This presentation considers the problem of optimally estimating the true offsets between state values at nodes of a network based on noisy measurements. A true value, taken to be an element of a Lie group, is associated with each vertex of a finite connected graph. Each edge is labeled with a noisy measurement of the offset between the values at the vertices it joins. The true values at the vertices are to be estimated based on the aggregated collection of measurements, thereby providing a means for registration of the values at the vertices.
The Fisher information for this estimation problem is derived and shown to depend on distribution of the measurement noise and standard descriptors of the graph structure in straightforward ways. This enables analysis of network configurations in terms of the quality of registration possible. Maximum likelihood estimators are derived, distributed estimation is considered, and connections to gauge field theory are noted.
In practical problems, the Lie group is often R or R^n (e.g., time or position in space), a circle or torus (e.g., the phase of one or more oscillators), or SO(n) (e.g., orientation of the local coordinate system for camera). In the first two cases, classical additive Gaussian and von Mises models for the noisy measurements present attractive features. Other cases raise interesting questions about the nature of maximum-entropy unimodal probability distributions on Lie groups, into which representation theoretic approaches provide significant insight.