  We propose a method for sensor array self-localization using a set of sources
at unknown locations. The sources produce signals whose times of arrival are
registered at the sensors. We look at the general case where neither the
emission times of the sources nor the reference time frames of the receivers
are known. Unlike previous work, our method directly recovers the array
geometry, instead of first estimating the timing information. The key component
is a new loss function which is insensitive to the unknown timings. We cast the
problem as a minimization of a non-convex functional of the Euclidean distance
matrix of microphones and sources subject to certain non-convex constraints.
After convexification, we obtain a semidefinite relaxation which gives an
approximate solution; subsequent refinement on the proposed loss via the
Levenberg-Marquardt scheme gives the final locations. Our method achieves
state-of-the-art performance in terms of reconstruction accuracy, speed, and
the ability to work with a small number of sources and receivers. It can also
handle missing measurements and exploit prior geometric and temporal knowledge,
for example if either the receiver offsets or the emission times are known, or
if the array contains compact subarrays with known geometry.
