a complex vector space ; {Li }i∈I be a ﬁxed family of linear forms on V . A phase retrieval problem is of the form Reconstruct x ∈ V from {|Li (x)|}i∈I . where |.| is the usual complex modulus ; the unknown x can be any element of V . If u ∈ C, |u| = 1, x and ux are indistinguishable. → We consider the reconstruction up to a global phase.
The element x may not be uniquely determined. ∀i, |Li (x)| = |Li (y)| ⇒ x = y ? Stability Reconstruction can be unstable under measurement noise. ∀i, |Li (x)| ≈ |Li (y)| ⇒ x ≈ y ? Algorithms It is diﬃcult to have simultaneously precision and reasonable complexity.
retrieval problems Well-behaved case : random or “generic” measurements Deterministic “physical” measurements Fourier transform Wavelet transform 2. Main two classes of algorithms Iterative algorithms Convexiﬁed algorithms PhaseLift PhaseCut 3. Numerical results : which algorithm for which problem ?
or “generic” measurements Reconstruct x ∈ V from {|Li (x)|}i∈I . Finite dimension : assume V = Cn, Li ∈ Cn. Choose the Li at random. → Uniqueness / stability results that hold with high probability / probability 1. Theorem If Card(I) ≥ 4n − 4, then the set of measurements for which there is no uniqueness has Lebesgue measure zero. [Balan, Casazza, and Edidin, 2006] [Conca, Edidin, Hering, and Vinzant, 2015] Random “generic” measurements → uniqueness with probability 1
hypotheses on the measurements distribution. Case 1 : the Li ’s are i.i.d. Gaussian, and isotropic. Case 2 : masked Fourier transform. M1 , ..., MJ ∈ Cn i.i.d. Lk,j : x → Mj x[k] (j ≤ L, k = 0, ..., n − 1) Theorem Uniqueness and stability hold with high probability, provided that : Card(I) ≥ Cn (case 1) J ≥ C log n (case 2) [Cand` es and Li, 2014] [Cand` es, Li, and Soltanolkotabi, 2015; Gross, Krahmer, and Kueng, 2015]
measurements Advantages : strong results (uniqueness, stability). Drawbacks : depending on the application, the measurements may or may not be possible to choose at random. But the questions of uniqueness and stability tend to be harder to analyze when the measurements are deterministic.
well-known case : the Fourier transform Reconstruct f ∈ L2(Rd ) from |ˆ f | s.t. f is compactly-supported Many applications in physics. → Has been studied for around 60 years. This case can be studied with harmonic analysis techniques. [Akutowicz, 1956; Walther, 1963] [Barakat and Newsam, 1984] Unfortunately, If d = 1, no uniqueness. If d ≥ 2, uniqueness “almost surely” but stability issues.
used for the Fourier transform can be adapted to more sophisticated cases. Example : fractional Fourier transform [Jaming, 2014] → Uniqueness, but no stability. The wavelet transform [Mallat and Waldspurger, 2014] The modulus of the wavelet transform is a widely-used representation of audio signals (scalogram ≈ spectrogram). Studying the phase retrieval problem allows to better understand why this representation is so useful in audio processing, modify the scalogram of a signal, then reconstruct a modiﬁed signal.
consider Cauchy wavelets. Uniqueness Any function f ∈ L2(R, R) is uniquely determined by the modulus of its wavelet transform, up to trivial ambiguities. Stability There is no “strong stability” : |Wf | ≈ |Wg| ⇒ f ≈ g. There is a form of “local stability”.
Random measurements Well-posed problems (uniqueness and stability). “Physical” case Not all problems are well-posed. There are stability issues, although they may not be dramatic.
Principle Start with an initial estimation. Reﬁne the estimation step after step. Hope to converge towards a solution. Most well-known : alternate projections. [Gerchberg and Saxton, 1972; Fienup, 1982] Advantages : relatively fast, simple to implement. Main drawback : the problem is not convex, and the algorithm may converge to a local optimum instead of the correct solution.
the initialization When the measurements are random, and follow a known probability law, a good estimation of x can be found with a spectral method. → Wirtinger Flow [Cand` es, Li, and Soltanolkotabi, 2015] [Chen and Cand` es, 2015] Theorem If the measurements are i.i.d. Gaussian and isotropic, then a variant of Wirtinger Flow succeds with high probability, provided that Card I ≥ Cn, for some C > 0. Drawback : a priori limited to random cases.
