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Distributed filesystems typically employ synchronous metadata updates, facing inherent challenges for access efficiency, load balancing, and directory contention, especially under dynamic and skewed workloads. This paper argues that synchronous updates are overly conservative for distributed filesystems. We propose AsyncFS with asynchronous metadata updates, allowing operations to return early and defer directory updates until respective read to enable latency hiding and conflict resolution. The key challenge is efficiently maintaining the synchronous semantics of metadata updates. To address this, AsyncFS is co-designed with a programmable switch, leveraging the constrained on-switch resources to holistically track directory states in the network with negligible cost. This allows AsyncFS to timely aggregate and efficiently apply delayed updates using batching and consolidation before directory reads. Evaluation shows that AsyncFS achieves up to 13.34$\times$ and 3.85$\times$ higher throughput, and 61.6% and 57.3% lower latency than two state-of-the-art distributed filesystems, InfiniFS and CFS-KV, respectively, on skewed workloads. For real-world workloads, AsyncFS improves end-to-end throughput by 21.1$\times$, 1.1$\times$ and 30.1% over Ceph, IndexFS and CFS-KV, respectively.

Large Language Models (LLMs) have demonstrated remarkable capabilities across a wide range of natural language processing and reasoning tasks. However, their performance in the foundational domain of arithmetic remains unsatisfactory. When dealing with arithmetic tasks, LLMs often memorize specific examples rather than learning the underlying computational logic, limiting their ability to generalize to new problems. In this paper, we propose a Composable Arithmetic Execution Framework (CAEF) that enables LLMs to learn to execute step-by-step computations by emulating Turing Machines, thereby gaining a genuine understanding of computational logic. Moreover, the proposed framework is highly scalable, allowing composing learned operators to significantly reduce the difficulty of learning complex operators. In our evaluation, CAEF achieves nearly 100% accuracy across seven common mathematical operations on the LLaMA 3.1-8B model, effectively supporting computations involving operands with up to 100 digits, a level where GPT-4o falls short noticeably in some settings., Comment: 30 pages

A graph $G$ is $k$-critical (list $k$-critical, DP $k$-critical) if $\chi(G)= k$ ($\chi_\ell(G)= k$, $\chi_\mathrm{DP}(G)= k$) and for every proper subgraph $G'$ of $G$, $\chi(G')\left(3 + \frac{1}{5} \right) \frac{n}{2}. $$ This is the first bound on $f_\mathrm{DP}(n,4)$ that is asymptotically better than the well-known bound $f(n,4)\geq \left(3 + \frac{1}{13} \right) \frac{n}{2}$ by Gallai from 1963. The result also yields a better bound on $f_{\ell}(n,4)$ than the one known before., Comment: 23 pages. arXiv admin note: substantial text overlap with arXiv:2409.00937

A graph $G$ is $k$-critical (list $k$-critical, DP $k$-critical) if $\chi(G)= k$ ($\chi_\ell(G)= k$, $\chi_\mathrm{DP}(G)= k$) and for every proper subgraph $G'$ of $G$, $\chi(G')\left(k - 1 + \left \lceil \frac{k^2 - 7}{2k-7} \right \rceil^{-1}\right)\frac{n}{2}.$$ This is the first bound on $f_\mathrm{DP}(n,k)$ that is asymptotically better than the well-known bound on $f(n,k)$ by Gallai from 1963. The result also yields a slightly better bound on $f_{\ell}(n,k)$ than the ones known before., Comment: 24 pages

Unveiling an empty street from crowded observations captured by in-car cameras is crucial for autonomous driving. However, removing all temporarily static objects, such as stopped vehicles and standing pedestrians, presents a significant challenge. Unlike object-centric 3D inpainting, which relies on thorough observation in a small scene, street scene cases involve long trajectories that differ from previous 3D inpainting tasks. The camera-centric moving environment of captured videos further complicates the task due to the limited degree and time duration of object observation. To address these obstacles, we introduce StreetUnveiler to reconstruct an empty street. StreetUnveiler learns a 3D representation of the empty street from crowded observations. Our representation is based on the hard-label semantic 2D Gaussian Splatting (2DGS) for its scalability and ability to identify Gaussians to be removed. We inpaint rendered image after removing unwanted Gaussians to provide pseudo-labels and subsequently re-optimize the 2DGS. Given its temporal continuous movement, we divide the empty street scene into observed, partial-observed, and unobserved regions, which we propose to locate through a rendered alpha map. This decomposition helps us to minimize the regions that need to be inpainted. To enhance the temporal consistency of the inpainting, we introduce a novel time-reversal framework to inpaint frames in reverse order and use later frames as references for earlier frames to fully utilize the long-trajectory observations. Our experiments conducted on the street scene dataset successfully reconstructed a 3D representation of the empty street. The mesh representation of the empty street can be extracted for further applications. The project page and more visualizations can be found at: https://streetunveiler.github.io, Comment: Project page: https://streetunveiler.github.io

