Optimal rates of convergence for noisy sparse phase retrieval via thresholded Wirtinger flow

Cai, Tianxi; Li, Xiaodong; Ma, Zongming

doi:10.1214/16-aos1443

Cited by 196 publications

(278 citation statements)

References 53 publications

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“…X * has rank r LRMC (first) [25] No left & right incoherence, m ≥ C(n/q)r 4.5 log(1/ ) C(n/q)r 6.5 log n log 2 (1/ ) X * has rank r LRMC (best) [33] No left & right incoherence m ≥ C(n/q)r 2 log 2 n log 2 (1/ ) C(n/q)r 3 log n log 2 (1/ ) X * has rank r Sparse PR (first) [6] Yes x * is s-sparse in canonical basis, m ≥ Cs 2 log n log(1/ ) Cns 3 log n log 2 (1/ ) min nonzero entry lower bounded Sparse PR (best) [17], [18] Yes x * is s-sparse in canonical basis m ≥ Cs 2 log n Cns 2 log n log(1/ )…”

Section: Problemmentioning

confidence: 99%

Provable Low Rank Phase Retrieval

Nayer

Narayanamurthy

Vaswani

2020

IEEE Trans. Inform. Theory

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We study the "Low Rank Phase Retrieval (LRPR)" problem defined as follows: recover an n × q matrix X * of rank r from a different and independent set of m phaseless (magnitude-only) linear projections of each of its columns. To be precise, we need to recover X * from y k := |A k x * k |, k = 1, 2, . . . , q when the measurement matrices A k are mutually independent. Here y k is an m length vector and denotes transpose. The question is when can we solve LRPR with m n? Our work introduces the first provably correct solution, Alternating Minimization for Low-Rank Phase Retrieval (AltMinLowRaP), for solving LRPR. We demonstrate its advantage over existing work via extensive simulation, and some partly real data, experiments. Our guarantee for AltMinLowRaP shows that it can solve LRPR to accuracy if mq ≥ Cnr 4 log(1/ ), the matrices A k contain i.i.d. standard Gaussian entries, the condition number of X * is bounded by a numerical constant, and its right singular vectors satisfy the incoherence (denseness) assumption from matrix completion literature. Its time complexity is only Cmqnr log 2 (1/ ). In the regime of small r, our sample complexity is much better than what standard PR methods need; and it is only about r 3 times worse than its order-optimal value of (n + q)r. Moreover, if we replace m by its lower bound for each approach, then the same can be said for the time complexity comparison with standard PR. We also briefly study the dynamic extension of LRPR.The LRPR problem occurs in phaseless dynamic imaging, e.g., Fourier ptychographic imaging of live biological specimens, where acquiring measurements is expensive. We should point out that LRPR is a very different problem than its A k = A version, or its A k = A and with-phase (linear) version, both of which have been extensively studied in the literature.for each k. If we assume κ is a constant, up to constant factors, (4) also implies (3). Thus, up to constant factors, requiring right incoherence is the same as requiring that the maximum energy of any signal x * k is within constant factors of the average.

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Section: Problemmentioning

confidence: 99%

Provable Low Rank Phase Retrieval

Nayer

Narayanamurthy

Vaswani

2020

IEEE Trans. Inform. Theory

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“…Proof. It follows from [20,Theorem 4,4] and the observation that when δ > 1, (1+δ) log(1+δ) > 4 3 δ. Lemma II.17. For v ∼ CN (0, I), Pr( 1 m | v 2 − 1| > t) < 2 exp −cm min t 2 C 2 , t C Proof.…”

Section: Axillary Lemmasmentioning

confidence: 99%

Phase Retrieval by Alternating Minimization With Random Initialization

Zhang

2020

IEEE Trans. Inform. Theory

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We consider a phase retrieval problem, where the goal is to reconstruct a n-dimensional complex vector from its phaseless scalar products with m sensing vectors, independently sampled from complex normal distributions. We show that, with a random initialization, the classical algorithm of alternating minimization succeeds with high probability as n, m → ∞ when m/log 3 m ≥ M n 3/2 log 1/2 n for some M > 0. This is a step toward proving the conjecture in [27], which conjectures that the algorithm succeeds when m = O(n). The analysis depends on an approach that enables the decoupling of the dependency between the algorithmic iterates and the sensing vectors.

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“…Employing a thresholded gradient descent algorithm to a non-convex empirical risk minimization problem that is derived from the phase retrieval problem, Cai et al [14] have established the minimax optimal rates of convergence for noisy sparse phase retrieval under sub-exponential noise.…”

Section: Related Work On Sparse Phase Retrievalmentioning

confidence: 99%

Sparse Phase Retrieval of One-Dimensional Signals by Prony's Method

Beinert

Plonka

2017

Front. Appl. Math. Stat.

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In this paper, we show that sparse signals f representable as a linear combination of a finite number N of spikes at arbitrary real locations or as a finite linear combination of B-splines of order m with arbitrary real knots can be almost surely recovered from2 up to trivial ambiguities. The constructive proof consists of two steps, where in the first step Prony's method is applied to recover all parameters of the autocorrelation function and in the second step the parameters of f are derived. Moreover, we present an algorithm to evaluate f from its Fourier intensities and illustrate it at different numerical examples.

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Optimal rates of convergence for noisy sparse phase retrieval via thresholded Wirtinger flow

Cited by 196 publications

References 53 publications

Provable Low Rank Phase Retrieval

Provable Low Rank Phase Retrieval

Phase Retrieval by Alternating Minimization With Random Initialization

Sparse Phase Retrieval of One-Dimensional Signals by Prony's Method

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