Eigenvector Based Block Vector Synchronization with Applications to Ptychographic Imaging

We consider the problem of recovering a complex vector (up to a global unimodular constant) given noisy and incomplete outer product measurements. Such problems arise when implementing distributed clock synchronization schemes, radar autofocus methods, and phaseless signal recovery. This prob-lem is known as vector synchronization and is a variant of the more common angular synchronization problem. In applications with windowed measurements and/or convolutional models - for example, phase retrieval from STFT magnitude data, the outer product measurement matrix is highly incomplete and has a block diagonal structure. We describe a vector synchronization technique which applies an eigenvector computation to blocks of this matrix followed by a block compatibility operation to piece together the final solution. We provide theoretical guarantees (in the noiseless case) and empirical simulations demonstrating the accuracy and efficiency of the method.