Computer vision research has long aimed to build systems that are robust to spatial transformations found in natural data. Traditionally, this is done using data augmentation or hard-coding invariances into the architecture. However, too much or too little invariance can hurt, and the correct amount is unknown a priori and dependent on the instance. Ideally, the appropriate invariance would be learned from data and inferred at test-time. We treat invariance as a prediction problem. Given any image, we use a normalizing flow to predict a distribution over transformations and average the predictions over them. Since this distribution only depends on the instance, we can align instances before classifying them and generalize invariance across classes. The same distribution can also be used to adapt to out-of-distribution poses. This normalizing flow is trained end-to-end and can learn a much larger range of transformations than Augerino and InstaAug. When used as data augmentation, our method shows accuracy and robustness gains on CIFAR 10, CIFAR10-LT, and TinyImageNet.

How much can you say about the gradient of a neural network without computing a loss or knowing the label? This may sound like a strange question: surely the answer is "very little." However, in this paper, we show that gradients are more structured than previously thought. Gradients lie in a predictable low-dimensional subspace which depends on the network architecture and incoming features. Exploiting this structure can significantly improve gradient-free optimization schemes based on directional derivatives, which have struggled to scale beyond small networks trained on toy datasets. We study how to narrow the gap in optimization performance between methods that calculate exact gradients and those that use directional derivatives. Furthermore, we highlight new challenges in overcoming the large gap between optimizing with exact gradients and guessing the gradients.

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.