The pretrain+fine-tune paradigm is foundational for deploying large language models (LLMs) across various downstream applications. Within this framework, Low-Rank Adaptation (LoRA) stands out for its parameter-efficient fine-tuning (PEFT), producing numerous reusable task-specific LoRA adapters. However, this approach requires explicit task intention selection, posing challenges for autonomous task sensing and switching during inference with multiple existing LoRA adapters embedded in a single LLM. In this work, we introduce MeteoRA (Multiple-tasks embedded LoRA), a scalable and efficient framework that reuses multiple task-specific LoRA adapters into the base LLM via a full-mode Mixture-of-Experts (MoE) architecture. This framework also includes novel MoE forward acceleration strategies to address the efficiency challenges of traditional MoE implementations. Our evaluation, using the LlaMA2-13B and LlaMA3-8B base models equipped with 28 existing LoRA adapters through MeteoRA, demonstrates equivalent performance with the traditional PEFT method. Moreover, the LLM equipped with MeteoRA achieves superior performance in handling composite tasks, effectively solving ten sequential problems in a single inference pass, thereby demonstrating the framework's enhanced capability for timely adapter switching., Comment: 26 pages

Recent studies in neuro-symbolic learning have explored the integration of logical knowledge into deep learning via encoding logical constraints as an additional loss function. However, existing approaches tend to vacuously satisfy logical constraints through shortcuts, failing to fully exploit the knowledge. In this paper, we present a new framework for learning with logical constraints. Specifically, we address the shortcut satisfaction issue by introducing dual variables for logical connectives, encoding how the constraint is satisfied. We further propose a variational framework where the encoded logical constraint is expressed as a distributional loss that is compatible with the model's original training loss. The theoretical analysis shows that the proposed approach bears salient properties, and the experimental evaluations demonstrate its superior performance in both model generalizability and constraint satisfaction., Comment: Published as a conference paper at ICLR 2023, and code is available at https://github.com/SoftWiser-group/NeSy-without-Shortcuts

Neuro-symbolic learning generally consists of two separated worlds, i.e., neural network training and symbolic constraint solving, whose success hinges on symbol grounding, a fundamental problem in AI. This paper presents a novel, softened symbol grounding process, bridging the gap between the two worlds, and resulting in an effective and efficient neuro-symbolic learning framework. Technically, the framework features (1) modeling of symbol solution states as a Boltzmann distribution, which avoids expensive state searching and facilitates mutually beneficial interactions between network training and symbolic reasoning;(2) a new MCMC technique leveraging projection and SMT solvers, which efficiently samples from disconnected symbol solution spaces; (3) an annealing mechanism that can escape from %being trapped into sub-optimal symbol groundings. Experiments with three representative neuro symbolic learning tasks demonstrate that, owining to its superior symbol grounding capability, our framework successfully solves problems well beyond the frontier of the existing proposals., Comment: Published as a conference paper at ICLR 2023. Code is available at https://github.com/SoftWiser-group/Soften-NeSy-learning

As foundation models (FMs) continue to shape the landscape of AI, the in-context learning (ICL) paradigm thrives but also encounters issues such as toxicity, hallucination, disparity, adversarial vulnerability, and inconsistency. Ensuring the reliability and responsibility of FMs is crucial for the sustainable development of the AI ecosystem. In this concise overview, we investigate recent advancements in enhancing the reliability and trustworthiness of FMs within ICL frameworks, focusing on four key methodologies, each with its corresponding subgoals. We sincerely hope this paper can provide valuable insights for researchers and practitioners endeavoring to build safe and dependable FMs and foster a stable and consistent ICL environment, thereby unlocking their vast potential., Comment: 18 pages, 15 figures

