Blaschke Product Neural Networks (BPNN): A Physics-Infused Neural Network for Phase Retrieval of Meromorphic Functions

1 INTRODUCTION . Numerous physical systems are described by ordinary or partial differential equations whose solutions are given by holomorphic or meromorphic functions f ( z ) = |f ( z ) | exp ( jθ ( z ) ) in the complex domain , where |f ( z ) | and θ ( z ) are respectively the magnitude and phase of f ( z ) . In many cases , only |f ( jω ) | , the magnitude of f ( z ) , is observed on a finite set of points ( Ω ) of the purely imaginary jω-axis1 , since the coherent measurement of their phases may be expensive ( see Ch . 13 of Stern ( 2016 ) ) or require significant expertise ( King , 2009 ) . However , it is desirable to retrieve the lost phase θ ( jω ) from measurements of the magnitudes |f ( jω ) | for ω ∈ Ω , since the phase often encodes important information . For example in both spectroscopy and optical imaging , estimation of θ ( jω ) is often not possible . However , phase information determines the optical constants in the spectroscopy of materials ( Lucarini , 2005 ) , and is more important than |f ( jω ) | in optical imaging ( Stern , 2016 ) . The phase retrieval problem is also important in a number of other disciplines , including holography ( Fienup , 1982 ) , x-ray crystallography ( Nugent , 2007 ) , astronomy ( Stark , 1987 ) , blind deconvolution ( Shechtman et al. , 2015 ) , and coherent diffractive imaging ( Latychevskaia , 2018 ) . Many techniques for phase retrieval have been developed in the literature . The Kramers–Kronig formula , see Lucarini ( 2005 ) ( also known as Sokhotski–Plemelj formula ) , is noteworthy and is derived by the application of the Hilbert Transform to log f ( jω ) = log |f ( jω ) | + jθ ( jω ) . Unfortunately , for phase retrieval , this elegant formula requires knowledge of the magnitude |f ( jω ) | along the entire jω-axis . However , as mentioned above , acquiring these values is often impossible since 1For readers unfamiliar with phase retrieval problems and have questions about why function values are only observed on the purely imaginary line , please refer to the explanation we have included in the Appendix . Ω is typically a finite set with a small number of elements . This severely limits the application of Kramers–Kronig ’ s formula in many applications . Another approach is based on compressed sensing , but it requires sparsity assumptions , and stringent assumptions on the design or projection matrix ( i.e. , Restricted Isometry Property ( RIP ) ( Candes , 2008 ) , mutual incoherence ( Foucart & Rauhut , 2013 ) , etc ) that often do not hold for meromorphic functions encountered in practice . To overcome the above limitations , phase retrieval can also be cast as a data-driven regression problem , where we wish to learn a mapping from the magnitude of a signal , X ∈ RN , to its phase , Θ ∈ CN . If the number of frequencies ( |Ω| ) measured is N then the phase can be represented as a vector of complex numbers Θ = [ exp ( θ ( jω1 ) ) , · · · , exp ( θ ( jωN ) ) ] ∈ CN , where exp ( θ ( jωi ) ) is the phase at i-th frequency . Similarly , the signal ’ s corresponding magnitude vector can be given by |f ( jωi ) | , i = 1 , 2 , · · · , N , and the goal is to infer Θ based upon X . We can then learn this mapping using a data-driven model ( see Figure 1 below for an example ) . With a sufficient amount of training data , it is likely that a universal function approximator ( i.e . neural network ) , can learn the underlying function well within a tolerance range . For instance , deep learning has shown great success on phase retrieval problems in the imaging area where abundant training data is available ( Sinha et al. , 2017 ; Metzler et al. , 2018 ) . But sufficient training data can be expensive to acquire in many practical applications , or sometimes even impossible . In this paper , we depart from the existing literature by exploring the applications of physics-infused neural networks to this setting . The infusion of physics reduces the number of required training examples to a small enough amount that is practical for many measurements and design applications of interest . The main idea of the approach is described next . Given the fact that any phase value has a magnitude of 1 ( i.e.