their top eigenvectors [LS91, ELW10]. However, many particular applications involve extra structure such as sparsity in the signal, and this structure is ignored by conventional PCA, leading to sub-optimal estimates. Thus, a major topic of recent interest in * The first two authors contributed equally. †
REF proposes an approximate message passing (AMP) algorithm, and analyzes its behavior by identifying phases where the problem is easy, computationally hard, and statistically impossible.
Message-passing algorithms for synchronization problems over compact groups
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Computer Science Mathematics