Definition: In this task, you are given a part of an article. Your task is to generate headline (title) for this text. Preferred headlines are under fifteen words.
Input: Phase retrieval problems involve solving linear equations, but with missing sign (or phase, for<lb>complex numbers) information. Over the last two decades, a popular generic empirical approach<lb>to the many variants of this problem has been one of alternating minimization; i.e. alternating<lb>between estimating the missing phase information, and the candidate solution. In this paper, we<lb>show that a simple alternating minimization algorithm geometrically converges to the solution<lb>of one such problem – finding a vector x from y,A, where y = |Ax| and |z| denotes a vector<lb>of element-wise magnitudes of z – under the assumption that A is Gaussian.<lb>Empirically, our algorithm performs similar to recently proposed convex techniques for this<lb>variant (which are based on “lifting” to a convex matrix problem) in sample complexity and<lb>robustness to noise. However, our algorithm is much more efficient and can scale to large<lb>problems. Analytically, we show geometric convergence to the solution, and sample complexity<lb>that is off by log factors from obvious lower bounds. We also establish close to optimal scaling<lb>for the case when the unknown vector is sparse. Our work represents the only known theoretical<lb>guarantee for alternating minimization for any variant of phase retrieval problems in the non-<lb>convex setting.

Phase Retrieval using Alternating Minimization