| exp ( jθ ( jω ) ) | = 1 for all ω ) , we observe that conditions of a Theorem of Helson and Sarason are then satisfied indicating that exp ( jθ ( jω ) ) can be approximated by a rational function whose numerator and denominator are both Blaschke products ( Garcia et al. , 2016 ) . This is then used to design a neural network for calculating coefficients of these Blaschke products which may be be used for phase retrieval . In Section 2 , we briefly review conventional phase retrieval methods , and existing applications of the neural network to phase retrieval problems . In Section 3 , we briefly review Blaschke products and the Helson-Sarason Theorem . In Section 4 , we propose our approach for baking these results into neural networks resulting in Blaschke Product Neural Networks . In Section 5 , we provide experimental results on various synthetic and metamaterials datasets along with a inclusive set of baselines to demonstrate the performance improvements of our construction . 2 RELATED WORK . We first briefly review existing methods of phase retrieval applicable to phase retrieval of analytic and meromorphic functions of interest . Perhaps the most important result in this area is the elegant calculation of Kramers and Kronig ( also known as Sokhotski–Plemelj formula ) obtained by the application of the Hilbert Transform to log f ( jω ) = log |f ( jω ) |+ jθ ( jω ) and is given by θ ( jω ) = 2ω π P.V . ∫ ∞ 0 log |f ( jβ ) | β2 − ω2 dβ , where P.V . denotes Cauchy ’ s principal value . We note that this can also be derived by invoking the Cauchy integration formula along a path containing the imaginary jω-axis . However , there is a major obstacle in the application of this elegant formula . In many important practical scenarios , the set Ω of frequencies ( points on jω-axis ) that we have measurements of |f ( jω ) | has a small cardinal number . In such cases , the approximation of Kramers-Kronig integral by a Riemann Sum produces unsatisfactory results . Other approaches used to analytically extend or extrapolate |f ( jω ) | have also demonstrated limited success ( Chalashkanov et al. , 2012 ) . Another approach to phase retrieval is by sparse representation . This approach assumes that it is a priori known that f ( z ) has a sparse representation in a known dictionary of functions D = { Φi ( z ) , i = 1 , 2 , · · · , M } i.e . it can be written as a linear sum of elements of D with only a small number of non-zero coefficients . Under this assumption , many techniques are developed to recover these coefficients and the phase of f ( z ) . Again , this is an interesting proposal but the underlying assumptions may not hold in many problems of interest . In particular , neither a dictionary D may be known nor classes of functions of interest may be sparse in any fixed dictionary . Recently , some new techniques based on deep learning have been developed to learn the structure of the underlying signal which we want to recover its phase . That is , deep neural networks are used to encode the prior knowledge on the structure of the signal instead of hard-coded assumptions such as sparsity ( Jagatap & Hegde , 2019 ) . However , these methods require a significant number of training samples from the magnitude of the underlying signal , which limits the applicability of these approaches on material science applications . Recently , DL has demonstrated its capability to solve the phase-retrieval problem , especially in the imaging area : Sinha et al . ( 2017 ) achieved lensless imaging for phase objects utilizing CNN . Xue et al . ( 2019 ) constructed a bayesian convolution neural network to achieve phase imaging while quantifying the uncertainty of the network output . Metzler et al . ( 2018 ) combined regularization-bydenoising framework with a convolution neural network for noisy phase retrieval for images . Goy et al . ( 2018 ) achieved low photon count phase retrieval with a Convolutional Neural Network ( CNN ) as well . To avoid the difficulty of getting a large set of paired phase and intensity measurements , Zhang et al . Zhang et al . ( 2021 ) used an unpaired dataset for phase-retrieval . Other works of Tayal et al . ( 2020 ) ; Houhou et al . ( 2020 ) have also used deep learning for end-to-end phase recovery . None of these works use the underlying physics in their deep learning approach or they require a large number of training samples ; hence , are not directly applicable to the physics applications we are interested in this paper . We took a different path of applying physical prior knowledge to our phase retrieval process . One closely related line of work to ours is solving partial differential equations using neural networks ( Dissanayake & Phan-Thien , 1994 ; Berg & Nyström , 2018 ; Zhang & Bilige , 2019 ; Chen et al. , 2018 ) . However , these methods typically need a lot of data and require careful selection of hyper-parameters ; additionally , they do not consider directly solving the phase retrieval problem . Our approach , on the other hand , needs no specific hyper-parameter search algorithm , and achieves high-accuracy result with a very limited training samples . 