[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"similar-bgavran--Category_Theory_Machine_Learning":3,"tool-bgavran--Category_Theory_Machine_Learning":61},[4,18,26,36,44,53],{"id":5,"name":6,"github_repo":7,"description_zh":8,"stars":9,"difficulty_score":10,"last_commit_at":11,"category_tags":12,"status":17},4358,"openclaw","openclaw\u002Fopenclaw","OpenClaw 是一款专为个人打造的本地化 AI 助手，旨在让你在自己的设备上拥有完全可控的智能伙伴。它打破了传统 AI 助手局限于特定网页或应用的束缚，能够直接接入你日常使用的各类通讯渠道，包括微信、WhatsApp、Telegram、Discord、iMessage 等数十种平台。无论你在哪个聊天软件中发送消息，OpenClaw 都能即时响应，甚至支持在 macOS、iOS 和 Android 设备上进行语音交互，并提供实时的画布渲染功能供你操控。\n\n这款工具主要解决了用户对数据隐私、响应速度以及“始终在线”体验的需求。通过将 AI 部署在本地，用户无需依赖云端服务即可享受快速、私密的智能辅助，真正实现了“你的数据，你做主”。其独特的技术亮点在于强大的网关架构，将控制平面与核心助手分离，确保跨平台通信的流畅性与扩展性。\n\nOpenClaw 非常适合希望构建个性化工作流的技术爱好者、开发者，以及注重隐私保护且不愿被单一生态绑定的普通用户。只要具备基础的终端操作能力（支持 macOS、Linux 及 Windows WSL2），即可通过简单的命令行引导完成部署。如果你渴望拥有一个懂你",349277,3,"2026-04-06T06:32:30",[13,14,15,16],"Agent","开发框架","图像","数据工具","ready",{"id":19,"name":20,"github_repo":21,"description_zh":22,"stars":23,"difficulty_score":10,"last_commit_at":24,"category_tags":25,"status":17},3808,"stable-diffusion-webui","AUTOMATIC1111\u002Fstable-diffusion-webui","stable-diffusion-webui 是一个基于 Gradio 构建的网页版操作界面，旨在让用户能够轻松地在本地运行和使用强大的 Stable Diffusion 图像生成模型。它解决了原始模型依赖命令行、操作门槛高且功能分散的痛点，将复杂的 AI 绘图流程整合进一个直观易用的图形化平台。\n\n无论是希望快速上手的普通创作者、需要精细控制画面细节的设计师，还是想要深入探索模型潜力的开发者与研究人员，都能从中获益。其核心亮点在于极高的功能丰富度：不仅支持文生图、图生图、局部重绘（Inpainting）和外绘（Outpainting）等基础模式，还独创了注意力机制调整、提示词矩阵、负向提示词以及“高清修复”等高级功能。此外，它内置了 GFPGAN 和 CodeFormer 等人脸修复工具，支持多种神经网络放大算法，并允许用户通过插件系统无限扩展能力。即使是显存有限的设备，stable-diffusion-webui 也提供了相应的优化选项，让高质量的 AI 艺术创作变得触手可及。",162132,"2026-04-05T11:01:52",[14,15,13],{"id":27,"name":28,"github_repo":29,"description_zh":30,"stars":31,"difficulty_score":32,"last_commit_at":33,"category_tags":34,"status":17},1381,"everything-claude-code","affaan-m\u002Feverything-claude-code","everything-claude-code 是一套专为 AI 编程助手（如 Claude Code、Codex、Cursor 等）打造的高性能优化系统。它不仅仅是一组配置文件，而是一个经过长期实战打磨的完整框架，旨在解决 AI 代理在实际开发中面临的效率低下、记忆丢失、安全隐患及缺乏持续学习能力等核心痛点。\n\n通过引入技能模块化、直觉增强、记忆持久化机制以及内置的安全扫描功能，everything-claude-code 能显著提升 AI 在复杂任务中的表现，帮助开发者构建更稳定、更智能的生产级 AI 代理。其独特的“研究优先”开发理念和针对 Token 消耗的优化策略，使得模型响应更快、成本更低，同时有效防御潜在的攻击向量。\n\n这套工具特别适合软件开发者、AI 研究人员以及希望深度定制 AI 工作流的技术团队使用。无论您是在构建大型代码库，还是需要 AI 协助进行安全审计与自动化测试，everything-claude-code 都能提供强大的底层支持。作为一个曾荣获 Anthropic 黑客大奖的开源项目，它融合了多语言支持与丰富的实战钩子（hooks），让 AI 真正成长为懂上",153609,2,"2026-04-13T11:34:59",[14,13,35],"语言模型",{"id":37,"name":38,"github_repo":39,"description_zh":40,"stars":41,"difficulty_score":32,"last_commit_at":42,"category_tags":43,"status":17},2271,"ComfyUI","Comfy-Org\u002FComfyUI","ComfyUI 是一款功能强大且高度模块化的视觉 AI 引擎，专为设计和执行复杂的 Stable Diffusion 图像生成流程而打造。它摒弃了传统的代码编写模式，采用直观的节点式流程图界面，让用户通过连接不同的功能模块即可构建个性化的生成管线。\n\n这一设计巧妙解决了高级 AI 绘图工作流配置复杂、灵活性不足的痛点。