Frames and Phase Retrieval

Transcription

1 Frames and Phase Retrieval Radu Balan University of Maryland Department of Mathematics and Center for Scientific Computation and Mathematical Modeling AMS Short Course: JMM San Antonio, TX

2 Current Collaborators and Sponsors Mark Lai (UMD) Max Scharrenbroich (UMD) Dongmian Zou (UMD) Yang Wang (MSU) Heiko Claussen (Siemens) Justinian Rosca (Siemens) 2

3 Overview Problem Statement Analysis Results Injectivity Results Deterministic Bounds: Lipschitz bounds Stochastic Bounds: CRLB The IRLS Algorithm Simulation Results 3

4 1. Problem Statement Consider the following nonlinear map: Where is a frame (spanning set) and. Replace by where iff there is a unit scalar, so that. Nonlinear map is well defined: 4

5 Nonlinear map: Problems: 1. Invertibility: When is injective? 2. Algorithms: How to efficiently invert? 3. Robustness of inversion: Lipschitz constants, stochastic performance bounds. Definition: The frame is said phase retrievable (or that it gives phase retrieval) if the nonlinear map is injective. 5

6 Relevance X Ray Crystallography I( k ) C e i k, r R( r ) dr 2 Audio and Signal processing Fiber optics data transmission THz Radar & imaging systems Beampattern design (from ray_crystallography) 6

7 2. Invertibility: Real Case where. Theorem [B.CasazzaEdidin 06,B. 12]The following are equivalent: 1. The frame is phase retrievable 2. For any disjoint partition either spans 3. There is a constant so that. Corollary [BCE 06] If is phase retrievable then. 7

8 2. Invertibility: Complex case Notations:, where. Note is a rank 2 real symmetric matrix. For any two vectors the symmetric outer product: For any denote 8

9 2: Invertibility: Complex Case (2) Theorem [B 13,BandeiraCahillMixonNelson 13] TFAE: 1. The frame is phase retrievable 2. There is a constant so that for all,,,,,,. 3. For any ξ 0, rank(v(ξ))=2n 1, where Φ Φ ; 4. There is a constant 0so that for all :Φ Φ : Remark. Critical redundancy is 4 log 423 9

10 3. Robustness Measures: CRLB Consider the following additive white Gaussian noise (AWGN) model: where are i.i.d. Theorem [B 12,B. 13,BCMN 13] The Fisher Information matrix for x is given by 10

11 3. Robustness Measures: CRLB (2) The No Oracle Case: Theorem [B 13,BCMN 13] Fix and denote. Then the covariance matrix of any unbiased estimator of an unknown vector known to be in, is bounded below by: (Real Case): (Complex Case): [CRLB Bounds] 11

12 3. Robustness Measures: CRLB (3) The Oracle Case: X ;, Oracle, x where: Assume,, (unbiased estimator). Problem: Find a lower bound for MSE: Still an open problem! 12

13 3. Robustness Measures: Lipschitz bounds (1) We insist on exact inversion on the range: for all x. Then we need to characterize bi Lipschitz maps. We shall consider two distances on : We want to compute: inf,,, inf,,, 13

14 3. Robustness Measures: Lipschitz bounds (2),,, This distance makes the embedding,, isometric, with nuclear norm on. Theorem [R.B.,Wang 13]. When is injective, then is bi Lipschitz: there are positive constants 0 so that for every,,, Explicitly:,,,, Same as in CRLB Mixed l 2,4 norm of the analysis operator 14

15 3. Robustness Measures: Lipschitz bounds (3) In the real case:,,, Hence: With:,, min, max Lower bound similar to [EldarMendelson 12] In the complex case:,,, 4, Hence: With:,, 4, min, max 15

16 3. Robustness Measures: Lipschitz bounds (4), min min,, 2,, Theorem [R.B.,Wang 13]. In the real case ( ) when is injective, then is also bi Lipschitz: there are positive constants 0 so that for every,,, Explicitly:,, min, with min and where A[S] denotes the lower frame bound of subset indexed by S, and B denotes the upper frame bound. 16 Complex Case: Still Open!

17 4. The Iterative Regularized Least Squares (IRLS) Algorithm The complex case (1) We want to minimize:, Φ, Instead we introduce an homotopic optimization procedure that uses regularization and splitting:, ;, Φ,,,;, Main iteration:, ;, 17

18 4. The IRLS Algorithm (2) 1. Initialization. For large, has global optimizer u=0. The largest for a nontrivial optimizer: Choose. Then initialize: 18

19 4. The IRLS Algorithm (3) 2. Iteration. Compute, ;, Adapt:, max, Solution to the optimization problem: Solve Φ Φ Φ 19

20 4. The IRLS Algorithm (4) 3. Stopping criterion Stop when regularization parameter reaches a preset minimal value: Alternatively: Choose a signal to noise level, say SNR, and stop when 20

21 5. Simulation Results Results for n=100, m=800 (redundancy 8) complex case 10 log, 10 log 21

22 Redundancy 6: n=100, m=

23 Redundancy 4: n=100, m=

24 Analysis for Redundancy=8, n=100, m= 800 Number of iterations: left plot; Components of the Mean Square error: Bias and Variance (right plot) 24

25 Redundancy 8: Trace of a realization; SNR = 0dB 25

26 Redundancy 8: Trace of a realization; SNR = 40dB 26

27 6. Conclusions We analyzed the phase retrieval problem (phaseless reconstruction): 1. We presented necessary and sufficient conditions for injectivity of the nonlinear map; 2. Explicit expressions of the Fisher Information Matrix and Cramer Rao Lower Bound; 3. Explicit expressions for bi Lipschitz constants 4. Analyzed performance of the Iterative Regularized Least Square (IRLS) algorithm. 27

28 Thank you! 28