Random Multi-Type Spanning Forests for Synchronization on Sparse Graphs

TL;DR

This work introduces diffusion estimators for the problems of angular synchronization and smoothing on graphs, in the presence of a rotation associated to each edge, and shows that these estimators outperform standard numerical-linear-algebra solvers in challenging instances, depending on the topology and density of the graph.

Abstract

Random diffusions are a popular tool in Monte-Carlo estimations, with well established algorithms such as Walk-on-Spheres (WoS) going back several decades. In this work, we introduce diffusion estimators for the problems of angular synchronization and smoothing on graphs, in the presence of a rotation associated to each edge. Unlike classical WoS algorithms that are point-wise estimators, our diffusion estimators allow for global estimations by propagating along the branches of random spanning subgraphs called multi-type spanning forests. Building upon efficient samplers based on variants of Wilson's algorithm, we show that our estimators outperform standard numerical-linear-algebra solvers in challenging instances, depending on the topology and density of the graph.

Figures (15)

• Figure 1: Propagations on the $4 \times 2$ grid graph, along different paths. The rotation angles $\theta_{i,j}$'s associated to each edge are represented as circular rotating arrows (only drawn along the paths we consider). Propagations always start from the large blue node, and propagated angles are represented as straight arrows (top left of each node). Left \ref{['fig:ex_propagation:basics']}: propagation along an arbitrary path. Center \ref{['fig:ex_propagation:tree']}: exact synchronization achieved by propagation along a spanning tree, in the absence of noise. Right \ref{['fig:ex_propagation:incoherent']}: synchronization impossible due to noise and incoherence along cycles.
• Figure 1: A connection $\Psi$ on the triangle graph $K_3$ (in the center). The connection maps $\psi_e$ are associated with angles $\theta_{(v_1,v_2)} = \theta_{(v_2,v_3)} = \theta_{(v_3,v_1)} = \frac{\pi}{4}$, and we represent their action on some $z_1 \in \mathbf{C}_{v_1}$. A cyclic path $C = ((v_1,v_2),(v_2,v_3),(v_3,v_1))$ is depicted using directed arrows, we denote by $\psi_C = \psi_{(v_3,v_1)} \circ \psi_{(v_2,v_3)} \circ \psi_{(v_1,v_2)}$ the composition of the connection maps along this cycle (a rotation of $\theta_C = \frac{3 \pi}{4})$. Top right: $z_1 \in \mathbf{C}_{v_1}$ (bold arrow) and $z_1' = \psi_C(z_1) \in \mathbf{C}_{v_1}$ (blue dotted arrow). Top left, bottom: $z_2 = \psi_{(v_1,v_2)}(z_2) \in \mathbf{C}_{v_2}$ and $z_3 = \psi_{(v_2,v_3)}(z_2) \in \mathbf{C}_{v_3}$.
• Figure 1: Theorem \ref{['th:fermionic']} illustrated on the $4 \times 4$ grid graph, on one MTSF.
• Figure 1: Runtime when varying $\overline{d}$, for different values of $q$.
• Figure 1: Examples of recovery from noisy images (left), with different noise levels, using both the exact solution to the spectral relaxation \ref{['eq:sr']} and the approach from Section \ref{['subsect:synchro_algo']}.

Theorems & Definitions (33)

• Theorem 1
• Remark 1.1
• Proposition 2.1
• Proposition 2.2
• Remark 2.3
• Theorem 2.4
• Remark 2.5
• Proposition 2.7
• Proposition 2.8
• Proposition 2.9