arXiv subjects

Chunmeng Liu

Publications and source records attributed to Chunmeng Liu.

9 recordsLinked to original sources

arXiv2026

A two-steps tensor eigenvector centrality for nodes and hyperedges in hypergraphs

Hypergraphs have been a powerful tool to represent higher-order interactions, where hyperedges can connect an arbitrary number of nodes. Quantifying the relative importance of nodes and hyperedges in hypergraphs is a fundamental problem in network analysis. In this paper, we propose a new tensor-based centrality measure for general hypergraphs. We use a third-order tensor to represent the relationship between nodes and hyperedges. The tensor's positive Perron vector is defined as the centrality vector of the hypergraph. The existence and uniqueness of this centrality vector are guaranteed by the Perron-Frobenius theorem for tensors. This new centrality measure captures a higher-order mutual reinforcement mechanism: a node's importance is determined by the importance of its incident hyperedges and the other nodes within these hyperedges; symmetrically, a hyperedge's importance is determined by the importance of its constituent nodes and the other hyperedges containing these nodes. We further provide a combinatorial interpretation by proving that the centrality vector represents the limit geometric capacity of two-steps expansion trees. We illustrate the centrality measure on real-world hypergraph datasets.

cs.SI↗

arXiv2025

The $k$-core of a graph and its high-order spectra

The $k$-core of a graph is its largest subgraph with minimum degree at least $k$, a fundamental concept for uncovering hierarchical structures. In this paper, we establish a connection between the $k$-core and the high-order spectra of graphs, a concept originally introduced by Cvetkovi\'{c}, Doob, and Sachs. Specifically, we consider the high-order spectra defined via the $k$-adjacency tensor. Within this framework, we prove that a graph admits a non-empty $k$-core if and only if the spectral radius of the $k$-adjacency tensor is greater than or equal to $1$. Moreover, when the $k$-core exists, vertices corresponding to positive entries in the Perron vector of the $k$-adjacency tensor belong to the $k$-core. We thus define the $k$-order eigenvector centrality via the Perron vector, which provides both membership identification and a measure of relative influence within the $k$-core. Numerical experiments confirm our theoretical findings and illustrate the properties of this centrality measure in some real-world networks.

math.CO↗

arXiv2025

Bollob\'{a}s-Nikiforov conjecture holds asymptotically almost surely

Bollob\'{a}s and Nikiforov (J. Combin. Theory Ser. B. 97 (2007) 859-865) conjectured that for a graph $G$ with $e(G)$ edges and the clique number $\omega(G)$, then $ \lambda_{1}^{2}+\lambda_{2}^{2}\leq 2e(G)\left(1-\frac{1}{\omega(G)}\right), $ where $\lambda_{1}$ and $\lambda_{2}$ are the largest and the second largest eigenvalues of the adjacency matrix of $G$, respectively. In this paper, we prove that for a sequence of random graphs the conjecture holds true with probability tending to one as the number of vertices tends to infinity.

math.CO↗

arXiv2024

A tensor's spectral bound on the clique number

In this paper, we study the spectral radius of the clique tensor A(G) associated with a graph G. This tensor is a higher-order extensions of the adjacency matrix of G. A lower bound of the clique number is given via the spectral radius of A(G). It is an extension of Nikiforov's spectral bound and tighter than the bound of Nikiforov in some classes of graphs. Furthermore, we obtain a spectral version of the Erdos-Simonovits stability theorem for clique tensors based on this bound.

math.CO↗

arXiv2024

Adaptive Prompt Learning with SAM for Few-shot Scanning Probe Microscope Image Segmentation

