Coordinate Descent for Unit-Modulus Least Squares Signal Processing
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Solution Overview
Problem
Current methods for solving unit-modulus least squares (UMLS) and unit-modulus quadratic program (UMQP) problems, such as semi-definite relaxation (SDR) and gradient projection (GP), face high computational complexity and slow convergence rates, making them unsuitable for large-scale scenarios in applications like MIMO detection, source localization, and phase synchronization.
Innovation Solution
The implementation of an iterative coordinate descent (CD) algorithmic framework with various coordinate index selection rules, including cyclic, randomized, randomly permuted, and greedy CD rules, to minimize an objective function subject to unit-modulus constraints, allowing for global convergence to a Karush-Kuhn-Tucker (KKT) point.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If semi-definite relaxation (SDR) approach is used to solve UMLS/UMQP problems, then the nonconvex constraint is relaxed to a convex optimization problem, but the problem dimension is lifted and computational complexity becomes high
Solution Approach 1:
The patent segments the high-dimensional matrix optimization problem into multiple low-dimensional vector optimization problems by applying coordinate descent. Instead of optimizing the entire matrix X simultaneously, the algorithm iteratively optimizes individual columns (vectors) of X, reducing the computational burden from O(n^7) to O(n^4) per iteration while maintaining convergence to a feasible solution.
Solution Approach 2:
The patent changes the optimization parameters from matrix-level operations to vector-level operations. By transforming the problem from optimizing matrix X to optimizing column vectors x_i sequentially, the computational complexity is reduced while preserving the unit-modulus constraints through parameterization x_i = e^{jθ_i}.
2Reliability
If gradient projection (GP) approach is used to solve UMLS/UMQP problems, then the unit-modulus constraint is maintained at each iteration, but the convergence rate is slow
Solution Approach 1:
The patent segments the gradient computation into coordinate-specific updates. Instead of computing and projecting the full gradient for all variables simultaneously, the algorithm computes gradients for individual coordinates and updates them sequentially, which accelerates convergence while maintaining constraint satisfaction through the unit-modulus projection at each step.
Solution Approach 2:
The patent implements continuous optimization by iteratively updating coordinates in a cyclic manner, ensuring that the optimization process continuously progresses toward the optimal solution. The algorithm maintains feasibility at each iteration while making monotonic progress in reducing the objective function value.
3Measurement precision
If traditional optimization methods are used for large-scale UMLS/UMQP problems, then solution accuracy can be achieved, but the methods are not suitable for large-scale scenarios due to high computational complexity
Solution Approach 1:
The patent applies segmentation by dividing the large-scale optimization problem into smaller sub-problems that can be solved independently. By optimizing one column vector at a time while keeping others fixed, the algorithm reduces the computational complexity from exponential to polynomial scale, making large-scale problems tractable while maintaining solution accuracy through iterative refinement.
Solution Approach 2:
The patent transitions from solving the problem in the full n-dimensional space simultaneously to solving it in a sequence of 1-dimensional subspaces. This dimensionality reduction approach allows the algorithm to handle large-scale problems efficiently by exploiting the problem structure and updating variables one at a time along coordinate axes.
Data Source
AI summary
The present disclosure relates to methods and systems for signal processing using coordinate descent technique for solving technical implementation problems that are expressed as unit-modulus least squares (UMLS) and unit-modulus quadratic program (UMQP) problems. Embodiments provide for iteratively minimizing an objective function of a signal vector associated with a UMLS/UMQP problem expression over a set of coordinates of the signal vector to a convergence point. The objective function is minimized with respect to a vector element corresponding to a selected coordinate index, while other vector elements that do not correspond to the selected coordinate index are fixed. Accordingly, at each iteration, minimizing the objective function involves a solution to a one-dimensional univariate quadratic minimization. Embodiments also provide various coordinate index selection rules that include a cyclic CD rule (CCD), a randomized CD rule (RCD), randomly permuted CD rule (RPCD), and a greedy CD rule (CCD).