The Geometry of Synchronization Problems and Learning Group Actions


We develop a geometric framework, based on the classical theory of fibre bundles, to characterize the cohomological nature of a large class of synchronization-type problems in the context of graph inference and combinatorial optimization. We identify each synchronization problem in topological group G on connected graph \(\Gamma \) with a flat principal G-bundle over \(\Gamma \), thus establishing a classification result for synchronization problems using the representation variety of the fundamental group of \(\Gamma \) into G. We then develop a twisted Hodge theory on flat vector bundles associated with these flat principal G-bundles, and provide a geometric realization of the graph connection Laplacian as the lowest-degree Hodge Laplacian in the twisted de Rham–Hodge cochain complex. Motivated by these geometric intuitions, we propose to study the problem of learning group actions—partitioning a collection of objects based on the local synchronizability of pairwise correspondence relations—and provide a heuristic synchronization-based algorithm for solving this type of problems. We demonstrate the efficacy of this algorithm on simulated and real datasets.

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  1. 1.

    Recall (see, e.g. [145, Sect. 2]) that a fibre bundle \(\pi :{\mathfrak {B}}\rightarrow X\), with total space \({\mathfrak {B}}\) and base space X, is said to be flat if it admits a system of local trivializations with locally constant bundle coordinate transformations.

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    Note that a flat bundle is not necessarily trivial (i.e. isomorphic to a product space)—the fundamental group of the base space plays a central role in this development (see e.g. [115, Chap. 2]).

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    Note this is not the case for jointly analyzing a collection of shapes in a landmark-based Procrustes analysis framework; see e.g. [34].


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The authors would like to thank Pankaj Agarwal, Douglas Boyer, Robert Bryant, Ingrid Daubechies, Pawe l D lotko, Kathryn Hess, Lek-Heng Lim, Vidit Nanda, Steve Smale, and Shmuel Weinberger for many inspirational discussions.

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T. Gao gratefully acknowledges partial support from Simons Math+X Investigators Award 400837, DARPA D15AP00109, NSF IIS 1546413, and an AMS-Simons Travel Grant; J. Brodzki would like to acknowledge the support for this work by the EPSRC grants EP/I016945/1 and EP/N014189/1; S. Mukherjee would like to acknowledge support from NSF DMS 16-13261, NSF IIS 1546331, NSF DMS-1418261, NSF IIS-1320357, NSF DMS-1045153, and HFSP RGP0051/2017.

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Appendix A: Proofs of Proposition 1.2 and Formula (37)

Proof of Proposition 1.2

The construction of \({\mathfrak {U}}\) using the stars of the vertices of \(\Gamma \) ensures that

  1. (1)

    \(U_i\cap U_j\ne \emptyset \) if and only if \(( i,j )\in E\);

  2. (2)

    \(U_i\cap U_j\cap U_k \ne \emptyset \) if and only if the 2-simplex (ijk) is in X.

For such pair (ij), define constant map \(g_{ij}:U_i\cap U_j\rightarrow G\) as

$$\begin{aligned} g_{ij} (x)=\rho _{ij}\quad \forall x\in U_i\cap U_j. \end{aligned}$$

Set \(g_{ii}=e\) for all \(1\le i\le | V |\), and note that \(g_{ij} (x)=g_{ji}^{-1} (x)\) for all \(x\in U_i\cap U_j\) by our assumption on \(\rho \). If \(\rho \) is synchronizable over G, let \(f:V\rightarrow G\) be a vertex potential satisfying \(\rho \), then \(\rho _{ij}=f_i f_j^{-1}\) for all \(( i,j )\in E\) from (1). Thus \(\rho _{kj}\rho _{ji}=\rho _{ki}\) for any triangle (ijk) in \(\Gamma \), or equivalently that \(g_{kj} ( x )g_{ji} (x)=g_{ki} (x)\) for all \(x\in U_i\cap U_j \cap U_k\). Therefore, \(\{ g_{ij}\,{|}\, 1\le i,j\le | V |\}\) defines a system of coordinate transformations [140, Sect. 2] with values in G. These data determine a principal fibre bundle \({\mathscr {P}}_{\rho }\) with base space \({\mathcal {X}}\) and structure group G—by a standard construction in the theory of fibre bundles (see e.g. [140, Sect. 3.2])—of which local trivializations are defined on the open sets in \({\mathfrak {U}}\) with constant transition functions \(g_{ij}\); this principal bundle is thus flat by definition. Furthermore, the vertex potential f and the compatibility constraints (1) ensure that the following global section \(s:{\mathcal {X}}\rightarrow {\mathscr {P}}_{\rho }\) is well-defined on this bundle:

$$\begin{aligned} s (x)=\phi _i( x, f_i),\quad x\in U_i \end{aligned}$$

where \(\phi _i:U_i\times G\rightarrow {\mathscr {P}}_{\rho }\) is the local trivialization of \({\mathscr {P}}_{\rho }\) over \(U_i\). The triviality of this principal bundle then follows from the existence of such a global section; see e.g. [140, Sect. 8.3]. The other direction of the proposition follows immediately from this triviality criterion for principal bundles. \(\square \)

