In many applications, an unknown vector is measured
according to the magnitude of its inner product with some known
vectors. It is desirable to design an ensemble of vectors for which
any unknown vector can be recovered from such measurements. This
measurement process is known to be injective for generic M-dimensional
vector ensembles of size at least 4M-2. Recently, semidefinite
programming was used to stably reconstruct from measurements with
random ensembles of size O(MlogM).
In this talk, we use the polarization identity, spectral graph theory
and new developments in angular synchronization to efficiently and
stably recover from phaseless measurements with specific ensembles of
size O(M).
The angular synchronization problem consists of estimating a set of
unknown angles (or higher dimensional rotations) from noisy
measurements of a subset of the pairwise ratios. We will conclude by
presenting a Cheeger-like inequality that relates how well it is
possible to solve this problem with the spectra of an operator, the
graph connection Laplacian.