We extra define the following notations for the proof. In Assumption 3.2, we assume the Lipschitz continuity and smoothness for all the activation functions. In the proof of lemmas, e.g., Lemma B.1 and B.2, we only use the fact that they are Lipschitz continuous and smooth, as well as bounded by a constant 0 > 0 at point 0, hence we use () to denote all the activation functions like what we do in Assumption 3.2 for simplicity. Additionally, in the following we introduce notations of the derivatives, mainly used in the proof of Lemma B.1 and Lemma B.2. By definition of feedforward neural networks in Section 2, different from the standard neural networks such as FCNs and CNNs in which the connection between neurons are generally only in adjacent layers, the neurons in feedforward neural networks can be arbitrarily connected as long as there is no loop.

As a representative continuous-depth neural network approach, stochastic differential equation (SDE)-based Bayesian neural networks (BNNs) have attracted considerable attention due to their solid theoretical foundations and strong potential for real-world applications. However, their reliance on numerical SDE solvers inevitably incurs a large number of function evaluations (NFEs), resulting in high computational cost and occasional convergence instability. To address these challenges, we propose a Nesterov-accelerated gradient (NAG) enhanced SDE-BNN model. By integrating NAG into the SDE-BNN framework along with an NFE-dependent residual skip connection, our method accelerates convergence and substantially reduces NFEs during both training and testing. Extensive empirical results show that our model consistently outperforms conventional SDE-BNNs across various tasks, including image classification and sequence modeling, achieving lower NFEs and improved predictive accuracy.

In this paper, we systematically analyze the connecting architectures of recurrent neural networks (RNNs). Our main contribution is twofold: first, we present a rigorous graph-theoretic framework describing the connecting architectures of RNNs in general. Second, we propose three architecture complexity measures of RNNs: (a) the recurrent depth, which captures the RNN's over-time nonlinear complexity, (b) the feedforward depth, which captures the local input-output nonlinearity (similar to the "depth" in feedforward neural networks (FNNs)), and (c) the recurrent skip coefficient which captures how rapidly the information propagates over time. We rigorously prove each measure's existence and computability. Our experimental results show that RNNs might benefit from larger recurrent depth and feedforward depth. We further demonstrate that increasing recurrent skip coefficient offers performance boosts on long term dependency problems.

In this paper, we propose a very deep fully convolutional encoding-decoding framework for image restoration such as denoising and super-resolution. The network is composed of multiple layers of convolution and deconvolution operators, learning end-to-end mappings from corrupted images to the original ones. The convolutional layers act as the feature extractor, which capture the abstraction of image contents while eliminating noises/corruptions. Deconvolutional layers are then used to recover the image details. We propose to symmetrically link convolutional and deconvolutional layers with skip-layer connections, with which the training converges much faster and attains a higher-quality local optimum. First, the skip connections allow the signal to be back-propagated to bottom layers directly, and thus tackles the problem of gradient vanishing, making training deep networks easier and achieving restoration performance gains consequently. Second, these skip connections pass image details from convolutional layers to deconvolutional layers, which is beneficial in recovering the original image. Significantly, with the large capacity, we can handle different levels of noises using a single model. Experimental results show that our network achieves better performance than recent state-of-the-art methods.

In the present study, we introduce an innovative structure for 3D medical image segmentation that effectively integrates 2D U-Net-derived skip connections into the architecture of 3D convolutional neural networks (3D CNNs). Conventional 3D segmentation techniques predominantly depend on isotropic 3D convolutions for the extraction of volumetric features, which frequently engenders inefficiencies due to the varying information density across the three orthogonal axes in medical imaging modalities such as computed tomography (CT) and magnetic resonance imaging (MRI). This disparity leads to a decline in axial-slice plane feature extraction efficiency, with slice plane features being comparatively underutilized relative to features in the time-axial. To address this issue, we introduce the U-shaped Connection (uC), utilizing simplified 2D U-Net in place of standard skip connections to augment the extraction of the axial-slice plane features while concurrently preserving the volumetric context afforded by 3D convolutions. Based on uC, we further present uC 3DU-Net, an enhanced 3D U-Net backbone that integrates the uC approach to facilitate optimal axial-slice plane feature utilization. Through rigorous experimental validation on five publicly accessible datasets--FLARE2021, OIMHS, FeTA2021, AbdomenCT-1K, and BTCV, the proposed method surpasses contemporary state-of-the-art models. Notably, this performance is achieved while reducing the number of parameters and computational complexity. This investigation underscores the efficacy of incorporating 2D convolutions within the framework of 3D CNNs to overcome the intrinsic limitations of volumetric segmentation, thereby potentially expanding the frontiers of medical image analysis.

Learning with recurrent neural networks (RNNs) on long sequences is a notoriously difficult task. There are three major challenges: 1) complex dependencies, 2) vanishing and exploding gradients, and 3) efficient parallelization. In this paper, we introduce a simple yet effective RNN connection structure, the DilatedRNN, which simultaneously tackles all of these challenges. The proposed architecture is characterized by multi-resolution dilated recurrent skip connections and can be combined flexibly with diverse RNN cells. Moreover, the DilatedRNN reduces the number of parameters needed and enhances training efficiency significantly, while matching state-of-the-art performance (even with standard RNN cells) in tasks involving very long-term dependencies. To provide a theory-based quantification of the architecture's advantages, we introduce a memory capacity measure, the mean recurrent length, which is more suitable for RNNs with long skip connections than existing measures. We rigorously prove the advantages of the DilatedRNN over other recurrent neural architectures.