Marco Montagna, Simone Scardapane, Lev Telyatnikov
{"title":"拓扑深度学习与状态空间模型:简约复合物的曼巴方法","authors":"Marco Montagna, Simone Scardapane, Lev Telyatnikov","doi":"arxiv-2409.12033","DOIUrl":null,"url":null,"abstract":"Graph Neural Networks based on the message-passing (MP) mechanism are a\ndominant approach for handling graph-structured data. However, they are\ninherently limited to modeling only pairwise interactions, making it difficult\nto explicitly capture the complexity of systems with $n$-body relations. To\naddress this, topological deep learning has emerged as a promising field for\nstudying and modeling higher-order interactions using various topological\ndomains, such as simplicial and cellular complexes. While these new domains\nprovide powerful representations, they introduce new challenges, such as\neffectively modeling the interactions among higher-order structures through\nhigher-order MP. Meanwhile, structured state-space sequence models have proven\nto be effective for sequence modeling and have recently been adapted for graph\ndata by encoding the neighborhood of a node as a sequence, thereby avoiding the\nMP mechanism. In this work, we propose a novel architecture designed to operate\nwith simplicial complexes, utilizing the Mamba state-space model as its\nbackbone. Our approach generates sequences for the nodes based on the\nneighboring cells, enabling direct communication between all higher-order\nstructures, regardless of their rank. We extensively validate our model,\ndemonstrating that it achieves competitive performance compared to\nstate-of-the-art models developed for simplicial complexes.","PeriodicalId":501301,"journal":{"name":"arXiv - CS - Machine Learning","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Topological Deep Learning with State-Space Models: A Mamba Approach for Simplicial Complexes\",\"authors\":\"Marco Montagna, Simone Scardapane, Lev Telyatnikov\",\"doi\":\"arxiv-2409.12033\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Graph Neural Networks based on the message-passing (MP) mechanism are a\\ndominant approach for handling graph-structured data. However, they are\\ninherently limited to modeling only pairwise interactions, making it difficult\\nto explicitly capture the complexity of systems with $n$-body relations. To\\naddress this, topological deep learning has emerged as a promising field for\\nstudying and modeling higher-order interactions using various topological\\ndomains, such as simplicial and cellular complexes. While these new domains\\nprovide powerful representations, they introduce new challenges, such as\\neffectively modeling the interactions among higher-order structures through\\nhigher-order MP. Meanwhile, structured state-space sequence models have proven\\nto be effective for sequence modeling and have recently been adapted for graph\\ndata by encoding the neighborhood of a node as a sequence, thereby avoiding the\\nMP mechanism. In this work, we propose a novel architecture designed to operate\\nwith simplicial complexes, utilizing the Mamba state-space model as its\\nbackbone. Our approach generates sequences for the nodes based on the\\nneighboring cells, enabling direct communication between all higher-order\\nstructures, regardless of their rank. We extensively validate our model,\\ndemonstrating that it achieves competitive performance compared to\\nstate-of-the-art models developed for simplicial complexes.\",\"PeriodicalId\":501301,\"journal\":{\"name\":\"arXiv - CS - Machine Learning\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - CS - Machine Learning\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.12033\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - CS - Machine Learning","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.12033","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Topological Deep Learning with State-Space Models: A Mamba Approach for Simplicial Complexes
Graph Neural Networks based on the message-passing (MP) mechanism are a
dominant approach for handling graph-structured data. However, they are
inherently limited to modeling only pairwise interactions, making it difficult
to explicitly capture the complexity of systems with $n$-body relations. To
address this, topological deep learning has emerged as a promising field for
studying and modeling higher-order interactions using various topological
domains, such as simplicial and cellular complexes. While these new domains
provide powerful representations, they introduce new challenges, such as
effectively modeling the interactions among higher-order structures through
higher-order MP. Meanwhile, structured state-space sequence models have proven
to be effective for sequence modeling and have recently been adapted for graph
data by encoding the neighborhood of a node as a sequence, thereby avoiding the
MP mechanism. In this work, we propose a novel architecture designed to operate
with simplicial complexes, utilizing the Mamba state-space model as its
backbone. Our approach generates sequences for the nodes based on the
neighboring cells, enabling direct communication between all higher-order
structures, regardless of their rank. We extensively validate our model,
demonstrating that it achieves competitive performance compared to
state-of-the-art models developed for simplicial complexes.