Sparse Phase Retrieval via Truncated Amplitude Flow

This paper develops a novel algorithm, termed <italic>SPARse Truncated Amplitude flow</italic> (SPARTA), to reconstruct a sparse signal from a small number of magnitude-only measurements. It deals with what is also known as sparse phase retrieval (PR), which is <italic>NP-hard</italic> in general and emerges in many science and engineering applications. Upon formulating sparse PR as an amplitude-based nonconvex optimization task, SPARTA works iteratively in two stages: In stage one, the support of the underlying sparse signal is recovered using an analytically well-justified rule, and subsequently a sparse orthogonality-promoting initialization is obtained via power iterations restricted on the support; and in the second stage, the initialization is successively refined by means of hard thresholding based gradient-type iterations. SPARTA is a simple yet effective, scalable, and fast sparse PR solver. On the theoretical side, for any <inline-formula><tex-math notation="LaTeX">$n$</tex-math></inline-formula>-dimensional <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-sparse (<inline-formula> <tex-math notation="LaTeX">$k\ll n$</tex-math></inline-formula>) signal <inline-formula><tex-math notation="LaTeX"> $\boldsymbol {x}$</tex-math></inline-formula> with minimum (in modulus) nonzero entries on the order of <inline-formula> <tex-math notation="LaTeX">$(1/\sqrt{k})\Vert \boldsymbol {x}\Vert _2$</tex-math></inline-formula>, SPARTA recovers the signal exactly (up to a global unimodular constant) from about <inline-formula><tex-math notation="LaTeX">$k^2\log n$ </tex-math></inline-formula> random Gaussian measurements with high probability. Furthermore, SPARTA incurs computational complexity on the order of <inline-formula><tex-math notation="LaTeX">$k^2n\log n$</tex-math> </inline-formula> with total runtime proportional to the time required to read the data, which improves upon the state of the art by at least a factor of <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>. Finally, SPARTA is robust against additive noise of bounded support. Extensive numerical tests corroborate markedly improved recovery performance and speedups of SPARTA relative to existing alternatives.

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