Inference on graphs via semidefinite programming

Abstract

Inference problems on graphs arise naturally when trying to make sense of network data. Oftentimes, these problems are formulated as intractable optimization programs. This renders the need for fast heuristics to find adequate solutions and for the study of their performance. For a certain class of problems, Javanmard et al. (1) successfully use tools from statistical physics to analyze the performance of semidefinite programming relaxations, an important heuristic for intractable problems. A particularly interesting class of inverse problems in graphs is that of synchronization problems (2, 3). These include registration of multiple pictures of the same scene, alignment of signals, community detection, and many others. After associating objects (for example, images or signals) to nodes of a graph, the goal is to estimate labels of the nodes (viewing direction of the image, shift of the signal, or community membership) from pairwise information about labels of nodes sharing edges (obtained, for example, by comparing two images). It is productive to think of the labels as being in a group (for example, of transformations) and the pairwise information being about group ratios (or relative transformations). An important example is synchronization of 3D rotations, which is a crucial step in the reconstruction problem in cryoelectron microscopy (4). Community detection under the binary stochastic block model has recently received significant attention (5⇓⇓⇓⇓–10). In this model, a random graph is drawn on vertices belonging to … [↵][1]1Email: bandeira{at}mit.edu. [1]: #xref-corresp-1-1