In the field of autonomous driving, online high-definition (HD) map reconstruction is crucial for planning tasks. Recent research has developed several high-performance HD map reconstruction models to meet this necessity. However, the point sequences within the instance vectors may be jittery or jagged due to prediction bias, which can impact subsequent tasks. Therefore, this paper proposes the Anti-disturbance Map reconstruction framework (ADMap). To mitigate point-order jitter, the framework consists of three modules: Multi-Scale Perception Neck, Instance Interactive Attention (IIA), and Vector Direction Difference Loss (VDDL). By exploring the point-order relationships between and within instances in a cascading manner, the model can monitor the point-order prediction process more effectively. ADMap achieves state-of-the-art performance on the nuScenes and Argoverse2 datasets. Extensive results demonstrate its ability to produce stable and reliable map elements in complex and changing driving scenarios. Code and more demos are available at https://github.com/hht1996ok/ADMap.

Transformer-based Large Language Models (LLMs) have been applied in diverse areas such as knowledge bases, human interfaces, and dynamic agents, and marking a stride towards achieving Artificial General Intelligence (AGI). However, current LLMs are predominantly pretrained on short text snippets, which compromises their effectiveness in processing the long-context prompts that are frequently encountered in practical scenarios. This article offers a comprehensive survey of the recent advancement in Transformer-based LLM architectures aimed at enhancing the long-context capabilities of LLMs throughout the entire model lifecycle, from pre-training through to inference. We first delineate and analyze the problems of handling long-context input and output with the current Transformer-based models. We then provide a taxonomy and the landscape of upgrades on Transformer architecture to solve these problems. Afterwards, we provide an investigation on wildly used evaluation necessities tailored for long-context LLMs, including datasets, metrics, and baseline models, as well as optimization toolkits such as libraries, frameworks, and compilers to boost the efficacy of LLMs across different stages in runtime. Finally, we discuss the challenges and potential avenues for future research. A curated repository of relevant literature, continuously updated, is available at https://github.com/Strivin0311/long-llms-learning., Comment: 40 pages, 3 figures, 4 tables

A graph $G$ is $(1,3)$-colorable if its vertices can be partitioned into subsets $V_1$ and $V_2$ so that every vertex in $G[V_1]$ has degree at most $1$ and every vertex in $G[V_2]$ has degree at most $3$. We prove that every graph with maximum average degree at most 28/9 is $(1, 3)$-colorable., Comment: 21 pages

Given a graph $G$, an injective edge-coloring of $G$ is a function $\psi:E(G) \rightarrow \mathbb N$ such that if $\psi(e) = \psi(e')$, then no third edge joins an endpoint of $e$ and an endpoint of $e'$. The injective chromatic index of a graph $G$, written $\chi_{inj}'(G)$, is the minimum number of colors needed for an injective edge coloring of $G$. In this paper, we investigate the injective chromatic index of certain classes of degenerate graphs. First, we show that if $G$ is a $d$-degenerate graph of maximum degree $\Delta$, then $\chi_{inj}'(G) = O(d^3 \log \Delta)$. Next, we show that if $G$ is a graph of Euler genus $g$, then $\chi_{inj}'(G) \leq (3+o(1))g$, which is tight when $G$ is a clique. Finally, we show that the oriented chromatic number of a graph is at most exponential in its injective chromatic index. Using this fact, we prove that the oriented chromatic number of a graph embedded on a surface of Euler genus $g$ has oriented chromatic number at most $O(g^{6400})$, improving the previously known upper bound of $2^{O(g^{\frac{1}{2} + \epsilon})}$ and resolving a conjecture of Aravind and Subramanian., Comment: 18 pages

In recent years, the neural implicit surface has emerged as a powerful representation for multi-view surface reconstruction due to its simplicity and state-of-the-art performance. However, reconstructing smooth and detailed surfaces in indoor scenes from multi-view images presents unique challenges. Indoor scenes typically contain large texture-less regions, making the photometric loss unreliable for optimizing the implicit surface. Previous work utilizes monocular geometry priors to improve the reconstruction in indoor scenes. However, monocular priors often contain substantial errors in thin structure regions due to domain gaps and the inherent inconsistencies when derived independently from different views. This paper presents \textbf{DebSDF} to address these challenges, focusing on the utilization of uncertainty in monocular priors and the bias in SDF-based volume rendering. We propose an uncertainty modeling technique that associates larger uncertainties with larger errors in the monocular priors. High-uncertainty priors are then excluded from optimization to prevent bias. This uncertainty measure also informs an importance-guided ray sampling and adaptive smoothness regularization, enhancing the learning of fine structures. We further introduce a bias-aware signed distance function to density transformation that takes into account the curvature and the angle between the view direction and the SDF normals to reconstruct fine details better. Our approach has been validated through extensive experiments on several challenging datasets, demonstrating improved qualitative and quantitative results in reconstructing thin structures in indoor scenes, thereby outperforming previous work., Comment: 14 pages, accepted by TPAMI