用户无需具备编程背景，也能自由组合模型、调整参数并实时预览效果，轻松实现从基础文生图到多步骤高清修复等各类复杂任务。ComfyUI 拥有极佳的兼容性，不仅支持 Windows、macOS 和 Linux 全平台，还广泛适配 NVIDIA、AMD、Intel 及苹果 Silicon 等多种硬件架构，并率先支持 SDXL、Flux、SD3 等前沿模型。\n\n无论是希望深入探索算法潜力的研究人员和开发者，还是追求极致创作自由度的设计师与资深 AI 绘画爱好者，ComfyUI 都能提供强大的支持。其独特的模块化架构允许社区不断扩展新功能，使其成为当前最灵活、生态最丰富的开源扩散模型工具之一，帮助用户将创意高效转化为现实。",108322,"2026-04-10T11:39:34",[14,15,13],{"id":45,"name":46,"github_repo":47,"description_zh":48,"stars":49,"difficulty_score":32,"last_commit_at":50,"category_tags":51,"status":17},6121,"gemini-cli","google-gemini\u002Fgemini-cli","gemini-cli 是一款由谷歌推出的开源 AI 命令行工具，它将强大的 Gemini 大模型能力直接集成到用户的终端环境中。对于习惯在命令行工作的开发者而言，它提供了一条从输入提示词到获取模型响应的最短路径，无需切换窗口即可享受智能辅助。\n\n这款工具主要解决了开发过程中频繁上下文切换的痛点，让用户能在熟悉的终端界面内直接完成代码理解、生成、调试以及自动化运维任务。无论是查询大型代码库、根据草图生成应用，还是执行复杂的 Git 操作，gemini-cli 都能通过自然语言指令高效处理。\n\n它特别适合广大软件工程师、DevOps 人员及技术研究人员使用。其核心亮点包括支持高达 100 万 token 的超长上下文窗口，具备出色的逻辑推理能力；内置 Google 搜索、文件操作及 Shell 命令执行等实用工具；更独特的是，它支持 MCP（模型上下文协议），允许用户灵活扩展自定义集成，连接如图像生成等外部能力。此外，个人谷歌账号即可享受免费的额度支持，且项目基于 Apache 2.0 协议完全开源，是提升终端工作效率的理想助手。",100752,"2026-04-10T01:20:03",[52,13,15,14],"插件",{"id":54,"name":55,"github_repo":56,"description_zh":57,"stars":58,"difficulty_score":32,"last_commit_at":59,"category_tags":60,"status":17},4721,"markitdown","microsoft\u002Fmarkitdown","MarkItDown 是一款由微软 AutoGen 团队打造的轻量级 Python 工具，专为将各类文件高效转换为 Markdown 格式而设计。它支持 PDF、Word、Excel、PPT、图片（含 OCR）、音频（含语音转录）、HTML 乃至 YouTube 链接等多种格式的解析，能够精准提取文档中的标题、列表、表格和链接等关键结构信息。\n\n在人工智能应用日益普及的今天，大语言模型（LLM）虽擅长处理文本，却难以直接读取复杂的二进制办公文档。MarkItDown 恰好解决了这一痛点，它将非结构化或半结构化的文件转化为模型“原生理解”且 Token 效率极高的 Markdown 格式，成为连接本地文件与 AI 分析 pipeline 的理想桥梁。此外，它还提供了 MCP（模型上下文协议）服务器，可无缝集成到 Claude Desktop 等 LLM 应用中。\n\n这款工具特别适合开发者、数据科学家及 AI 研究人员使用，尤其是那些需要构建文档检索增强生成（RAG）系统、进行批量文本分析或希望让 AI 助手直接“阅读”本地文件的用户。虽然生成的内容也具备一定可读性，但其核心优势在于为机器",93400,"2026-04-06T19:52:38",[52,14],{"id":62,"github_repo":63,"name":64,"description_en":65,"description_zh":66,"ai_summary_zh":66,"readme_en":67,"readme_zh":68,"quickstart_zh":69,"use_case_zh":70,"hero_image_url":71,"owner_login":72,"owner_name":73,"owner_avatar_url":74,"owner_bio":75,"owner_company":76,"owner_location":77,"owner_email":78,"owner_twitter":76,"owner_website":79,"owner_url":80,"languages":81,"stars":86,"forks":85,"last_commit_at":87,"license":76,"difficulty_score":88,"env_os":89,"env_gpu":90,"env_ram":90,"env_deps":91,"category_tags":94,"github_topics":95,"view_count":32,"oss_zip_url":76,"oss_zip_packed_at":76,"status":17,"created_at":100,"updated_at":101,"faqs":102,"releases":103},7182,"bgavran\u002FCategory_Theory_Machine_Learning","Category_Theory_Machine_Learning","List of papers studying machine learning through the lens of category theory","Category_Theory_Machine_Learning 是一个专注于整理“范畴论与机器学习交叉领域”研究论文的开源知识库。它旨在解决该前沿方向文献分散、难以系统追踪的问题，将大量探讨如何利用范畴论这一高度抽象的数学语言来重新审视和构建机器学习理论的学术成果，按主题进行了清晰分类。