Fast Quadratic Sensing Via Nonconvex Optimization

This paper concerns the problem of recovering an unknown but rank-r signal from quadratic measurements. We formulate the recovery problem as two-stage nonconvex optimization approach where an efficient adaptive accelerated gradient-like iterative refinement starting from a careful initialization. Moreover, an efficient initial scheme is proposed for this problem. The average relative error of our initialization estimator reduces exponentially as the oversampling ratio grows, which can also boost the performance of existing two-stage approach for rank-
$-r$
matrix recovery from vector measurements. Experimental results not only clearly demonstrate the superiority of introduced initialization estimator but also show the advantages of our adaptive accelerated gradient-like approach in terms of both sample complexity and computational complexity.