The last decade has witnessed an experimental revolution in data science and machine learning, epitomised by deep learning methods. Indeed, many high-dimensional learning tasks previously thought to be beyond reach -- such as computer vision, playing Go, or protein folding -- are in fact feasible with appropriate computational scale. Remarkably, the essence of deep learning is built from two simple algorithmic principles: first, the notion of representation or feature learning, whereby adapted, often hierarchical, features capture the appropriate notion of regularity for each task, and second, learning by local gradient-descent type methods, typically implemented as backpropagation. While learning generic functions in high dimensions is a cursed estimation problem, most tasks of interest are not generic, and come with essential pre-defined regularities arising from the underlying low-dimensionality and structure of the physical world. This text is concerned with exposing these regularities through unified geometric principles that can be applied throughout a wide spectrum of applications. Such a 'geometric unification' endeavour, in the spirit of Felix Klein's Erlangen Program, serves a dual purpose: on one hand, it provides a common mathematical framework to study the most successful neural network architectures, such as CNNs, RNNs, GNNs, and Transformers. On the other hand, it gives a constructive procedure to incorporate prior physical knowledge into neural architectures and provide principled way to build future architectures yet to be invented.
Rate-Distortion Optimized Quantization (RDOQ) has played an important role in the coding performance of recent video compression standards such as H.264/AVC, H.265/HEVC, VP9 and AV1. This scheme yields significant reductions in bit-rate at the expense of relatively small increases in distortion. Typically, RDOQ algorithms are prohibitively expensive to implement on real-time hardware encoders due to their sequential nature and their need to frequently obtain entropy coding costs. This work addresses this limitation using a neural network-based approach, which learns to trade-off rate and distortion during offline supervised training. As these networks are based solely on standard arithmetic operations that can be executed on existing neural network hardware, no additional area-on-chip needs to be reserved for dedicated RDOQ circuitry. We train two classes of neural networks, a fully-convolutional network and an auto-regressive network, and evaluate each as a post-quantization step designed to refine cheap quantization schemes such as scalar quantization (SQ). Both network architectures are designed to have a low computational overhead. After training they are integrated into the HM 16.20 implementation of HEVC, and their video coding performance is evaluated on a subset of the H.266/VVC SDR common test sequences. Comparisons are made to RDOQ and SQ implementations in HM 16.20. Our method achieves 1.64% BD-rate savings on luminosity compared to the HM SQ anchor, and on average reaches 45% of the performance of the iterative HM RDOQ algorithm.
Conventional neural message passing algorithms are invariant under permutation of the messages and hence forget how the information flows through the network. Studying the local symmetries of graphs, we propose a more general algorithm that uses different kernels on different edges, making the network equivariant to local and global graph isomorphisms and hence more expressive. Using elementary category theory, we formalize many distinct equivariant neural networks as natural networks, and show that their kernels are 'just' a natural transformation between two functors. We give one practical instantiation of a natural network on graphs which uses a equivariant message network parameterization, yielding good performance on several benchmarks.
In this paper, we present a novel adversarial lossy video compression model. At extremely low bit-rates, standard video coding schemes suffer from unpleasant reconstruction artifacts such as blocking, ringing etc. Existing learned neural approaches to video compression have achieved reasonable success on reducing the bit-rate for efficient transmission and reduce the impact of artifacts to an extent. However, they still tend to produce blurred results under extreme compression. In this paper, we present a deep adversarial learned video compression model that minimizes an auxiliary adversarial distortion objective. We find this adversarial objective to correlate better with human perceptual quality judgement relative to traditional quality metrics such as MS-SSIM and PSNR. Our experiments using a state-of-the-art learned video compression system demonstrate a reduction of perceptual artifacts and reconstruction of detail lost especially under extremely high compression.
A common approach to define convolutions on meshes is to interpret them as a graph and apply graph convolutional networks (GCNs). Such GCNs utilize isotropic kernels and are therefore insensitive to the relative orientation of vertices and thus to the geometry of the mesh as a whole. We propose Gauge Equivariant Mesh CNNs which generalize GCNs to apply anisotropic gauge equivariant kernels. Since the resulting features carry orientation information, we introduce a geometric message passing scheme defined by parallel transporting features over mesh edges. Our experiments validate the significantly improved expressivity of the proposed model over conventional GCNs and other methods.
Group equivariant convolutional neural networks (G-CNNs) have recently emerged as a very effective model class for learning from signals in the context of known symmetries. A wide variety of equivariant layers has been proposed for signals on 2D and 3D Euclidean space, graphs, and the sphere, and it has become difficult to see how all of these methods are related, and how they may be generalized. In this paper, we present a fairly general theory of equivariant convolutional networks. Convolutional feature spaces are described as fields over a homogeneous base space, such as the plane $\mathbb{R}^2$, sphere $S^2$ or a graph $\mathcal{G}$. The theory enables a systematic classification of all existing G-CNNs in terms of their group of symmetry, base space, and field type (e.g. scalar, vector, or tensor field, etc.). In addition to this classification, we use Mackey theory to show that convolutions with equivariant kernels are the most general class of equivariant maps between such fields, thus establishing G-CNNs as a universal class of equivariant networks. The theory also explains how the space of equivariant kernels can be parameterized for learning, thereby simplifying the development of G-CNNs for new spaces and symmetries. Finally, the theory introduces a rich geometric semantics to learned feature spaces, thus improving interpretability of deep networks, and establishing a connection to central ideas in mathematics and physics.
We present a convolutional network that is equivariant to rigid body motions. The model uses scalar-, vector-, and tensor fields over 3D Euclidean space to represent data, and equivariant convolutions to map between such representations. These SE(3)-equivariant convolutions utilize kernels which are parameterized as a linear combination of a complete steerable kernel basis, which is derived analytically in this paper. We prove that equivariant convolutions are the most general equivariant linear maps between fields over R^3. Our experimental results confirm the effectiveness of 3D Steerable CNNs for the problem of amino acid propensity prediction and protein structure classification, both of which have inherent SE(3) symmetry.
We propose a new model for digital pathology segmentation, based on the observation that histopathology images are inherently symmetric under rotation and reflection. Utilizing recent findings on rotation equivariant CNNs, the proposed model leverages these symmetries in a principled manner. We present a visual analysis showing improved stability on predictions, and demonstrate that exploiting rotation equivariance significantly improves tumor detection performance on a challenging lymph node metastases dataset. We further present a novel derived dataset to enable principled comparison of machine learning models, in combination with an initial benchmark. Through this dataset, the task of histopathology diagnosis becomes accessible as a challenging benchmark for fundamental machine learning research.
We propose Teacher-Student Curriculum Learning (TSCL), a framework for automatic curriculum learning, where the Student tries to learn a complex task and the Teacher automatically chooses subtasks from a given set for the Student to train on. We describe a family of Teacher algorithms that rely on the intuition that the Student should practice more those tasks on which it makes the fastest progress, i.e. where the slope of the learning curve is highest. In addition, the Teacher algorithms address the problem of forgetting by also choosing tasks where the Student's performance is getting worse. We demonstrate that TSCL matches or surpasses the results of carefully hand-crafted curricula in two tasks: addition of decimal numbers with LSTM and navigation in Minecraft. Using our automatically generated curriculum enabled to solve a Minecraft maze that could not be solved at all when training directly on solving the maze, and the learning was an order of magnitude faster than uniform sampling of subtasks.