The Segment Anything Model (SAM) has demonstrated strong performance in image segmentation of natural scene images. However, its effectiveness diminishes markedly when applied to specific scientific domains, such as Scanning Probe Microscope (SPM) images. This decline in accuracy can be attributed to the distinct data distribution and limited availability of the data inherent in the scientific images. On the other hand, the acquisition of adequate SPM datasets is both time-intensive and laborious as well as skill-dependent. To address these challenges, we propose an Adaptive Prompt Learning with SAM (APL-SAM) framework tailored for few-shot SPM image segmentation. Our approach incorporates two key innovations to enhance SAM: 1) An Adaptive Prompt Learning module leverages few-shot embeddings derived from limited support set to learn adaptively central representatives, serving as visual prompts. This innovation eliminates the need for time-consuming online user interactions for providing prompts, such as exhaustively marking points and bounding boxes slice by slice; 2) A multi-source, multi-level mask decoder specifically designed for few-shot SPM image segmentation is introduced, which can effectively capture the correspondence between the support and query images. To facilitate comprehensive training and evaluation, we introduce a new dataset, SPM-Seg, curated for SPM image segmentation. Extensive experiments on this dataset reveal that the proposed APL-SAM framework significantly outperforms the original SAM, achieving over a 30% improvement in terms of Dice Similarity Coefficient with only one-shot guidance. Moreover, APL-SAM surpasses state-of-the-art few-shot segmentation methods and even fully supervised approaches in performance. Code and dataset used in this study will be made available upon acceptance.

cs.CV↗

arXiv2024

An Erdős-Stone type result for high-order spectra of graphs

Erdős-Stone Theorem is a well-known result in extremal graph theory which determines the asymptotic behaviour of maximum number of edges in an $n$-vertex $H$-free graph. In 2009, Nikiforov gave a spectral version of Erdős-Stone Theorem. In this paper, we obtain a tensor's spectral version of Erdős-Stone Theorem.

math.CO↗

arXiv2023

MECPformer: Multi-estimations Complementary Patch with CNN-Transformers for Weakly Supervised Semantic Segmentation

The initial seed based on the convolutional neural network (CNN) for weakly supervised semantic segmentation always highlights the most discriminative regions but fails to identify the global target information. Methods based on transformers have been proposed successively benefiting from the advantage of capturing long-range feature representations. However, we observe a flaw regardless of the gifts based on the transformer. Given a class, the initial seeds generated based on the transformer may invade regions belonging to other classes. Inspired by the mentioned issues, we devise a simple yet effective method with Multi-estimations Complementary Patch (MECP) strategy and Adaptive Conflict Module (ACM), dubbed MECPformer. Given an image, we manipulate it with the MECP strategy at different epochs, and the network mines and deeply fuses the semantic information at different levels. In addition, ACM adaptively removes conflicting pixels and exploits the network self-training capability to mine potential target information. Without bells and whistles, our MECPformer has reached new state-of-the-art 72.0% mIoU on the PASCAL VOC 2012 and 42.4% on MS COCO 2014 dataset. The code is available at https://github.com/ChunmengLiu1/MECPformer.

cs.CV↗

arXiv2023

On a generalization of the spectral Mantel's theorem

Mantel's theorem is a classical result in extremal graph theory which implies that the maximum number of edges of a triangle-free graph of order $n$. In 1970, E. Nosal obtained a spectral version of Mantel's theorem which gave the maximum spectral radius of a triangle-free graph of order $n$. In this paper, the clique tensor of a graph $G$ is proposed and the spectral Mantel's theorem is extended via the clique tensor. Furthermore, a sharp upper bound of the number of cliques in $G$ via the spectral radius of the clique tensor is obtained. And we show that the results of this paper implies that a result of Erdős [Magyar Tud. Akad. Mat. Kutató Int. Közl. 7 (1962)] under certain conditions.

math.CO↗

arXiv2022

WegFormer: Transformers for Weakly Supervised Semantic Segmentation

Although convolutional neural networks (CNNs) have achieved remarkable progress in weakly supervised semantic segmentation (WSSS), the effective receptive field of CNN is insufficient to capture global context information, leading to sub-optimal results. Inspired by the great success of Transformers in fundamental vision areas, this work for the first time introduces Transformer to build a simple and effective WSSS framework, termed WegFormer. Unlike existing CNN-based methods, WegFormer uses Vision Transformer (ViT) as a classifier to produce high-quality pseudo segmentation masks. To this end, we introduce three tailored components in our Transformer-based framework, which are (1) a Deep Taylor Decomposition (DTD) to generate attention maps, (2) a soft erasing module to smooth the attention maps, and (3) an efficient potential object mining (EPOM) to filter noisy activation in the background. Without any bells and whistles, WegFormer achieves state-of-the-art 70.5% mIoU on the PASCAL VOC dataset, significantly outperforming the previous best method. We hope WegFormer provides a new perspective to tap the potential of Transformer in weakly supervised semantic segmentation. Code will be released.

cs.CV↗