Let $p$ be an odd prime. The factorization of the polynomial $x^{p+1}-1$ over the integer residue ring $\mathbb{Z}_{p^e}$ is pivotal for constructing cyclic codes with Hermitian symmetry, a critical resource for Linear Complementary Dual (LCD) codes and Entanglement-Assisted Quantum Error-Correcting Codes (EAQECC). Traditionally, lifting factorizations relies on the generic Hensel's Lemma, masking the underlying algebraic structure. In this paper, we establish a structural isomorphism between the lifting process and the roots of a special auxiliary polynomial $V(x)$, unveiling a deterministic link to Dickson polynomials. Based on this theory, we develop \texttt{Dickson-Engine}, a linear-time algorithm ($O(ep)$) that outperforms standard libraries by orders of magnitude. Applying this engine to $\mathbb{Z}_{169}$, we explicitly construct a family of classical LCD codes of length $n=182$ via the isometric Gray map. Our search reveals codes with parameters (e.g., $[182, 1, 168]_{13}$ and $[182, 2, 144]_{13}$) that are \textbf{near-optimal} with respect to the theoretical Griesmer Bound. Notably, we discover a ``robustness plateau'' starting from non-trivial dimensions ($k=4$), where the minimum distance remains stable ($d=120$) even as the dimension triples ($k=4 \rightarrow 12$). These codes provide exceptional resources for post-quantum cryptography and quantum error correction without entanglement consumption ($c=0$).

Abstract. This paper is concerned with the problem of inverse scattering of time-harmonic acoustic plane waves by a two-layered medium with a locally rough interface in two dimensions. A direct imaging method is proposed to reconstruct the locally rough interface from the phaseless total-field data measured on the upper half of the circle with a large radius at a fixed frequency or from the phased far-field data measured on the upper half of the unit circle at a fixed frequency. The presence of the locally rough interface poses challenges in the theoretical analysis of the imaging methods. To address these challenges, a technically involved asymptotic analysis is provided for the relevant oscillatory integrals involved in the imaging methods, based mainly on the techniques and results in our recent work [L. Li, J. Yang, B. Zhang, and H. Zhang, arXiv:2208.00456 , 2022] on the uniform far-field asymptotics of the scattered field for acoustic scattering in a two-layered medium. Finally, extensive numerical experiments are conducted to demonstrate the feasibility and robustness of our imaging algorithms.

Adversarial Training (AT) has become arguably the state-of-the-art algorithm for extracting robust features. However, researchers recently notice that AT suffers from severe robust overfitting problems, particularly after learning rate (LR) decay. In this paper, we explain this phenomenon by viewing adversarial training as a dynamic minimax game between the model trainer and the attacker. Specifically, we analyze how LR decay breaks the balance between the minimax game by empowering the trainer with a stronger memorization ability, and show such imbalance induces robust overfitting as a result of memorizing non-robust features. We validate this understanding with extensive experiments, and provide a holistic view of robust overfitting from the dynamics of both the two game players. This understanding further inspires us to alleviate robust overfitting by rebalancing the two players by either regularizing the trainer's capacity or improving the attack strength. Experiments show that the proposed ReBalanced Adversarial Training (ReBAT) can attain good robustness and does not suffer from robust overfitting even after very long training. Code is available at https://github.com/PKU-ML/ReBAT.

Recent efforts on learning-based image denoising approaches use unrolled architectures with a fixed number of repeatedly stacked blocks. However, due to difficulties in training networks corresponding to deeper layers, simply stacking blocks may cause performance degradation, and the number of unrolled blocks needs to be manually tuned to find an appropriate value. To circumvent these problems, this paper describes an alternative approach with implicit models. To our best knowledge, our approach is the first attempt to model iterative image denoising through an implicit scheme. The model employs implicit differentiation to calculate gradients in the backward pass, thus avoiding the training difficulties of explicit models and elaborate selection of the iteration number. Our model is parameter-efficient and has only one implicit layer, which is a fixed-point equation that casts the desired noise feature as its solution. By simulating infinite iterations of the model, the final denoising result is given by the equilibrium that is achieved through accelerated black-box solvers. The implicit layer not only captures the non-local self-similarity prior for image denoising, but also facilitates training stability and thereby boosts the denoising performance. Extensive experiments show that our model leads to better performances than state-of-the-art explicit denoisers with enhanced qualitative and quantitative results.

On large-scale graphs, many graph neural networks are problematic in capturing long-range dependencies due to the oversmoothing problem. Recently, Graph Equilibrium Models (GEQs) arise as a promising solution to this issue. Their output is the equilibrium of a fixed-point equation, which can be seen as the result of iterating a GNN layer for infinite times, so that they inherently have global receptive fields. However, to find the equilibrium, GEQs require running costly full-batch root-finding algorithms from scratch during each model update, which leads to severe efficiency and scalability issues that prevent them from scaling to large graphs. To address these limitations, we propose VEQ, an efficient learning method to scale GEQs to large graphs. Instead of initializing the equilibrium from scratch in full-batch training, VEQ uses the latest equilibrium of in-batch nodes and their 1-hop neighbors (dubbed Virtual Equilibrium) to accelerate and calibrate the root-finding process in mini-batch training. With virtual equilibrium as an informative prior, VEQ is able to reach the equilibrium in fewer steps while still capturing global dependencies. Theoretically, we provide convergence analysis for the forward and backward pass of VEQ. Empirically, VEQ significantly outperforms existing GEQs by a large margin (more than 1.5%) on all benchmark datasets, with much less training time and memory. Also, VEQ achieves competitive and even superior performance to many highly engineered explicit GNNs on large-scale benchmark datasets like ogbn-arxiv and ogbn-products. VEQ shows that after we resolve the efficiency and scalability issues, GEQs are indeed favorable on large graphs due to their advantage of capturing long-range dependencies.

