In many applications, the main goal is to obtain a global low dimensional representation of the data, given some local noisy geometric constraints. In this talk we will show how all (seemingly unrelated) problems listed below can be solved by constructing suitable operators on their data. Those operators are different from the graph Laplacian, and can be regarded as its extension. The solutions involve only the computation of a few eigenvectors of sparse matrices corresponding to the data operators.
Cryo Electron Microscopy for protein structuring: reconstructing the three-dimensional structure of a molecule from projection images taken at random unknown orientations (unlike classical tomography, where orientations are known).
NMR spectroscopy for protein structuring: finding the global positioning of all hydrogen atoms in a molecule from their local distances. Distances between neighboring hydrogen atoms are estimated from the spectral lines corresponding to the short ranged spin-spin interaction.
Sensor networks: finding the global positioning from local distances.
Detecting the slow manifold in stochastic chemical reactions: finding the slow coordinates in large multi-scaled dynamical systems from experimental or simulation data.
Non linear independent component analysis: de-mixing statistically independent processes that were mixed by an unknown smooth non-linear functions.
Numerical integration: integrating a multivariate function given its noisy gradient field, for example, in surface reconstruction.