SurReal: Complex-Valued Learning as Principled Transformations on a Scaling and Rotation Manifold

Area of Science:

• Deep Learning
• Signal Processing
• Image Processing
• Complex Analysis

Background:

• Complex-valued data are prevalent in signal and image processing.
• Current complex-valued deep learning models lag behind real-valued counterparts.
• Existing methods often treat complex values as independent real channels, ignoring their geometric properties.

Purpose of the Study:

• To propose a principled geometric approach for complex-valued deep learning.
• To develop a model that accounts for the underlying geometry of complex numbers.
• To improve the performance and efficiency of complex-valued deep learning models.

Main Methods:

• Modeling the space of complex numbers as a product manifold of scaling and rotations.
• Defining convolution as a weighted Fréchet mean equivariant to scaling/rotation actions.
• Developing manifold-aware nonlinear activation functions and residual connections.

Main Results:

• The proposed model, SurReal, achieves high performance on MSTAR and RadioML datasets.
• SurReal models demonstrate superior performance with significantly smaller model sizes compared to baselines.
• The geometric approach effectively handles complex-valued data properties like scaling and rotation.

Conclusions:

• A geometric manifold perspective offers a powerful framework for complex-valued deep learning.
• SurReal provides an efficient and effective solution for complex-valued deep learning tasks.
• This approach advances the field by better leveraging the properties of complex-valued data.