Analysis of Spectral Methods for Phase Retrieval With Random Orthogonal Matrices

Dudeja, Rishabh; Bakhshizadeh, Milad; Ma, Junjie; Maleki, Arian

doi:10.1109/tit.2020.2981910

Cited by 21 publications

(19 citation statements)

References 26 publications

(43 reference statements)

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“…• Importantly, our analysis is essentially not rigorous (hence the use of conjectures). An interesting perspective would be to establish rigorously our statements, in similarity with what is proven in [DBMM20] on the analysis of [MDX + 19] for column-unitary matrices. This would require a random matrix theory analysis of the "BBP" 1 transition in matrices of the form of eq.…”

Section: Perspectivesmentioning

confidence: 56%

“…The asymptotic optimal performances in a large class of problems including phase retrieval were conjectured using the non-rigorous replica method of statistical physics in [Kab08,TK20], and these results were extended and partly proven in [BKM + 19, MLKZ20]. Specifically for the phase retrieval problem, the limits of weak-recovery were analyzed for Gaussian matrices Φ in [LL20, MM19,LAL19], and for column-unitary Φ in [MDX + 19, DMM20,DBMM20]. In this work we derive the optimal spectral methods for a more generic assumption of right orthogonal (or unitary) invariance, that is we assume: Hypothesis 1 (Matrix ensemble).…”

Section: Introduction 1setting Of the Problem And Related Workmentioning

confidence: 99%

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Construction of optimal spectral methods in phase retrieval

Maillard¹,

Krząkała²,

Lu³

et al. 2020

Preprint

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We consider the phase retrieval problem, in which the observer wishes to recover a n-dimensional real or complex signal X from the (possibly noisy) observation of |ΦX |, in which Φ is a matrix of size m × n. We consider a high-dimensional setting where n, m → ∞ with m/n = O(1), and a large class of (possibly correlated) random matrices Φ and observation channels. Spectral methods are a powerful tool to obtain approximate observations of the signal X which can be then used as initialization for a subsequent algorithm, at a low computational cost. In this paper, we extend and unify previous results and approaches on spectral methods for the phase retrieval problem. More precisely, we combine the linearization of message-passing algorithms and the analysis of the Bethe Hessian, a classical tool of statistical physics. Using this toolbox, we show how to derive optimal spectral methods for arbitrary channel noise and right-unitarily invariant matrix Φ, in an automated manner (i.e. with no optimization over any hyperparameter or preprocessing function).

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Section: Perspectivesmentioning

confidence: 56%

Section: Introduction 1setting Of the Problem And Related Workmentioning

confidence: 99%

Construction of optimal spectral methods in phase retrieval

Maillard¹,

Krząkała²,

Lu³

et al. 2020

Preprint

View full text Add to dashboard Cite

show abstract

“…For this reason, one needs to study which initialization schemes work for the measurements considered in this paper. A natural approach will be to try spectral initializations and recent generalizations that have been shown to be feasible for a basically minimal number of measurements [15,29,30,33]. We expect that the analysis provided in this paper will prove useful for this endeavour as the spectral initialization is somewhat connected to trace-norm minimization.…”

Section: Phaseliftmentioning

confidence: 99%

Complex Phase Retrieval from Subgaussian Measurements

Krahmer

Stöger

2020

J Fourier Anal Appl

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Phase retrieval refers to the problem of reconstructing an unknown vector $$x_0 \in {\mathbb {C}}^n$$ x 0 ∈ C n or $$x_0 \in {\mathbb {R}}^n $$ x 0 ∈ R n from m measurements of the form $$y_i = \big \vert \langle \xi ^{\left( i\right) }, x_0 \rangle \big \vert ^2 $$ y i = | ⟨ ξ i , x 0 ⟩ | 2 , where $$ \left\{ \xi ^{\left( i\right) } \right\} ^m_{i=1} \subset {\mathbb {C}}^m $$ ξ i i = 1 m ⊂ C m are known measurement vectors. While Gaussian measurements allow for recovery of arbitrary signals provided the number of measurements scales at least linearly in the number of dimensions, it has been shown that ambiguities may arise for certain other classes of measurements $$ \left\{ \xi ^{\left( i\right) } \right\} ^{m}_{i=1}$$ ξ i i = 1 m such as Bernoulli measurements or Fourier measurements. In this paper, we will prove that even when a subgaussian vector $$ \xi ^{\left( i\right) } \in {\mathbb {C}}^m $$ ξ i ∈ C m does not fulfill a small-ball probability assumption, the PhaseLift method is still able to reconstruct a large class of signals $$x_0 \in {\mathbb {R}}^n$$ x 0 ∈ R n from the measurements. This extends recent work by Krahmer and Liu from the real-valued to the complex-valued case. However, our proof strategy is quite different and we expect some of the new proof ideas to be useful in several other measurement scenarios as well. We then extend our results $$x_0 \in {\mathbb {C}}^n $$ x 0 ∈ C n up to an additional assumption which, as we show, is necessary.

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“…A few lower complexity alternatives to VAMP have been proposed recently, including convolutional AMP [Tak21a], Memory AMP for linear models [LHK20], and Generalized Memory AMP for GLMs [TLC21]. The phase retrieval problem (which is a special case of a GLM) has been studied for design matrices with orthogonal columns, a model distinct from the rotationally invariant one considered here [DBMM20,DMM20]. Finally, we mention that AMP has also been applied to low-rank matrix estimation with rotationally invariant noise [OCW16, C ¸O19, Fan21, ZSF21, MV21b].…”

Section: Introductionmentioning

confidence: 99%

Estimation in Rotationally Invariant Generalized Linear Models via Approximate Message Passing

Venkataramanan¹,

Kögler²,

Mondelli³

2021

Preprint

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We consider the problem of signal estimation in generalized linear models defined via rotationally invariant design matrices. Since these matrices can have an arbitrary spectral distribution, this model is well suited to capture complex correlation structures which often arise in applications. We propose a novel family of approximate message passing (AMP) algorithms for signal estimation, and rigorously characterize their performance in the high-dimensional limit via a state evolution recursion. Assuming knowledge of the design matrix spectrum, our rotationally invariant AMP has complexity of the same order as the existing AMP for Gaussian matrices; it also recovers the existing AMP as a special case. Numerical results showcase a performance close to Vector AMP (which is conjectured to be Bayes-optimal in some settings), but obtained with a much lower complexity, as the proposed algorithm does not require a computationally expensive singular value decomposition.

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Analysis of Spectral Methods for Phase Retrieval With Random Orthogonal Matrices

Cited by 21 publications

References 26 publications

Construction of optimal spectral methods in phase retrieval

Construction of optimal spectral methods in phase retrieval

Complex Phase Retrieval from Subgaussian Measurements

Estimation in Rotationally Invariant Generalized Linear Models via Approximate Message Passing

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