Co-domain Symmetry for Complex-Valued Deep Learning

TL;DR

The paper addresses the vulnerability of complex-valued deep networks to complex-valued scaling by treating it as a co-domain transformation and introduces a cohesive framework of complex-scale equivariant and invariant layers. It delivers a suite of architectural primitives—Econv, E{N}, magnitude-based pooling, phase-equivariant BN, division/conjugate invariant layers, and a generalized Tangent ReLU—along with two complex color encodings (Sliding and LAB) and two model types (Type I and Type E) to preserve information while achieving scale invariance. Empirically, CDS classifiers achieve higher accuracy, better generalization, and robustness to co-domain transformations across MSTAR, CIFAR10/100, and SVHN with significantly fewer parameters than prior complex-valued models like DCN and SurReal, plus favorable bias-variance characteristics. The work demonstrates practical impact in robust complex-valued processing for both specialized (SAR) and standard vision tasks, and it provides insights and ablations that guide future design of co-domain-aware architectures and color representations.

Abstract

We study complex-valued scaling as a type of symmetry natural and unique to complex-valued measurements and representations. Deep Complex Networks (DCN) extends real-valued algebra to the complex domain without addressing complex-valued scaling. SurReal takes a restrictive manifold view of complex numbers, adopting a distance metric to achieve complex-scaling invariance while losing rich complex-valued information. We analyze complex-valued scaling as a co-domain transformation and design novel equivariant and invariant neural network layer functions for this special transformation. We also propose novel complex-valued representations of RGB images, where complex-valued scaling indicates hue shift or correlated changes across color channels. Benchmarked on MSTAR, CIFAR10, CIFAR100, and SVHN, our co-domain symmetric (CDS) classifiers deliver higher accuracy, better generalization, robustness to co-domain transformations, and lower model bias and variance than DCN and SurReal with far fewer parameters.

Figures (9)

• Figure 1: An image is a function from the domain $\mathbb{R}^2$ to the co-domain $\mathbb{C}^N$. Image transformations like rotation and translation act on the domain, mapping points in $\mathbb{R}^2$ to other points, while leaving the underlying function values intact. Previous works like lieconvcohen2016group aim to produce architectures invariant to domain transformations. Co-domain transformations like color distortion or complex-valued scaling, on the other hand, act on the function values only.
• Figure 2: Our method learns invariant features with respect to complex-scaling of the input. All examples are from CIFAR 10 with our LAB encoding, undergoing multiplication by a unit complex number. (b, e) tSNE embedding trajectories from DCN trabelsi2017deep and our model. Each color represents a different example. Embeddings form tight clusters for our model, and irregular overlapping curves for DCN. (c) Visualization of our complex-valued embedding of LAB information. The $L^*$ channel is visualized as a grayscale image, and the complex-valued $a^* + \textit{i} b^*$ visualized as a color image. (d) Model confidence of the correct class for a single example. Higher confidence means larger radius. DCN predictions are highly variable, while our model is robust to complex-scaling and thus constant. (f) Accuracy under complex-scaling and color jitter. Red bars represent complex-rotations sampled from different rotation ranges. Blue bars represent color jitter (as used in mean_teach). Our method maintains high accuracy across complex-rotations and color jitter, whereas DCN and Real-valued CNN fail. SurReal chakraborty2019surreal is robust, but has low overall accuracy. Our method combines high accuracy with robustness. (g) Average accuracy under different rotation ranges, comparing DCN with phase normalization (dotted blue line) and without phase normalization (solid blue line) against our method. The color encoding has a complicated phase distribution, and phase normalization fails to estimate the amount of rotation, resulting in poor accuracy. Our method is thus more suitable for complicated phase distributions.
• Figure 3: Our equivariant non-linearity, $\mathcal{E}\{\mathcal{N}\}$, works in four stages. We compute the channel mean $m$ of the input feature $f$ and normalize it to retain only phase information. This normalized mean vector $\hat{m}$ is equivariant to input phase. We multiply $f$ by $\hat{m}^*$ to cancel the input phase, resulting in a phase-invariant feature $f \odot \hat{m}^*$. We feed this feature to the non-linearity $\mathcal{N}$ and multiply by $\hat{m}$ to restore the previously cancelled input phase. The result is equivariant in phase and also equivariant in magnitude if $\mathcal{N}$ is.
• Figure 4: Our Division Layer is complex-scale invariant, and preserves more information than the Distance Transform chakraborty2019surreal. Consider 4 complex numbers (colored dots) and the reference (black dot). Complex-scaling changes the orientation and size of the trapezoid. While the manifold distances chakraborty2019surreal from the black dot to the colored dots are identical, the Division Layer output is invariant and distinct for each dot (the same colored dots around $0^\circ$).
• Figure 5: Our Generalized Tangent ReLU transforms the input in three stages: (a) given input complex vectors, it multiplies each channel with a learned scaling factor, (b) thresholds the input magnitude and phase with hyperparameter $r$, and (c) scales the phase to adapt the output distribution.