The Geometry of Synchronization Problems and Learning Group Actions

Abstract

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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Notes

  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.

  2. 2.

    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]).

  3. 3.

    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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Acknowledgements

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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Appendices

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 https://github.com/trgao10/GOS-SynCut.

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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). https://doi.org/10.1007/s00454-019-00100-2

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Keywords

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

Mathematics Subject Classification

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