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Equivariance of linear neural network layers is well studied. In this work, we relax the equivariance condition to only be true in a projective sense. We propose a way to construct a projectively equivariant neural network through building a standard equivariant network where the linear group representations acting on each intermediate feature space are "multiplicatively modified lifts" of projective group representations. By theoretically studying the relation of projectively and linearly equivariant linear layers, we show that our approach is the most general possible when building a network out of linear layers. The theory is showcased in two simple experiments.

In this paper, we study random neural networks which are single-hidden-layer feedforward neural networks whose weights and biases are randomly initialized. After this random initialization, only the linear readout needs to be trained, which can be performed efficiently, e.g., by the least squares method. By viewing random neural networks as Banach space-valued random variables, we prove a universal approximation theorem within a large class of Bochner spaces. Hereby, the corresponding Banach space can be significantly more general than the space of continuous functions over a compact subset of a Euclidean space, namely, e.g., an $L^p$-space or a Sobolev space, where the latter includes the approximation of the derivatives. Moreover, we derive approximation rates and an explicit algorithm to learn a deterministic function by a random neural network. In addition, we provide a full error analysis and study when random neural networks overcome the curse of dimensionality in the sense that the training costs scale at most polynomially in the input and output dimension. Furthermore, we show in two numerical examples the empirical advantages of random neural networks compared to fully trained deterministic neural networks.

In this article, we create an artificial neural network (ANN) that combines both classical and modern techniques for determining the key length of a Vigen\`{e}re cipher. We provide experimental evidence supporting the accuracy of our model for a wide range of parameters. We also discuss the creation and features of this ANN along with a comparative analysis between our ANN, the index of coincidence, and the twist-based algorithms.

In this paper, we study the problem of efficiently and effectively embedding the high-dimensional spatio-spectral information of hyperspectral (HS) images, guided by feature diversity. Specifically, based on the theoretical formulation that feature diversity is correlated with the rank of the unfolded kernel matrix, we rectify 3D convolution by modifying its topology to enhance the rank upper-bound. This modification yields a rank-enhanced spatial-spectral symmetrical convolution set (ReS$^3$-ConvSet), which not only learns diverse and powerful feature representations but also saves network parameters. Additionally, we also propose a novel diversity-aware regularization (DA-Reg) term that directly acts on the feature maps to maximize independence among elements. To demonstrate the superiority of the proposed ReS$^3$-ConvSet and DA-Reg, we apply them to various HS image processing and analysis tasks, including denoising, spatial super-resolution, and classification. Extensive experiments show that the proposed approaches outperform state-of-the-art methods both quantitatively and qualitatively to a significant extent. The code is publicly available at //github.com/jinnh/ReSSS-ConvSet.

Recently, graph neural networks have been gaining a lot of attention to simulate dynamical systems due to their inductive nature leading to zero-shot generalizability. Similarly, physics-informed inductive biases in deep-learning frameworks have been shown to give superior performance in learning the dynamics of physical systems. There is a growing volume of literature that attempts to combine these two approaches. Here, we evaluate the performance of thirteen different graph neural networks, namely, Hamiltonian and Lagrangian graph neural networks, graph neural ODE, and their variants with explicit constraints and different architectures. We briefly explain the theoretical formulation highlighting the similarities and differences in the inductive biases and graph architecture of these systems. We evaluate these models on spring, pendulum, gravitational, and 3D deformable solid systems to compare the performance in terms of rollout error, conserved quantities such as energy and momentum, and generalizability to unseen system sizes. Our study demonstrates that GNNs with additional inductive biases, such as explicit constraints and decoupling of kinetic and potential energies, exhibit significantly enhanced performance. Further, all the physics-informed GNNs exhibit zero-shot generalizability to system sizes an order of magnitude larger than the training system, thus providing a promising route to simulate large-scale realistic systems.

