Provable Low Rank Phase Retrieval

We study the Low Rank Phase Retrieval (LRPR) problem defined as follows: recover an <inline-formula> <tex-math notation="LaTeX">$n \times q$ </tex-math></inline-formula> matrix <inline-formula> <tex-math notation="LaTeX">${ \boldsymbol {X}^{*}}$ </tex-math></inline-formula> of rank <inline-formula> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula> from a different and independent set of <inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> phaseless (magnitude-only) linear projections of each of its columns. To be precise, we need to recover <inline-formula> <tex-math notation="LaTeX">${ \boldsymbol {X}^{*}}$ </tex-math></inline-formula> from <inline-formula> <tex-math notation="LaTeX">$\boldsymbol {y}_{k}:= | \boldsymbol {A}_{k}{}' \boldsymbol {x}^{*}_{k}|, k=1,2, {\dots }, q$ </tex-math></inline-formula> when the measurement matrices <inline-formula> <tex-math notation="LaTeX">$\boldsymbol {A}_{k}$ </tex-math></inline-formula> are mutually independent. Here <inline-formula> <tex-math notation="LaTeX">$\boldsymbol {y}_{k}$ </tex-math></inline-formula> is an <inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> length vector, <inline-formula> <tex-math notation="LaTeX">$\boldsymbol {A}_{k}$ </tex-math></inline-formula> is an <inline-formula> <tex-math notation="LaTeX">$n \times m$ </tex-math></inline-formula> matrix, and <inline-formula> <tex-math notation="LaTeX">$'$ </tex-math></inline-formula> denotes matrix transpose. The question is when can we solve LRPR with <inline-formula> <tex-math notation="LaTeX">$m \ll n$ </tex-math></inline-formula>? A reliable solution can enable fast and low-cost phaseless dynamic imaging, e.g., Fourier ptychographic imaging of live biological specimens. In this work, we develop the first provably correct approach for solving this LRPR problem. Our proposed algorithm, Alternating Minimization for Low-Rank Phase Retrieval (AltMinLowRaP), is an AltMin based solution and hence is also provably fast (converges geometrically). Our guarantee shows that AltMinLowRaP solves LRPR to <inline-formula> <tex-math notation="LaTeX">$\epsilon $ </tex-math></inline-formula> accuracy, with high probability, as long as <inline-formula> <tex-math notation="LaTeX">$m q \ge C n r^{4} \log (1/\epsilon)$ </tex-math></inline-formula>, the matrices <inline-formula> <tex-math notation="LaTeX">$\boldsymbol {A}_{k}$ </tex-math></inline-formula> contain i.i.d. standard Gaussian entries, and the right singular vectors of <inline-formula> <tex-math notation="LaTeX">${ \boldsymbol {X}^{*}}$ </tex-math></inline-formula> satisfy the incoherence assumption from matrix completion literature. Here <inline-formula> <tex-math notation="LaTeX">$C$ </tex-math></inline-formula> is a numerical constant that only depends on the condition number of <inline-formula> <tex-math notation="LaTeX">${ \boldsymbol {X}^{*}}$ </tex-math></inline-formula> and on its incoherence parameter. Its time complexity is only <inline-formula> <tex-math notation="LaTeX">$C mq nr \log ^{2}(1/\epsilon)$ </tex-math></inline-formula>. Since even the linear (with phase) version of the above problem is not fully solved, the above result is also the first complete solution and guarantee for the linear case. Finally, we also develop a simple extension of our results for the dynamic LRPR setting.

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