3 MATHEMATICAL BACKGROUND . We briefly review the mathematical background required in the rest of this paper . In particular , we review Blaschke products and the Helson-Sarason Theorem . We note that the standard presentation of the Blaschke products is motivated by inner functions in Hardy spaces presented over the unit disk whose boundary is the unit circle ( Rudin , 1987 ; Garcia et al. , 2016 ) . In our case , we are interested in the right hand half plane whose boundary is the jω-axis . These two are equivalent using the complex Mobius transformation ( z + 1 ) / ( 1− z ) . In this light , our exposition is closer to Akutowicz ( 1956 ) . Let a1 , a2 , a3 , · · · be an infinite sequence of numbers for which < ( ak ) ≥ 0 for all k for which ∞∑ j=1 < ( aj ) |aj |2 + 1 < ∞ , then for any integer n , any complex function given by B ( z ) = ( 1 + z 1− z ) n∏ j |aj − 1| aj − 1 |aj + 1| aj + 1 z − aj z + āj is referred to as a infinite Blaschke Product . If the set { a1 , a2 , · · · , ak } is finite , B ( z ) is referred to as a finite Blaschke product . For a finite Blaschke Product , it is clear that ∑k j=1 < ( aj ) |aj |2+1 < ∞ since there are only finite number of coefficients aj , and therefore this condition can be dropped . Note that roots with < ( ak ) = 0 do not contribute to the product . For both finite and infinite Blaschke products , we can see that |B ( z ) | = 1 for z = jω on the purely imaginary axis 2 . In this paper , we are only interested in finite Blaschke products because of the following approximation Theorem of Helson and Sarason . Theorem 1 Let u ( jω ) be any continuous function on the jω-axis for which |u ( jω ) | = 1 for all ω . Let > 0 . Then there exists finite Blaschke products B1 ( jω ) and B2 ( jω ) such that ‖u ( jω ) − B1 ( jω ) B2 ( jω ) ‖∞ < , where the ‖ · ‖∞ denotes the standard L∞ norm computed over the jω axis . Let b ( jω ) = exp ( jθ ( jω ) ) . We know that |b ( jω ) | = 1 because |b ( jω ) | = | exp ( jθ ( jω ) ) | = | cos ( θ ( jω ) ) + j sin ( θ ( jω ) ) | = √ cos ( θ ( jω ) ) 2 + sin ( θ ( jω ) ) 2 = 1 . Applying the Helson-Sarason Theorem , for analytic or meromorphic functions f ( jω ) ( with no poles on the imaginary axis ) , the phase b ( jω ) can be approximated by a rational function of finite Blaschke products . The main theme of this paper is that these Blaschke products can be in turn computed using neural networks with a small amount of data . From the expression of finite Blaschke products , we can see that Bi ( jω ) = AiPi ( jω ) /P ∗i ( jω ) for i = 1 , 2 where Ai a complex number with |Ai| = 1 and Pi ( jω ) is a complex polynomial of ω and P ∗i ( z ) = P̄i ( −z̄ ) , where P̄i ( jω ) is constructed from Pi ( jω ) by replacing all its coefficients with their respective conjugates . Note that [ P ∗ ( jω ) ] ∗ = P ( jω ) . Let A = A1/A2 and represent the Blaschke products B1 and B2 in the above theorem with the complex polynomials , we have the approximating rational function written asAP1 ( jω ) P ∗2 ( jω ) /P ∗ 1 ( jω ) P2 ( jω ) with |A| = 1 . Thus we have the estimation for phase function b ( jw ) as : b ( jw ) ≈ AP1 ( jw ) P ∗ 2 ( jw ) P ∗1 ( jw ) P2 ( jw ) = A P ( jw ) P ∗ ( jw ) , ( 1 ) where P ( z ) = P1 ( z ) P ∗2 ( z ) and P1 ( z ) and P ∗ 2 ( z ) correspond to respectively factors of P ( z ) with roots with positive and negative real parts .

Summary: In complex analysis, the Blaschke product is an important family of bounded analytic functions in the open unit disk that is constructed to have roots at prescribed complex numbers. Finite Blaschke products are classically used for constructing rational functions with prescribed poles or zeros, which are then used in the approximation theory, differential equations, dynamical systems, and harmonic analysis.  Here, the authors use finite Blaschke products for phase retrieval of holomorphic and meromorphic functions. By looking at the argument of the approximant, the authors are able to estimate the phase at a set of frequencies. The authors use an L2 loss function to learn the parameters in the Blaschke product.