What is learned by sophisticated neural network agents such as AlphaZero? This question is of both scientific and practical interest. If the representations of strong neural networks bear no resemblance to human concepts, our ability to understand faithful explanations of their decisions will be restricted, ultimately limiting what we can achieve with neural network interpretability. In this work we provide evidence that human knowledge is acquired by the AlphaZero neural network as it trains on the game of chess. By probing for a broad range of human chess concepts we show when and where these concepts are represented in the AlphaZero network. We also provide a behavioural analysis focusing on opening play, including qualitative analysis from chess Grandmaster Vladimir Kramnik. Finally, we carry out a preliminary investigation looking at the low-level details of AlphaZero's representations, and make the resulting behavioural and representational analyses available online.

This paper does not describe a working system. Instead, it presents a single idea about representation which allows advances made by several different groups to be combined into an imaginary system called GLOM. The advances include transformers, neural fields, contrastive representation learning, distillation and capsules. GLOM answers the question: How can a neural network with a fixed architecture parse an image into a part-whole hierarchy which has a different structure for each image? The idea is simply to use islands of identical vectors to represent the nodes in the parse tree. If GLOM can be made to work, it should significantly improve the interpretability of the representations produced by transformer-like systems when applied to vision or language

In this paper, we focus on the self-supervised learning of visual correspondence using unlabeled videos in the wild. Our method simultaneously considers intra- and inter-video representation associations for reliable correspondence estimation. The intra-video learning transforms the image contents across frames within a single video via the frame pair-wise affinity. To obtain the discriminative representation for instance-level separation, we go beyond the intra-video analysis and construct the inter-video affinity to facilitate the contrastive transformation across different videos. By forcing the transformation consistency between intra- and inter-video levels, the fine-grained correspondence associations are well preserved and the instance-level feature discrimination is effectively reinforced. Our simple framework outperforms the recent self-supervised correspondence methods on a range of visual tasks including video object tracking (VOT), video object segmentation (VOS), pose keypoint tracking, etc. It is worth mentioning that our method also surpasses the fully-supervised affinity representation (e.g., ResNet) and performs competitively against the recent fully-supervised algorithms designed for the specific tasks (e.g., VOT and VOS).

Deep neural networks have achieved remarkable success in computer vision tasks. Existing neural networks mainly operate in the spatial domain with fixed input sizes. For practical applications, images are usually large and have to be downsampled to the predetermined input size of neural networks. Even though the downsampling operations reduce computation and the required communication bandwidth, it removes both redundant and salient information obliviously, which results in accuracy degradation. Inspired by digital signal processing theories, we analyze the spectral bias from the frequency perspective and propose a learning-based frequency selection method to identify the trivial frequency components which can be removed without accuracy loss. The proposed method of learning in the frequency domain leverages identical structures of the well-known neural networks, such as ResNet-50, MobileNetV2, and Mask R-CNN, while accepting the frequency-domain information as the input. Experiment results show that learning in the frequency domain with static channel selection can achieve higher accuracy than the conventional spatial downsampling approach and meanwhile further reduce the input data size. Specifically for ImageNet classification with the same input size, the proposed method achieves 1.41% and 0.66% top-1 accuracy improvements on ResNet-50 and MobileNetV2, respectively. Even with half input size, the proposed method still improves the top-1 accuracy on ResNet-50 by 1%. In addition, we observe a 0.8% average precision improvement on Mask R-CNN for instance segmentation on the COCO dataset.

Nowadays, the Convolutional Neural Networks (CNNs) have achieved impressive performance on many computer vision related tasks, such as object detection, image recognition, image retrieval, etc. These achievements benefit from the CNNs outstanding capability to learn the input features with deep layers of neuron structures and iterative training process. However, these learned features are hard to identify and interpret from a human vision perspective, causing a lack of understanding of the CNNs internal working mechanism. To improve the CNN interpretability, the CNN visualization is well utilized as a qualitative analysis method, which translates the internal features into visually perceptible patterns. And many CNN visualization works have been proposed in the literature to interpret the CNN in perspectives of network structure, operation, and semantic concept. In this paper, we expect to provide a comprehensive survey of several representative CNN visualization methods, including Activation Maximization, Network Inversion, Deconvolutional Neural Networks (DeconvNet), and Network Dissection based visualization. These methods are presented in terms of motivations, algorithms, and experiment results. Based on these visualization methods, we also discuss their practical applications to demonstrate the significance of the CNN interpretability in areas of network design, optimization, security enhancement, etc.