# Convex Quaternion Optimization for Signal Processing: Theory and Applications

arXiv: https://arxiv.org/abs/2305.06879 · HTML: https://mlsanity.com/paper/2305.06879/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 46 references that are in the map.
- First posted on arXiv 9 May 2023.

## Builds on (most cited first, 6 of 6)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Quaternion Gradient and Hessian](https://mlsanity.com/paper/1406.3587/) (2014)
- [A general framework for constrained convex quaternion optimization](https://mlsanity.com/paper/2102.02763/) (2021)
- [Quaternion Non-negative Matrix Factorization: definition, uniqueness and algorithm](https://mlsanity.com/paper/1903.10593/) (2019)
- [Time-frequency analysis of bivariate signals](https://mlsanity.com/paper/1609.02463/) (2016)
- [An inexact restoration-nonsmooth algorithm with variable accuracy for stochastic nonsmooth convex optimization problems in machine learning and stochastic linear complementarity problems](https://mlsanity.com/paper/2103.13651/) (2021)

## Built on by (most cited first, 1 of 1)

- [Quaternionic Reweighted Amplitude Flow for Phase Retrieval in Image Reconstruction](https://mlsanity.com/paper/2501.02180/) (2025)

## Abstract (arXiv)

Convex optimization methods have been extensively used in the fields of communications and signal processing. However, the theory of quaternion optimization is currently not as fully developed and systematic as that of complex and real optimization. To this end, we establish an essential theory of convex quaternion optimization for signal processing based on the generalized Hamilton-real (GHR) calculus. This is achieved in a way which conforms with traditional complex and real optimization theory. For rigorous, We present five discriminant theorems for convex quaternion functions, and four discriminant criteria for strongly convex quaternion functions. Furthermore, we provide a fundamental theorem for the optimality of convex quaternion optimization problems, and demonstrate its utility through three applications in quaternion signal processing. These results provide a solid theoretical foundation for convex quaternion optimization and open avenues for further developments in signal processing applications.