Proof of Formula (37)

$$\begin{aligned} \langle \omega , \eta \rangle&= \frac{1}{2}\sum _{( i,j )\in E}\Big [w_{ij} \big \langle p_i\big (\omega ^{( i )}_{ij}\big ), p_i\big (\eta ^{( i )}_{ij}\big ) \big \rangle _F+w_{ji} \big \langle p_j\big (\omega ^{( j )}_{ji}\big ), p_j\big (\eta ^{( j )}_{ji}\big ) \big \rangle _F\Big ]\\&=\frac{1}{2}\sum _{( i,j )\in E}\Big [w_{ij} \big \langle p_i\big (\omega ^{( i )}_{ij}\big ), p_i\big (\eta ^{( i )}_{ij}\big )\big \rangle _F+w_{ij} \big \langle \rho _{ij} p_j\big (\omega ^{( j )}_{ji}\big ), \rho _{ij}p_j\big (\eta ^{( j )}_{ji}\big ) \big \rangle _F\Big ]\\&=\frac{1}{2}\sum _{( i,j )\in E}\Big [w_{ij} \big \langle p_i\big (\omega ^{( i )}_{ij}\big ), p_i\big (\eta ^{( i )}_{ij}\big ) \big \rangle _F+w_{ij} \big \langle p_i\big (\omega ^{( i )}_{ji}\big ), \rho _{ij}p_i\big (\eta ^{( i )}_{ji}\big ) \big \rangle _F\Big ] \\&\quad \quad \quad \quad \quad \quad \quad \quad \text {(see compatibility condition } (31))\\&=\frac{1}{2}\sum _{( i,j )\in E}\Big [w_{ij} \big \langle p_i\big (\omega ^{( i )}_{ij}\big ), p_i\big (\eta ^{( i )}_{ij}\big ) \big \rangle _F+w_{ij} \big \langle p_i\big (\omega ^{( i )}_{ij}\big ), \rho _{ij}p_i\big (\eta ^{( i )}_{ij}\big ) \big \rangle _F\Big ] \\&\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \text {(skew-symmetry)}\\&=\sum _{( i,j )\in E}w_{ij} \big \langle p_i\big (\omega ^{( i )}_{ij}\big ), p_i\big (\eta ^{( i )}_{ij}\big ) \big \rangle _F. \end{aligned}$$

\(\square \)

Appendix B: Graph Laplacian in Discrete Hodge Theory

Define \({\mathbb {K}}\)-valued 0- and 1-forms on weighted graph \(\Gamma =( V,E,w )\) as

$$\begin{aligned} \Omega ^0 ( \Gamma ) := \{ f:V\rightarrow {\mathbb {K}} \},\quad \Omega ^1 ( \Gamma ) := \{ \omega :E\rightarrow {\mathbb {K}}\mid \omega _{ij}=-\omega _{ji}\,\,\forall ( i,j )\in E \}, \end{aligned}$$

equipped with natural inner products

$$\begin{aligned} \begin{aligned} \langle f,g \rangle&:=\sum _id_i \langle f_i, g_i \rangle _{{\mathbb {K}}}, \quad \forall f,g\in \Omega ^0 ( \Gamma ),\\ \langle \omega ,\eta \rangle&:=\sum _{( i,j )\in E}w_{ij}\langle \omega _{ij}, \eta _{ij} \rangle _{{\mathbb {K}}},\quad \forall \omega ,\eta \in \Omega ^1 ( \Gamma ), \end{aligned} \end{aligned}$$

where \(\langle \cdot ,\cdot \rangle _{{\mathbb {K}}}\) is an inner product on \({\mathbb {K}}\), and \(d_i=\sum _{j:( i,j )\in E}w_{ij}\) is the weighted degree at vertex \(i\in V\). Analogous to the study of differential forms on a smooth manifold, one can define the differential \(d:\Omega ^0 ( \Gamma )\rightarrow \Omega ^1 ( \Gamma )\) and codifferential \(\delta :\Omega ^1 ( \Gamma )\rightarrow \Omega ^0 ( \Gamma )\) operators that are formal adjoints of each other:

$$\begin{aligned} (df)_{ij}=f_i-f_j,\quad \forall f\in \Omega ^0 ( \Gamma ),\quad (\delta \omega )_i:=\frac{1}{d_i}\sum _{j:( i,j )\in E}w_{ij}\omega _{ij},\quad \forall \omega \in \Omega ^1 ( \Gamma ). \end{aligned}$$

These constructions can be encoded into a de Rham cochain complex

which realizes \(L_0^{\mathrm {rw}}\), the graph random walk Laplacian, as the Hodge Laplacian of degree zero:

$$\begin{aligned} \big (\Delta ^{( 0 )}f\big )_i:=(\delta df)_i=\frac{1}{d_i}\sum _{j:( i,j )\in E}w_{ij} ( f_i-f_j )=(L_0^{\mathrm {rw}}f)_i,\quad \forall i\in V,\,\,\forall f\in \Omega ^0 ( \Gamma ). \end{aligned}$$

It is well known that \(L_0^{\mathrm {rw}}\) differs from the normalized graph Laplacian \(L_0\) by a similarity transform \(L_0=D^{-1/2}L_0^{\mathrm {rw}}D^{1/2}\), where D is a diagonal matrix with weighted degrees of each vertex on its diagonal.

Software MATLAB code implementing SynCut for the numerical simulations and application in automated geometric morphometrics is publicly available at

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Gao, T., Brodzki, J. & Mukherjee, S. The Geometry of Synchronization Problems and Learning Group Actions. Discrete Comput Geom 65, 150–211 (2021).

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  • Synchronization problem
  • Fibre bundle
  • Holonomy
  • Hodge theory
  • Graph connection Laplacian

Mathematics Subject Classification

  • 05C50
  • 62-07
  • 57R22
  • 58A14