  We present a novel approach for recovering a sparse signal from
cross-correlated data. Cross-correlations naturally arise in many fields of
imaging, such as optics, holography and seismic interferometry. Compared to the
sparse signal recovery problem that uses linear measurements, the unknown is
now a matrix formed by the cross correlation of the unknown signal. Hence, the
bottleneck for inversion is the number of unknowns that grows quadratically.
The main idea of our proposed approach is to reduce the dimensionality of the
problem by recovering only the diagonal of the unknown matrix, whose dimension
grows linearly with the size of the problem. The keystone of the methodology is
the use of an efficient {\em Noise Collector} that absorbs the data that come
from the off-diagonal elements of the unknown matrix and that do not carry
extra information about the support of the signal. This results in a linear
problem whose cost is similar to the one that uses linear measurements. Our
theory shows that the proposed approach provides exact support recovery when
the data is not too noisy, and that there are no false positives for any level
of noise. Moreover, our theory also demonstrates that when using
cross-correlated data, the level of sparsity that can be recovered increases,
scaling almost linearly with the number of data. The numerical experiments
presented in the paper corroborate these findings.