\n\n这份清单涵盖了从深度学习基础组件、梯度下降的函子解释，到注意力机制的拓扑空间分析等广泛议题。其独特的技术亮点在于揭示了神经网络架构背后的代数结构与组合性原理，例如通过“反向导数上升”或“透镜（Lenses）”等概念，为理解模型内部运作提供了全新的数学视角。\n\n该资源主要适合人工智能研究人员、理论计算机科学家以及对数学基础有浓厚兴趣的高级开发者使用。对于希望突破传统工程视角，从更本质的数学层面探索下一代机器学习架构的学者而言，Category_Theory_Machine_Learning 提供了一份极具价值的入门指南与研究地图，帮助用户高效定位关键文献，把握学科融合的最新动态。","# Category Theory ∩ Machine Learning\n\nCategory theory has been finding increasing applications in machine learning. This repository aims to list all of the relevant papers, grouped by fields.\n\nFor an introduction to the ideas behind category theory, check out [this link](https:\u002F\u002Fgithub.com\u002Fbgavran\u002FCategory_Theory_Resources).\n\n\u003Cp align=\"center\">\u003Cimg width=\"100%\" src=\"https:\u002F\u002Foss.gittoolsai.com\u002Fimages\u002Fbgavran_Category_Theory_Machine_Learning_readme_121b5441bf8c.png\" \u002F>\u003C\u002Fp>\n\nThere might be papers missing, and some papers are in multiple fields. Feel free to contribute to this list or suggest changes to it by creating a pull request or opening an issue.\n\n### Theses\n\n* [Fundamental Components of Deep Learning: A category-theoretic approach](https:\u002F\u002Farxiv.org\u002Fabs\u002F2403.13001)\n* [Robust Diagrams for Deep Learning Architectures: Applications and Theory](https:\u002F\u002Fwww.vtabbott.io\u002Fcontent\u002Ffiles\u002F2023\u002F11\u002FRobust-Diagrams-for-Deep-Learning-Architectures.pdf)\n* [Category-Theoretic Datastructures and Algorithms for Learning Polynomial Circuits](https:\u002F\u002Feprints.soton.ac.uk\u002F483757\u002F1\u002Fpaul_wilson_thesis_acrobat_fixup.pdf)\n* [Category Theory for Quantum Natural Language Processing](https:\u002F\u002Farxiv.org\u002Fabs\u002F2212.06615)\n* [Towards a Categorical Foundation of Deep Learning: A Survey](https:\u002F\u002Farxiv.org\u002Fabs\u002F2410.05353)\n  \n---\n\n### General Deep Learning\n* [Categorical Foundations of Gradient-Based Learning](https:\u002F\u002Farxiv.org\u002Fabs\u002F2103.01931)\n* [Position: Categorical Deep Learning is an Algebraic Theory of All Architectures](https:\u002F\u002Farxiv.org\u002Fabs\u002F2402.15332)\n* [Coalgebras for categorical deep learning: Representability and universal approximation](https:\u002F\u002Farxiv.org\u002Fabs\u002F2603.03227)\n* [Position: Topological Deep Learning is the New Frontier for Relational Learning](https:\u002F\u002Farxiv.org\u002Fabs\u002F2402.08871)\n* [Backprop as Functor](https:\u002F\u002Farxiv.org\u002Fabs\u002F1711.10455)\n* [Lenses and Learners](https:\u002F\u002Farxiv.org\u002Fabs\u002F1903.03671)\n* [Reverse Derivative Ascent](https:\u002F\u002Farxiv.org\u002Fabs\u002F2101.10488)\n* [Dioptics](http:\u002F\u002Fevents.cs.bham.ac.uk\u002Fsyco\u002Fstrings3-syco5\u002Fpapers\u002Fdalrymple.pdf)\n* [Learning Functors using Gradient