Sample-Efficient Low Rank Phase Retrieval

This work studies the Low Rank Phase Retrieval (LRPR) problem: recover an $n \times q$ rank- notation="LaTeX">$r$ matrix notation="LaTeX">${ \boldsymbol {X}^{\ast}}$ from notation="LaTeX">$\boldsymbol {y}_{k} = | {A}_{k}^\top {x}^{\ast} _{k}|$ , notation="LaTeX">$k=1, 2,\ldots, when each {y}_{k}$ is m-length vector containing independent phaseless linear projections of {x}^{\ast}_{k}$ . Here notation="LaTeX">$|.|$ takes element-wise magnitudes a vector. The different matrices {A}_{k}$ are i.i.d. and contains standard Gaussian entries. We obtain improved guarantee for AltMinLowRaP, which Alternating Minimization solution to LRPR that was introduced studied in our recent work. As long as right singular vectors satisfy incoherence assumption, we can show AltMinLowRaP estimate converges geometrically if total number measurements notation="LaTeX">$mq \gtrsim nr^{2} (r + \log (1/\epsilon))$ In addition, also need notation="LaTeX">$m max(r, q, n)$ because specific asymmetric nature problem. Compared work, improve sample complexity AltMin iterations by factor notation="LaTeX">$r^{2}$ initialization argue, based on comparison with related well-studied problems, why above cannot be any further non-convex solutions LRPR. extend result noisy case; prove stability corruption small additive noise.