An interesting generalization of list coloring is so called DP-coloring (named after Dvo\v{r}\'ak and Postle). We study $(i,j)$-defective DP-colorings of simple graphs. Define $g_{DP}(i,j,n)$ to be the minimum number of edges in an $n$-vertex DP-$(i,j)$-critical graph. We prove sharp bounds on $g_{DP}(i,j,n)$ for $i=1,2$ and $j\geq 2i$ for infinitely many $n$., Comment: 19 pages, 1 figure. arXiv admin note: text overlap with arXiv:2006.10244

In recent years, great progress has been made in the Lift-Splat-Shot-based (LSS-based) 3D object detection method. However, inaccurate depth estimation remains an important constraint to the accuracy of camera-only and multi-model 3D object detection models, especially in regions where the depth changes significantly (i.e., the "depth jump" problem). In this paper, we proposed a novel Edge-aware Lift-splat-shot (EA-LSS) framework. Specifically, edge-aware depth fusion (EADF) module is proposed to alleviate the "depth jump" problem and fine-grained depth (FGD) module to further enforce refined supervision on depth. Our EA-LSS framework is compatible for any LSS-based 3D object detection models, and effectively boosts their performances with negligible increment of inference time. Experiments on nuScenes benchmarks demonstrate that EA-LSS is effective in either camera-only or multi-model models. It is worth mentioning that EA-LSS achieved the state-of-the-art performance on nuScenes test benchmarks with mAP and NDS of 76.5% and 77.6%, respectively.

Being a powerful tool for linear time-invariant (LTI) systems, system response analysis can also be applied to the so-called linear space-invariant (LSI) but time-varying systems, which is a dual of the conventional LTI problems. In this paper, we propose a system response analysis method for LSI problems by conducting Fourier transform of the field distribution on the space instead of time coordinate. Specifically, input and output signals can be expressed in the wavenumber (spatial frequency) domain. In this way, the system function in wavenumber domain can also be obtained for LSI systems. Given an arbitrary input and temporal profile of the medium, the output can be easily predicted using the system function. Moreover, for a complex temporal system, the proposed method allows for decomposing it into multiple simpler subsystems that appear in sequence in time. The system function of the whole system can be efficiently calculated by multiplying those of the individual subsystems., Comment: This article has been withdrawn due to an unresolvable internal author dispute

Time-varying materials bring an extra degree of design freedom compared to their conventional time-invariant counterparts. However, few discussions have focused on the underlying physical difference between spatial and temporal boundaries. In this letter, we thoroughly investigate those differences from the perspective of conservation laws. By doing so, the building blocks of optics and electromagnetics such as the reflection law, Snell's law, and Fresnel's equations can be analogously derived in a temporal context, but with completely different interpretations. Furthermore, we study the unique features of temporal boundaries, such as their nonconformance to energy conservation and causality., Comment: This article has been withdrawn due to an unresolvable internal author dispute

We consider a new problem of adapting a human mesh reconstruction model to out-of-domain streaming videos, where performance of existing SMPL-based models are significantly affected by the distribution shift represented by different camera parameters, bone lengths, backgrounds, and occlusions. We tackle this problem through online adaptation, gradually correcting the model bias during testing. There are two main challenges: First, the lack of 3D annotations increases the training difficulty and results in 3D ambiguities. Second, non-stationary data distribution makes it difficult to strike a balance between fitting regular frames and hard samples with severe occlusions or dramatic changes. To this end, we propose the Dynamic Bilevel Online Adaptation algorithm (DynaBOA). It first introduces the temporal constraints to compensate for the unavailable 3D annotations, and leverages a bilevel optimization procedure to address the conflicts between multi-objectives. DynaBOA provides additional 3D guidance by co-training with similar source examples retrieved efficiently despite the distribution shift. Furthermore, it can adaptively adjust the number of optimization steps on individual frames to fully fit hard samples and avoid overfitting regular frames. DynaBOA achieves state-of-the-art results on three out-of-domain human mesh reconstruction benchmarks., Comment: 14 pages, 13 figures; code repositoty: https://github.com/syguan96/DynaBOA

