Coordinate Descent Phase Retrieval for Signal Recovery
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Solution Overview
Problem
Phase retrieval, which involves recovering a complete signal from intensity information, is challenging due to its nonconvex optimization nature and high computational complexity, particularly in being NP-hard, making existing methods like Gerchberg-Saxton, Wirtinger Flow, PhaseLift, and PhaseCut inefficient with lengthy observations and high computational requirements.
Innovation Solution
The use of cyclic, randomized, and greedy coordinate descent techniques to minimize multivariate quartic polynomials, allowing for the recovery of signals by solving a single unknown value at each iteration, reducing the problem to univariate quartic polynomial minimization and achieving closed-form roots of cubic polynomials, thereby simplifying the computational complexity.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If conventional phase retrieval techniques (Gerchberg-Saxton, Wirtinger Flow, PhaseLift, PhaseCut) are used to recover signals from intensity information, then signal recovery can be achieved, but the computational complexity becomes extremely high and the number of iterations required becomes very large
Solution Approach 1:
The patent applies coordinate descent by segmenting the multivariate optimization problem into univariate subproblems. At each iteration, only one variable is updated while others are held constant, transforming the complex multivariate quartic minimization into a sequence of simpler univariate quartic minimizations. This segmentation dramatically reduces computational complexity per iteration while maintaining signal recovery capability.
Solution Approach 2:
The patent changes the approach from global optimization methods to coordinate-wise optimization. By altering the optimization strategy to focus on single-variable updates with closed-form solutions for univariate quartics, the method reduces the computational burden. The closed-form solution for univariate quartic minimization further simplifies each step compared to iterative numerical methods.
2Reliability
If convex relaxation methods (PhaseLift, PhaseCut) are used to handle the nonconvex phase retrieval problem, then the problem becomes solvable, but the number of iterations and computational resources required increase significantly
Solution Approach 1:
The patent segments the optimization into coordinate-wise updates, where each update step involves minimizing a univariate quartic polynomial. This segmentation allows for closed-form solutions at each step, avoiding the need for numerous iterations required by convex relaxation methods. The coordinate descent approach directly targets the nonconvex problem structure rather than relying on convex approximations.
3Reliability
If the Gerchberg-Saxton and Wirtinger Flow techniques directly address the nonconvex formulation, then phase retrieval can be performed, but the computational complexity and iteration count become excessively high
Solution Approach 1:
The patent segments the multivariate optimization into independent univariate problems. By updating one coordinate at a time with closed-form minimization of univariate quartics, the method achieves much faster convergence than gradient-based methods. This segmentation transforms the computationally intensive global optimization into a series of simple local minimizations.
Solution Approach 2:
The patent replaces iterative numerical optimization mechanisms with closed-form analytical solutions. Instead of using gradient descent or alternating minimization that require numerous iterative steps, the coordinate descent method uses closed-form solutions for univariate quartic minimization, substituting mechanical iteration with direct calculation for each coordinate update.
Data Source
AI summary
Coordinate descent is applied to recover a signal-of-interest from only magnitude information. In doing so, a single unknown value is solved at each iteration, while all other variables are held constant. As a result, only minimization of a univariate quartic polynomial is required, which is efficiently achieved by finding the closed-form roots of a cubic polynomial. Cyclic, randomized, and/or a greedy coordinate descent technique can be used. Each coordinate descent technique globally converges to a stationary point of the nonconvex problem, and specifically, the randomized coordinate descent technique locally converges to the global minimum and attains exact recovery of the signal-of-interest at a geometric rate with high probability when the sample size is sufficiently large. The cyclic and randomized coordinate descent techniques can also be modified via minimization of the l1-regularized quartic polynomial for phase retrieval of sparse signals-of-interest, i.e., those signals with only a few nonzero elements.