Descent](http:\u002F\u002Fwww.cs.ox.ac.uk\u002FACT2019\u002Fpreproceedings\u002FBruno%20Gavranovic.pdf) (longer version [here](https:\u002F\u002Farxiv.org\u002Fabs\u002F1907.08292))\n* [Compositionality for Recursive Neural Networks](https:\u002F\u002Farxiv.org\u002Fabs\u002F1901.10723)\n* [Deep neural networks as nested dynamical systems](https:\u002F\u002Farxiv.org\u002Fabs\u002F2111.01297)\n* [Category Theory in Machine Learning](https:\u002F\u002Farxiv.org\u002Fabs\u002F2106.07032)\n* [Neural network layers as parametric spans](https:\u002F\u002Farxiv.org\u002Fabs\u002F2208.00809)\n* [Categories of Differentiable Polynomial Circuits for Machine Learning](https:\u002F\u002Farxiv.org\u002Fabs\u002F2203.06430)\n* [Learners are almost Free Compact Closed](https:\u002F\u002Foxford24.github.io\u002Fassets\u002Fact-papers\u002F39_learners_are_almost_free_compa.pdf)\n* [Going Beyond Neural Network Feature Similarity: The Network Feature Complexity and Its Interpretation Using Category Theory](https:\u002F\u002Farxiv.org\u002Fabs\u002F2310.06756)\n* [Attending to Topological Spaces: The Cellular Transformer](https:\u002F\u002Farxiv.org\u002Fabs\u002F2405.14094)\n* [On the Anatomy of Attention](https:\u002F\u002Farxiv.org\u002Fabs\u002F2407.02423)\n* [Algebraic Dynamical Systems in Machine Learning](https:\u002F\u002Farxiv.org\u002Fabs\u002F2311.03118)\n* [Can neural operators always be continuously discretized?](https:\u002F\u002Farxiv.org\u002Fabs\u002F2412.03393)\n* [Order Theory in the Context of Machine Learning: an application](https:\u002F\u002Farxiv.org\u002Fabs\u002F2412.06097)\n* [An Invitation to Neuroalgebraic Geometry](https:\u002F\u002Farxiv.org\u002Fabs\u002F2501.18915)\n* [Algebraic Positional Encodings](https:\u002F\u002Farxiv.org\u002Fabs\u002F2312.16045)\n* [Learning Structure-Aware Representations of Dependent Types](https:\u002F\u002Farxiv.org\u002Fabs\u002F2402.02104)\n* [Accelerating Machine Learning Systems via Category Theory: Applications to Spherical Attention for Gene Regulatory Networks](https:\u002F\u002Farxiv.org\u002Fabs\u002F2505.09326)\n* [Copresheaf Topological Neural Networks: A Generalized Deep Learning Framework](https:\u002F\u002Farxiv.org\u002Fabs\u002F2505.21251)\n* [Categorical Invariants of Learning Dynamics](https:\u002F\u002Farxiv.org\u002Fabs\u002F2510.04376)\n* [Product Interaction: An Algebraic Formalism for Deep Learning Architectures](https:\u002F\u002Farxiv.org\u002Fabs\u002F2602.02573)\n\n\n#### Equivariance\n* [Equivariant neural networks and piecewise linear representation theory](https:\u002F\u002Farxiv.org\u002Fabs\u002F2408.00949)\n* [Local Permutation Equivariance For Graph Neural Networks](https:\u002F\u002Farxiv.org\u002Fabs\u002F2111.11840)\n* [Stochastic Neural Network