With the increased availability of rich tactile sensors, there is an equally proportional need for open-source and integrated software capable of efficiently and effectively processing raw touch measurements into high-level signals that can be used for control and decision-making. In this paper, we present PyTouch -- the first machine learning library dedicated to the processing of touch sensing signals. PyTouch, is designed to be modular, easy-to-use and provides state-of-the-art touch processing capabilities as a service with the goal of unifying the tactile sensing community by providing a library for building scalable, proven, and performance-validated modules over which applications and research can be built upon. We evaluate PyTouch on real-world data from several tactile sensors on touch processing tasks such as touch detection, slip and object pose estimations. PyTouch is open-sourced at https://github.com/facebookresearch/pytouch ., Comment: 7 pages. Accepted at ICRA 2021

This paper considers a new problem of adapting a pre-trained model of human mesh reconstruction to out-of-domain streaming videos. However, most previous methods based on the parametric SMPL model \cite{loper2015smpl} underperform in new domains with unexpected, domain-specific attributes, such as camera parameters, lengths of bones, backgrounds, and occlusions. Our general idea is to dynamically fine-tune the source model on test video streams with additional temporal constraints, such that it can mitigate the domain gaps without over-fitting the 2D information of individual test frames. A subsequent challenge is how to avoid conflicts between the 2D and temporal constraints. We propose to tackle this problem using a new training algorithm named Bilevel Online Adaptation (BOA), which divides the optimization process of overall multi-objective into two steps of weight probe and weight update in a training iteration. We demonstrate that BOA leads to state-of-the-art results on two human mesh reconstruction benchmarks., Comment: CVPR 2021, the project page: https://sites.google.com/view/humanmeshboa

Deep neural networks (DNNs) have become a key part of many modern software applications. After training and validating, the DNN is deployed as an irrevocable component and applied in real-world scenarios. Although most DNNs are built meticulously with huge volumes of training data, data in the real world still remain unknown to the DNN model, which leads to the crucial requirement of runtime out-of-distribution (OOD) detection. However, many existing approaches 1) need OOD data for classifier training or parameter tuning, or 2) simply combine the scores of each hidden layer as an ensemble of features for OOD detection. In this paper, we present a novel outlook on in-distribution data in a generative manner, which takes their latent features generated from each hidden layer as a joint distribution across representation spaces. Since only the in-distribution latent features are comprehensively understood in representation space, the internal difference between in-distribution and OOD data can be naturally revealed without the intervention of any OOD data. Specifically, We construct a generative model, called Latent Sequential Gaussian Mixture (LSGM), to depict how the in-distribution latent features are generated in terms of the trace of DNN inference across representation spaces. We first construct the Gaussian Mixture Model (GMM) based on in-distribution latent features for each hidden layer, and then connect GMMs via the transition probabilities of the inference traces. Experimental evaluations on popular benchmark OOD datasets and models validate the superiority of the proposed method over the state-of-the-art methods in OOD detection., Comment: Submitted to a conference of computer vision

Synthesizing 3D human motion plays an important role in many graphics applications as well as understanding human activity. While many efforts have been made on generating realistic and natural human motion, most approaches neglect the importance of modeling human-scene interactions and affordance. On the other hand, affordance reasoning (e.g., standing on the floor or sitting on the chair) has mainly been studied with static human pose and gestures, and it has rarely been addressed with human motion. In this paper, we propose to bridge human motion synthesis and scene affordance reasoning. We present a hierarchical generative framework to synthesize long-term 3D human motion conditioning on the 3D scene structure. Building on this framework, we further enforce multiple geometry constraints between the human mesh and scene point clouds via optimization to improve realistic synthesis. Our experiments show significant improvements over previous approaches on generating natural and physically plausible human motion in a scene.

