Riemannian Residual Neural Networks

Recent methods in geometric deep learning have introduced various neural networks to operate over data that lie on Riemannian manifolds. Such networks are often necessary to learn well over graphs with a hierarchical structure or to learn over manifold-valued data encountered in the natural sciences. These networks are often inspired by and directly generalize standard Euclidean neural networks. However, extending Euclidean networks is difficult and has only been done for a select few manifolds. In this work, we examine the residual neural network (ResNet) and show how to extend this construction to general Riemannian manifolds in a geometrically principled manner. Originally introduced to help solve the vanishing gradient problem, ResNets have become ubiquitous in machine learning due to their beneficial learning properties, excellent empirical results, and easy-to-incorporate nature when building varied neural networks. We find that our Riemannian ResNets mirror these desirable properties: when compared to existing manifold neural networks designed to learn over hyperbolic space and the manifold of symmetric positive definite matrices, we outperform both kinds of networks in terms of relevant testing metrics and training dynamics.

Equivariant Manifold Flows

Tractably modelling distributions over manifolds has long been an important goal in the natural sciences. Recent work has focused on developing general machine learning models to learn such distributions. However, for many applications these distributions must respect manifold symmetries -- a trait which most previous models disregard. In this paper, we lay the theoretical foundations for learning symmetry-invariant distributions on arbitrary manifolds via equivariant manifold flows. We demonstrate the utility of our approach by using it to learn gauge invariant densities over $SU(n)$ in the context of quantum field theory.

Neural Manifold Ordinary Differential Equations

To better conform to data geometry, recent deep generative modelling techniques adapt Euclidean constructions to non-Euclidean spaces. In this paper, we study normalizing flows on manifolds. Previous work has developed flow models for specific cases; however, these advancements hand craft layers on a manifold-by-manifold basis, restricting generality and inducing cumbersome design constraints. We overcome these issues by introducing Neural Manifold Ordinary Differential Equations, a manifold generalization of Neural ODEs, which enables the construction of Manifold Continuous Normalizing Flows (MCNFs). MCNFs require only local geometry (therefore generalizing to arbitrary manifolds) and compute probabilities with continuous change of variables (allowing for a simple and expressive flow construction). We find that leveraging continuous manifold dynamics produces a marked improvement for both density estimation and downstream tasks.

Differentiating through the Fréchet Mean

Recent advances in deep representation learning on Riemannian manifolds extend classical deep learning operations to better capture the geometry of the manifold. One possible extension is the Fr\'echet mean, the generalization of the Euclidean mean; however, it has been difficult to apply because it lacks a closed form with an easily computable derivative. In this paper, we show how to differentiate through the Fr\'echet mean for arbitrary Riemannian manifolds. Then, focusing on hyperbolic space, we derive explicit gradient expressions and a fast, accurate, and hyperparameter-free Fr\'echet mean solver. This fully integrates the Fr\'echet mean into the hyperbolic neural network pipeline. To demonstrate this integration, we present two case studies. First, we apply our Fr\'echet mean to the existing Hyperbolic Graph Convolutional Network, replacing its projected aggregation to obtain state-of-the-art results on datasets with high hyperbolicity. Second, to demonstrate the Fr\'echet mean's capacity to generalize Euclidean neural network operations, we develop a hyperbolic batch normalization method that gives an improvement parallel to the one observed in the Euclidean setting.

Enhancing Adversarial Example Transferability with an Intermediate Level Attack

Neural networks are vulnerable to adversarial examples, malicious inputs crafted to fool trained models. Adversarial examples often exhibit black-box transfer, meaning that adversarial examples for one model can fool another model. However, adversarial examples are typically overfit to exploit the particular architecture and feature representation of a source model, resulting in sub-optimal black-box transfer attacks to other target models. We introduce the Intermediate Level Attack (ILA), which attempts to fine-tune an existing adversarial example for greater black-box transferability by increasing its perturbation on a pre-specified layer of the source model, improving upon state-of-the-art methods. We show that we can select a layer of the source model to perturb without any knowledge of the target models while achieving high transferability. Additionally, we provide some explanatory insights regarding our method and the effect of optimizing for adversarial examples in intermediate feature maps.

Fashion++: Minimal Edits for Outfit Improvement

Given an outfit, what small changes would most improve its fashionability? This question presents an intriguing new vision challenge. We introduce Fashion++, an approach that proposes minimal adjustments to a full-body clothing outfit that will have maximal impact on its fashionability. Our model consists of a deep image generation neural network that learns to synthesize clothing conditioned on learned per-garment encodings. The latent encodings are explicitly factorized according to shape and texture, thereby allowing direct edits for both fit/presentation and color/patterns/material, respectively. We show how to bootstrap Web photos to automatically train a fashionability model, and develop an activation maximization-style approach to transform the input image into its more fashionable self. The edits suggested range from swapping in a new garment to tweaking its color, how it is worn (e.g., rolling up sleeves), or its fit (e.g., making pants baggier). Experiments demonstrate that Fashion++ provides successful edits, both according to automated metrics and human opinion. Project page is at http://vision.cs.utexas.edu/projects/FashionPlus.

Adversarial Example Decomposition

Research has shown that widely used deep neural networks are vulnerable to carefully crafted adversarial perturbations. Moreover, these adversarial perturbations often transfer across models. We hypothesize that adversarial weakness is composed of three sources of bias: architecture, dataset, and random initialization. We show that one can decompose adversarial examples into an architecture-dependent component, data-dependent component, and noise-dependent component and that these components behave intuitively. For example, noise-dependent components transfer poorly to all other models, while architecture-dependent components transfer better to retrained models with the same architecture. In addition, we demonstrate that these components can be recombined to improve transferability without sacrificing efficacy on the original model.

Intermediate Level Adversarial Attack for Enhanced Transferability

Neural networks are vulnerable to adversarial examples, malicious inputs crafted to fool trained models. Adversarial examples often exhibit black-box transfer, meaning that adversarial examples for one model can fool another model. However, adversarial examples may be overfit to exploit the particular architecture and feature representation of a source model, resulting in sub-optimal black-box transfer attacks to other target models. This leads us to introduce the Intermediate Level Attack (ILA), which attempts to fine-tune an existing adversarial example for greater black-box transferability by increasing its perturbation on a pre-specified layer of the source model. We show that our method can effectively achieve this goal and that we can decide a nearly-optimal layer of the source model to perturb without any knowledge of the target models.

Semantic Segmentation with Scarce Data

Semantic segmentation is a challenging vision problem that usually necessitates the collection of large amounts of finely annotated data, which is often quite expensive to obtain. Coarsely annotated data provides an interesting alternative as it is usually substantially more cheap. In this work, we present a method to leverage coarsely annotated data along with fine supervision to produce better segmentation results than would be obtained when training using only the fine data. We validate our approach by simulating a scarce data setting with less than 200 low resolution images from the Cityscapes dataset and show that our method substantially outperforms solely training on the fine annotation data by an average of 15.52% mIoU and outperforms the coarse mask by an average of 5.28% mIoU.