Symmetrisation in Markov Categories](https:\u002F\u002Farxiv.org\u002Fabs\u002F2406.11814)\n* [Metric Learning for Clifford Group Equivariant Neural Networks](https:\u002F\u002Farxiv.org\u002Fabs\u002F2407.09926)\n* [Categorification of Group Equivariant Neural Networks](https:\u002F\u002Farxiv.org\u002Fabs\u002F2304.14144)\n* [Equivariant Single View Pose Prediction Via Induced and Restricted Representations](https:\u002F\u002Farxiv.org\u002Fabs\u002F2307.03704)\n* [Characterizing the invariances of learning algorithms using category theory](https:\u002F\u002Farxiv.org\u002Fabs\u002F1905.02072)\n* [Mathematical Foundation of Interpretable Equivariant Surrogate Models](https:\u002F\u002Farxiv.org\u002Fabs\u002F2503.01942)\n* [Filter Equivariant Functions: A symmetric account of length-general extrapolation on lists](https:\u002F\u002Farxiv.org\u002Fabs\u002F2507.08796)\n* [Relational inductive biases on attention mechanisms](https:\u002F\u002Farxiv.org\u002Fabs\u002F2507.04117)\n* [Identifiable Equivariant Networks are Layerwise Equivariant](https:\u002F\u002Farxiv.org\u002Fabs\u002F2601.21645)\n\n#### Graph Neural Networks\n* [Graph Neural Networks are Dynamic Programmers](https:\u002F\u002Farxiv.org\u002Fabs\u002F2203.15544)\n* [Natural Graph Networks](https:\u002F\u002Farxiv.org\u002Fabs\u002F2007.08349)\n* [Sheaf Representation Learning](https:\u002F\u002Fwww.proquest.com\u002Fopenview\u002F5ab4860c57313d5f1a02fbe90a505a7f\u002F1?pq-origsite=gscholar&cbl=18750&diss=y)\n* [Sheaf Neural Networks](https:\u002F\u002Farxiv.org\u002Fabs\u002F2012.06333)\n* [Sheaf Neural Networks with Connection Laplacians](https:\u002F\u002Farxiv.org\u002Fabs\u002F2206.08702)\n* [Neural Sheaf Diffusion: A Topological Perspective on Heterophily and Oversmoothing in GNNs](https:\u002F\u002Farxiv.org\u002Fabs\u002F2202.04579)\n* [Nonlinear Sheaf Diffusion in Graph Neural Networks](https:\u002F\u002Farxiv.org\u002Fabs\u002F2403.00337)\n* [Graph Convolutional Neural Networks as Parametric CoKleisli morphisms](https:\u002F\u002Farxiv.org\u002Fabs\u002F2212.00542)\n* [Learnable Commutative Monoids for Graph Neural Networks](https:\u002F\u002Farxiv.org\u002Fabs\u002F2212.08541)\n* [Sheaf Neural Networks for Graph-based Recommender Systems](https:\u002F\u002Farxiv.org\u002Fabs\u002F2304.09097)\n* [Sheaf theory: from deep geometry to deep learning](https:\u002F\u002Farxiv.org\u002Fabs\u002F2502.15476)\n* [Asynchronous Algorithmic Alignment with Cocycles](https:\u002F\u002Farxiv.org\u002Fabs\u002F2306.15632)\n* [Topologically Attributed Graphs for Shape Discrimination](https:\u002F\u002Farxiv.org\u002Fabs\u002F2306.17805)\n* [Grothendieck Graph Neural Networks Framework: An Algebraic Platform for Crafting Topology-Aware GNNs](https:\u002F\u002Farxiv.org\u002Fabs\u002F2412.08835)\n* [Don't