Due to the over-smoothing issue, most existing graph neural networks can only capture limited dependencies with their inherently finite aggregation layers. To overcome this limitation, we propose a new kind of graph convolution, called Graph Implicit Nonlinear Diffusion (GIND), which implicitly has access to infinite hops of neighbors while adaptively aggregating features with nonlinear diffusion to prevent over-smoothing. Notably, we show that the learned representation can be formalized as the minimizer of an explicit convex optimization objective. With this property, we can theoretically characterize the equilibrium of our GIND from an optimization perspective. More interestingly, we can induce new structural variants by modifying the corresponding optimization objective. To be specific, we can embed prior properties to the equilibrium, as well as introducing skip connections to promote training stability. Extensive experiments show that GIND is good at capturing long-range dependencies, and performs well on both homophilic and heterophilic graphs with nonlinear diffusion. Moreover, we show that the optimization-induced variants of our models can boost the performance and improve training stability and efficiency as well. As a result, our GIND obtains significant improvements on both node-level and graph-level tasks.

In this paper, a flexible and adaptive block-wise image compression architecture with restoration network is proposed to improve the rate-distortion (R-D) performance of learned image compression. In contrast to existing learned compression algorithms which perform the image based transform coding, our approach splits the input image into non-overlapped blocks and compresses each block adaptively based on their local content characteristics. To reduce the artifacts caused by the block partitioning, block fusion network is subsequently proposed to overcome the drawbacks of block-wise independent coding. Further, quantization fine-tuning training process is proposed and utilized to reduce the difference of quantization operations between the training process and testing process. Experimental results show that the proposed network architecture has the ability to obtain significant coding gain compared with existing conventional compression standards including Versatile Video Coding (VVC) and achieves comparable R-D performance with state-of-the-art learned image codec using less parameters.

Multi-view methods learn representations by aligning multiple views of the same image and their performance largely depends on the choice of data augmentation. In this paper, we notice that some other useful augmentations, such as image rotation, are harmful for multi-view methods because they cause a semantic shift that is too large to be aligned well. This observation motivates us to relax the exact alignment objective to better cultivate stronger augmentations. Taking image rotation as a case study, we develop a generic approach, Pretext-aware Residual Relaxation (Prelax), that relaxes the exact alignment by allowing an adaptive residual vector between different views and encoding the semantic shift through pretext-aware learning. Extensive experiments on different backbones show that our method can not only improve multi-view methods with existing augmentations, but also benefit from stronger image augmentations like rotation.

Graph Convolutional Networks (GCNs) have attracted more and more attentions in recent years. A typical GCN layer consists of a linear feature propagation step and a nonlinear transformation step. Recent works show that a linear GCN can achieve comparable performance to the original non-linear GCN while being much more computationally efficient. In this paper, we dissect the feature propagation steps of linear GCNs from a perspective of continuous graph diffusion, and analyze why linear GCNs fail to benefit from more propagation steps. Following that, we propose Decoupled Graph Convolution (DGC) that decouples the terminal time and the feature propagation steps, making it more flexible and capable of exploiting a very large number of feature propagation steps. Experiments demonstrate that our proposed DGC improves linear GCNs by a large margin and makes them competitive with many modern variants of non-linear GCNs.

In this paper, we propose an enhanced bi-prediction scheme based on the convolutional neural network (CNN) to improve the rate-distortion performance in video compression. In contrast to the traditional bi-prediction strategy which computes the linear superposition as the predictive signals with pixel-to-pixel correspondence, the proposed scheme employs CNN to directly infer the predictive signals in a data-driven manner. As such, the predicted blocks are fused in a nonlinear fashion to improve the coding performance. Moreover, the patch-to-patch inference strategy with CNN also improves the prediction accuracy since the patch-level information for the prediction of each individual pixel can be exploited. The proposed enhanced bi-prediction scheme is further incorporated into the high-efficiency video coding standard, and the experimental results exhibit a significant performance improvement under different coding configurations.

In this paper, we propose a novel light field image compression scheme by exploiting the intrinsic similarity of light field images with deep learning. In particular, instead of conveying all LF sub-views, only sparsely sampled LF sub-views are compressed and the remaining sub-views are reconstructed from the coded sub-views in the neighbourhood with convolutional neural network (CNN). To jointly suppress the artifacts induced in compression and reconstruct the un-coded views with high geometric accuracy, a multi-view joint enhancement network is introduced to improve the coding performance. Extensive experiments show the superior compression performance of our scheme compared with the state-of-the-art methods.

The state-of-the-art High Efficiency Video Coding (HEVC) standard adopts the bi-prediction to improve the coding efficiency for B frame. However, the underlying assumption of this technique is that the motion field is characterized by the block-wise translational motion model, which may not be efficient in the challenging scenarios such as rotation and deformation. Inspired by the excellent signal level prediction capability of deep learning, we propose a bi-directional motion compensation algorithm with convolutional neural network, which is further incorporated into the video coding pipeline to improve the performance of video compression. Our network consists of six convolutional layers and a skip connection, which integrates the prediction error detection and non-linear signal prediction into an end-to-end framework. Experimental results show that by incorporating the proposed scheme into HEVC, up to 10.5% BD-rate savings and 3.1% BD-rate savings on average for random access (RA) configuration have been observed.