[Paper Review] SurReal: Complex-Valued Learning as Principled Transformations on a Scaling and Rotation Manifold

Rudrasis Chakraborty, Yifei Xing+1 more|arXiv (Cornell University)|Oct 18, 2019

Cognitive Science and Education Research47 references4 citations

TL;DR

This paper proposes SurReal, a geometric deep learning framework for complex-valued data that models complex numbers as a Riemannian manifold of scaling and rotation, enabling equivariant convolutions and invariant distance transforms. The method achieves state-of-the-art performance on MSTAR and RadioML with only 0.7% the size of real-valued baselines and 58% of complex-valued baselines.

ABSTRACT

Complex-valued data is ubiquitous in signal and image processing applications, and complex-valued representations in deep learning have appealing theoretical properties. While these aspects have long been recognized, complex-valued deep learning continues to lag far behind its real-valued counterpart. We propose a principled geometric approach to complex-valued deep learning. Complex-valued data could often be subject to arbitrary complex-valued scaling; as a result, real and imaginary components could co-vary. Instead of treating complex values as two independent channels of real values, we recognize their underlying geometry: We model the space of complex numbers as a product manifold of non-zero scaling and planar rotations. Arbitrary complex-valued scaling naturally becomes a group of transitive actions on this manifold. We propose to extend the property instead of the form of real-valued functions to the complex domain. We define convolution as weighted Fréchet mean on the manifold that is equivariant to the group of scaling/rotation actions, and define distance transform on the manifold that is invariant to the action group. The manifold perspective also allows us to define nonlinear activation functions such as tangent ReLU and G-transport, as well as residual connections on the manifold-valued data. We dub our model SurReal, as our experiments on MSTAR and RadioML deliver high performance with only a fractional size of real-valued and complex-valued baseline models.

Motivation & Objective

• To address the limitation of real-valued deep learning in handling complex-valued data, which often involves co-varying real and imaginary components due to arbitrary complex scaling.
• To develop a principled geometric framework that treats complex numbers as points on a non-Euclidean manifold of scaling and rotation, rather than independent real/imaginary channels.
• To embed invariance to complex-valued scaling directly into the network architecture, avoiding data augmentation and improving generalization.
• To design novel layer functions—such as weighted Fréchet mean convolution, distance transform, and manifold-valued activation—tailored to the complex manifold geometry.

Proposed method

• Model the space of complex numbers as a product manifold of non-zero scaling (R>0) and planar rotation (S¹), forming a Lie group action.
• Define convolution as a weighted Fréchet mean on this manifold, ensuring equivariance to complex-valued scaling and rotation.
• Introduce a distance transform layer that computes manifold distance between feature maps and their Fréchet mean, achieving invariance to group actions.
• Propose tangent ReLU and G-transport as nonlinear activation functions defined on the manifold, preserving geometric structure.
• Construct residual connections on the manifold using exponential and logarithmic maps, enabling deep architectures.
• Use the resulting SurReal model to classify complex-valued data in MSTAR and RadioML, with real-valued layers applied post-distance transform.

Experimental results

Research questions

• RQ1 How can complex-valued deep learning be made invariant to arbitrary complex-valued scaling without relying on data augmentation?
• RQ2 What geometric structure underlies complex-valued data that can be exploited for more efficient and principled learning?
• RQ3 Can convolution and activation functions be redefined on a Riemannian manifold of complex numbers to achieve equivariance and invariance?
• RQ4 How do manifold-based operations compare to standard real-valued or complex-valued baselines in terms of accuracy and model efficiency?
• RQ5 Can a deep learning model built on geometric principles achieve high performance with significantly reduced parameter count?

Key findings

• SurReal achieves 76.1% test accuracy on the RadioML dataset at SNR = 10, outperforming the real-valued baseline (72.7%) while using only 0.7% of its model size.
• At the same SNR, SurReal matches the performance of the complex-valued baseline (76.3%) but uses only 58% of its parameter count.
• The model demonstrates built-in invariance to complex-valued scaling, eliminating the need for data augmentation and reducing training time.
• On the MSTAR SAR dataset, SurReal delivers high performance with a significantly smaller architecture compared to both real-valued and complex-valued baselines.
• The use of manifold-based convolution and distance transform enables efficient representation learning that captures both magnitude and phase information effectively.
• The framework enables the design of geometrically consistent nonlinearities like tangent ReLU and G-transport, which are not possible in standard Euclidean space.