Principle Minimization of a non-convex functional. Approximate as the minimization of a convex functional on a larger space. Hope that the solutions on the small and large spaces coincide. First example : PhaseLift. [Cand` es, Eldar, Strohmer, and Voroninski, 2011] [Chai, Moscoso, and Papanicolaou, 2011] Advantage : much more precise (theoretical guarantees in some cases). Drawback : high complexity.
PhaseLift M∗ def = tM V = Cn and ∀i = 1, ..., p, Li ∈ M1,n (C). ﬁnd x ∈ Cn s.t. ∀i, |Li (x)| = bi ⇐⇒ ﬁnd x ∈ Cn s.t. ∀i, x∗L∗ i Li x = b2 i ﬁnd X ∈ Mn (C) s.t. ∀i, Tr(L∗ i Li X) = b2 i § ¦ ¤ ¥ rank (X) = 1 ⇐⇒ ﬁnd x ∈ Cn s.t. ∀i, Tr(L∗ i Li xx∗) = b2 i
PhaseLift M∗ def = tM V = Cn and ∀i = 1, ..., p, Li ∈ M1,n (C). ﬁnd x ∈ Cn s.t. ∀i, |Li (x)| = bi ⇐⇒ ﬁnd x ∈ Cn s.t. ∀i, x∗L∗ i Li x = b2 i ﬁnd X ∈ Mn (C) s.t. ∀i, Tr(L∗ i Li X) = b2 i § ¦ ¤ ¥ rank (X) = 1 ⇐⇒ ﬁnd x ∈ Cn s.t. ∀i, Tr(L∗ i Li xx∗) = b2 i Not convex
PhaseLift M∗ def = tM V = Cn and ∀i = 1, ..., p, Li ∈ M1,n (C). ﬁnd x ∈ Cn s.t. ∀i, |Li (x)| = bi ⇐⇒ ﬁnd x ∈ Cn s.t. ∀i, x∗L∗ i Li x = b2 i ﬁnd X ∈ Mn (C) s.t. ∀i, Tr(L∗ i Li X) = b2 i § ¦ ¤ ¥ rank (X) = 1 ⇐⇒ ﬁnd x ∈ Cn s.t. ∀i, Tr(L∗ i Li xx∗) = b2 i Approximation − − − − − − − − − − → min Tr X s.t. ∀i, Tr(L∗ i Li X) = b2 i X 0
PhaseLift M∗ def = tM V = Cn and ∀i = 1, ..., p, Li ∈ M1,n (C). ﬁnd x ∈ Cn s.t. ∀i, |Li (x)| = bi ⇐⇒ ﬁnd x ∈ Cn s.t. ∀i, x∗L∗ i Li x = b2 i ﬁnd X ∈ Mn (C) s.t. ∀i, Tr(L∗ i Li X) = b2 i § ¦ ¤ ¥ rank (X) = 1 ⇐⇒ ﬁnd x ∈ Cn s.t. ∀i, Tr(L∗ i Li xx∗) = b2 i Approximation − − − − − − − − − − → min Tr X s.t. ∀i, Tr(L∗ i Li X) = b2 i X 0 (Convex)
principle, but we focus on the reconstruction of {Li (x)}, instead of the direct reconstruction of x. Advantages In terms of precision, there is an equivalence between PhaseCut and a modiﬁed version of PhaseLift. → PhaseCut is “as precise” as PhaseLift. Diﬀerent complexity. → PhaseCut is more eﬃcient is some cases.
PhaseCut min Tr X s.t. ∀i, Tr(L∗ i Li X) = b2 i X 0 min Tr(MU) s.t. diag(U) = 1 U 0 X is of size n × n ; U is of size p × p, with p ∼ 4n. → U is larger than X. The constraint diag(U) = 1 is simpler. It makes PhaseCut and instance of Max-Cut. → Possible optimizations for PhaseCut.
depends on the algorithm used to minimize the functional. With an interior-point method, PhaseLift : O a2n4.5 log 1 ; PhaseCut : O a3.5n3.5 log 1 . 10 20 30 0 2 4 6 8 x 108 PhaseLift PhaseCut (a = p/n ≈ 4 ; is the precision.) Interior-point methods are adapted to problems where high precision is required.
transform. (No uniqueness.) Masked Fourier transform, with 4 masks. (Uniqueness and strong stability, with high probability.) Wavelet transform. (Uniqueness ; weak stability.) Signals of size 128. Four algorithms Alternate projections Wirtinger Flow PhaseLift PhaseCut
modulus of the wavelet transform for audio signals : n ∼ 10000. The cost of convexiﬁed methods is prohibitive. Alternative Iterative algorithms that use the particular structure of the signal or the measurements.
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