Robust Group Synchronization via Cycle-Edge Message Passing

Foundations of Computational Mathematics https://doi.org/10.1007/s10208-021-09532-w Robust Group Synchronization via Cycle-Edge Message Passing Gilad Lerman1 · Yunpeng Shi2 Received: 8 January 2020 / Revised: 27 May 2021 / Accepted: 30 June 2021 © The Author(s) 2021 Abstract We propose a general framework for solving the group synchronization problem, where we focus on the setting of adversarial or uniform corruption and sufficiently small noise. Specifically, we apply a novel message passing procedure that uses cycle consistency information in order to estimate the corruption levels of group ratios and consequently solve the synchronization problem in our setting. We first explain why the group cycle consistency information is essential for effectively solving group synchronization problems. We then establish exact recovery and linear convergence guarantees for the proposed message passing procedure under a deterministic setting with adversarial corruption. These guarantees hold as long as the ratio of corrupted cycles per edge is bounded by a reasonable constant. We also establish the stability of the proposed procedure to sub-Gaussian noise. We further establish exact recovery with high probability under a common uniform corruption model. Keywords Group synchronization · Robust estimation · Exact recovery · Message passing Mathematics Subject Classification 90-08 · 62G35 · 68Q25 · 68W40 · 68Q87 · 93E10 Communicated by Thomas Strohmer. This work was supported by NSF award DMS-1821266. B Gilad Lerman Yunpeng Shi 1 School of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church Street SE, Minneapolis, MN 55455, USA 2 Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, USA 123 Foundations of Computational Mathematics 1 Introduction The problem of synchronization arises in important data-related tasks, such as structure from motion (SfM), simultaneous localization and mapping (SLAM), cryo-EM, community detection and sensor network localization. The underlying setting of the problem includes objects with associated states, where examples of states are locations, rotations and binary labels. The main problem is estimating the states of objects from the relative state measurements between pairs of objects. One example is rotation synchronization, which aims to recover rotations of objects from the relative rotations between pairs of objects. The problem is simple when one has the correct measurements of all relative states. However, in practice the measurements of some relative states can be erroneous or missing. The main goal of this paper is to establish a theoretically guaranteed solution for general compact group synchronization that can tolerate large amounts of measurement error. We mathematically formulate the general problem in Sect. 1.1 and discuss common special cases of this problem in Sect. 1.2. Section 1.3 briefly mentions the computational difficulties in solving this problem and the disadvantages of the common convex relaxation approach. Section 1.4 nontechnically describes our method, and Sect. 1.5 highlights its contributions. At last, Sect. 1.6 provides a roadmap for the rest of the paper. 1.1 Problem Formulation The most common mathematical setting of synchronization is group synchronization, which asks to recover group elements from their noisy group ratios. It assumes a group n and a graph G([n], E) with n vertices indexed by G, a subset of this group {gi∗ }i=1 [n] = {1, . . . , n}. The group ratio between gi∗ and g ∗j is defined as gi∗j = gi∗ g ∗−1 j . We use the star superscript to emphasize original elements of G, since the actual measurements can be corrupted or noisy. We remark that since g ∗ji = gi∗j −1 , our setting of an undirected graph, G([n], E), is fine. We say that a ratio gi∗j is corrupted when it is replaced by g̃i j ∈ G \ {gi∗j }, either deterministically or probabilistically. We partition E into the sets of uncorrupted (good) and corrupted (bad) edges, which we denote by E g and E b , respectively. We denote the group identity by eG . We assume a metric dG on G, which is biinvariant. This means that for any g1 , g2 , g3 ∈ G, dG (g1 , g2 ) = dG (g3 g1 , g3 g2 ) = dG (g1 g3 , g2 g3 ). We further assume that G is bounded with respect to dG , and we thus restrict our theory to compact groups. We appropriately scale dG so that the diameter of G is at most 1. Additional noise can be applied to the group ratios associated with edges in E g . For i j ∈ E g , the noise model replaces gi∗j with gi∗j gij , where gij is a G-valued random variable such that dG (gij , eG ) is sub-Gaussian. We denote the corrupted and noisy group ratios by {gi j }i j∈E and summarize their form as follows: 123 Foundations of Computational Mathematics  gi j = gi∗j gij , i j ∈ E g ; g̃i j , i j ∈ Eb . (1) We refer to the case where gij = eG for all i j ∈ E as the noiseless case. We view (1) as an adversarial corruption model since the corrupted group ratios and the corrupted edges in E b can be arbitrarily chosen; however, our theory introduces some restrictions on both of them. The problem of group synchronization asks to recover the original group elements {gi∗ }i∈[n] given the graph G([n], E) and corrupted and noisy group ratios {gi j }i j∈E . One can only recover, or approximate, the original group elements {gi∗ }i∈[n] up to a right group action. Indeed, for any g0 ∈ G, gi∗j can also be written as gi∗ g0 (g ∗j g0 )−1 and thus {gi∗ g0 }i∈[n] is also a solution. It is natural to assume that G([n], E g ) is connected, since in this case the arbitrary right multiplication is the only degree of freedom of the solution. In the noiseless case, one aims to exactly recover the original group elements under certain conditions on the corruption and the graph. In the noisy case, one aims to nearly recover the original group elements with recovery error depending on the distribution of dG (gij , eG ). At last, we remark that for similar models where the measurement gi j may not be in G but in an embedding space, one can first project gi j onto G and then apply our proposed method. Any theory developed for our model can extend for the latter one by projecting onto G. 1.2 Examples of Group Synchronization We review the three common instances of group synchronization. 1.2.1 Z2 Synchronization This is the simplest and most widely known problem of group synchronization. The underlying group, Z2 , is commonly represented in this setting by {−1, 1} with direct multiplication. A natural motivation for this problem is binary graph clustering, where one wishes to recover the labels in {−1, 1} of two different clusters of graph nodes from corrupted measurements of signed interactions between pairs of nodes connected by edges. Namely, the signed interaction of two nodes is 1 if they are in the same cluster and -1 if they are in a different cluster. Note that without any erroneous measurement, the signed int (...truncated)