On Approximation Algorithms for Commutative Quaternion Polynomial Optimization
Chang He, Bo Jiang, Hongye Wang, Xihua Zhu
TL;DR
This paper expands optimization in the quaternion setting by studying sphere-constrained, homogeneous polynomials over the commutative quaternion domain, linking them to best rank-one quaternion tensor approximation. It develops a novel quaternion probability inequality and two levels of randomized algorithms based on tensor relaxation and random sampling to achieve provable approximation guarantees. The main results provide explicit approximation ratios that depend on the degree d and sampling parameter γ, with distinct guarantees for odd and even d, and are validated through numerical experiments. These contributions offer a principled, intrinsic approach to quaternion polynomial optimization with practical implications for related tensor problems.
Abstract
Quaternion optimization has attracted significant interest due to its broad applications, including color face recognition, video compression, and signal processing. Despite the growing literature on quadratic and matrix quaternion optimization, to the best of our knowledge, the study on quaternion polynomial optimization still remains blank. In this paper, we introduce the first investigation into this fundamental problem, and focus on the sphere-constrained homogeneous polynomial optimization over the commutative quaternion domain, which includes the best rank-one tensor approximation as a special case. Our study proposes a polynomial-time randomized approximation algorithm that employs tensor relaxation and random sampling techniques to tackle this problem. Theoretically, we prove an approximation ratio for the algorithm providing a worst-case performance guarantee