This technical report presents a solution for the 2020 Traffic4Cast Challenge. We consider the traffic forecasting problem as a future frame prediction task with relatively weak temporal dependencies (might be due to stochastic urban traffic dynamics) and strong prior knowledge, \textit{i.e.}, the roadmaps of the cities. For these reasons, we use the U-Net as the backbone model, and we propose a roadmap generation method to make the predicted traffic flows more rational. Meanwhile, we use a fine-tuning strategy based on the validation set to prevent overfitting, which effectively improves the prediction results. At the end of this report, we further discuss several approaches that we have considered or could be explored in future work: (1) harnessing inherent data patterns, such as seasonality; (2) distilling and transferring common knowledge between different cities. We also analyze the validity of the evaluation metric., Comment: Code is available at \

A coloring of edges of a graph $G$ is injective if for any two distinct edges $e_1$ and $e_2$, the colors of $e_1$ and $e_2$ are distinct if they are at distance $1$ in $G$ or in a common triangle. Naturally, the injective chromatic index of $G$, $\chi'_{inj}(G)$, is the minimum number of colors needed for an injective edge-coloring of $G$. We study how large can be the injective chromatic index of $G$ in terms of maximum degree of $G$ when we have restrictions on girth and/or chromatic number of $G$. We also compare our bounds with analogous bounds on the strong chromatic index., Comment: 14 pages

Generating diverse and natural human motion is one of the long-standing goals for creating intelligent characters in the animated world. In this paper, we propose a self-supervised method for generating long-range, diverse and plausible behaviors to achieve a specific goal location. Our proposed method learns to model the motion of human by decomposing a long-range generation task in a hierarchical manner. Given the starting and ending states, a memory bank is used to retrieve motion references as source material for short-range clip generation. We first propose to explicitly disentangle the provided motion material into style and content counterparts via bi-linear transformation modelling, where diverse synthesis is achieved by free-form combination of these two components. The short-range clips are then connected to form a long-range motion sequence. Without ground truth annotation, we propose a parameterized bi-directional interpolation scheme to guarantee the physical validity and visual naturalness of generated results. On large-scale skeleton dataset, we show that the proposed method is able to synthesise long-range, diverse and plausible motion, which is also generalizable to unseen motion data during testing. Moreover, we demonstrate the generated sequences are useful as subgoals for actual physical execution in the animated world., Comment: ECCV 2020, Project Page:\

In video prediction tasks, one major challenge is to capture the multi-modal nature of future contents and dynamics. In this work, we propose a simple yet effective framework that can efficiently predict plausible future states. The key insight is that the potential distribution of a sequence could be approximated with analogous ones in a repertoire of training pool, namely, expert examples. By further incorporating a novel optimization scheme into the training procedure, plausible predictions can be sampled efficiently from distribution constructed from the retrieved examples. Meanwhile, our method could be seamlessly integrated with existing stochastic predictive models; significant enhancement is observed with comprehensive experiments in both quantitative and qualitative aspects. We also demonstrate the generalization ability to predict the motion of unseen class, i.e., without access to corresponding data during training phase., Comment: Project Page: https://sites.google.com/view/vpeg-supp/home

Follow-up serves an important role in the management of pulmonary nodules for lung cancer. Imaging diagnostic guidelines with expert consensus have been made to help radiologists make clinical decision for each patient. However, tumor growth is such a complicated process that it is difficult to stratify high-risk nodules from low-risk ones based on morphologic characteristics. On the other hand, recent deep learning studies using convolutional neural networks (CNNs) to predict the malignancy score of nodules, only provides clinicians with black-box predictions. To this end, we propose a unified framework, named Nodule Follow-Up Prediction Network (NoFoNet), which predicts the growth of pulmonary nodules with high-quality visual appearances and accurate quantitative results, given any time interval from baseline observations. It is achieved by predicting future displacement field of each voxel with a WarpNet. A TextureNet is further developed to refine textural details of WarpNet outputs. We also introduce techniques including Temporal Encoding Module and Warp Segmentation Loss to encourage time-aware and shape-aware representation learning. We build an in-house follow-up dataset from two medical centers to validate the effectiveness of the proposed method. NoFoNet significantly outperforms direct prediction by a U-Net in terms of visual quality; more importantly, it demonstrates accurate differentiating performance between high- and low-risk nodules. Our promising results suggest the potentials in computer aided intervention for lung nodule management., Comment: MICCAI 2020

DP-coloring (also known as correspondence coloring) is a generalization of list coloring developed recently by Dvo\v{r}\'ak and Postle. We introduce and study $(i,j)$-defective DP-colorings of simple graphs. Let $g_{DP}(i,j,n)$ be the minimum number of edges in an $n$-vertex DP-$(i,j)$-critical graph. In this paper we determine sharp bound on $g_{DP}(i,j,n)$ for each $i\geq3$ and $j\geq 2i+1$ for infinitely many $n$., Comment: 17 pages