  Coherent techniques for searches of gravitational-wave bursts effectively
combine data from several detectors, taking into account differences in their
responses. The efforts are now focused on the maximum likelihood principle as
the most natural way to combine data, which can also be used without prior
knowledge of the signal. Recent studies however have shown that straightforward
application of the maximum likelihood method to gravitational waves with
unknown waveforms can lead to inconsistencies and unphysical results such as
discontinuity in the residual functional, or divergence of the variance of the
estimated waveforms for some locations in the sky. So far the solutions to
these problems have been based on rather different physical arguments.
Following these investigations, we now find that all these inconsistencies stem
from rank deficiency of the underlying network response matrix. In this paper
we show that the detection of gravitational-wave bursts with a network of
interferometers belongs to the category of ill-posed problems. We then apply
the method of Tikhonov regularization to resolve the rank deficiency and
introduce a minimal regulator which yields a well-conditioned solution to the
inverse problem for all locations on the sky.