be Afraid of Cell Complexes! An Introduction from an Applied Perspective](https:\u002F\u002Farxiv.org\u002Fabs\u002F2506.09726)\n* [Graph Lineages and Skeletal Graph Products](https:\u002F\u002Farxiv.org\u002Fabs\u002F2508.00197)\n* [On the Sheafification of Higher-Order Message Passing](https:\u002F\u002Farxiv.org\u002Fabs\u002F2509.23020)\n* [Disentangling Hyperedges through the Lens of Category Theory](https:\u002F\u002Farxiv.org\u002Fabs\u002F2510.16289)\n* [Weisfeiler and Lehman Go Categorical](https:\u002F\u002Farxiv.org\u002Fabs\u002F2602.06787)\n* [Modeling Topological Impact on Node Attribute Distributions in Attributed Graphs](https:\u002F\u002Farxiv.org\u002Fabs\u002F2602.01454)\n\n---\n\n### Differentiable programming \u002F automatic differentiation\n* [CHAD: Combinatory Homomorphic Automatic Differentiation](https:\u002F\u002Farxiv.org\u002Fabs\u002F2103.15776)\n* [CHAD for Expressive Total Languages](https:\u002F\u002Farxiv.org\u002Fabs\u002F2110.00446)\n* [The Differentiable Curry](https:\u002F\u002Fdimitriv.github.io\u002Fpapers\u002Fhoad-workshop.pdf)\n* [Functorial String Diagrams for Reverse-Mode Automatic Differentiation](https:\u002F\u002Farxiv.org\u002Fabs\u002F2107.13433)\n* [Differentiable Causal Computations via Delayed Trace](https:\u002F\u002Farxiv.org\u002Fabs\u002F1903.01093)\n* [Simple Essence of Automatic Differentiation](https:\u002F\u002Farxiv.org\u002Fabs\u002F1804.00746)\n* [Reverse Derivative Categories](https:\u002F\u002Farxiv.org\u002Fabs\u002F1910.07065)\n* [Reverse Faà di Bruno's Formula for Cartesian Reverse Differential Categories](https:\u002F\u002Farxiv.org\u002Fabs\u002F2509.20931)\n* [Towards formalizing and extending differential programming using tangent categories](http:\u002F\u002Fwww.cs.ox.ac.uk\u002FACT2019\u002Fpreproceedings\u002FJonathan%20Gallagher,%20Geoff%20Cruttwell%20and%20Ben%20MacAdam.pdf)\n* [Correctness of Automatic Differentiation via Diffeologies and Categorical Gluing](https:\u002F\u002Farxiv.org\u002Fabs\u002F2001.02209)\n* [Denotationally Correct, Purely Functional, Efficient Reverse-mode Automatic Differentiation](https:\u002F\u002Farxiv.org\u002Fabs\u002F2212.09801)\n* [Higher Order Automatic Differentiation of Higher Order Functions](https:\u002F\u002Farxiv.org\u002Fabs\u002F2101.06757)\n* [Space-time tradeoffs of lenses and optics via higher category theory](https:\u002F\u002Farxiv.org\u002Fabs\u002F2209.09351)\n* [Using Rewrite Strategies for Efficient Functional Automatic Differentiation](https:\u002F\u002Farxiv.org\u002Fabs\u002F2307.02447)\n\n### CuTe Layouts\n\n* [CuTe Layout Representation and Algebra](https:\u002F\u002Farxiv.org\u002Fabs\u002F2603.02298v1)\n* [Categorical Foundations for CuTe Layouts](https:\u002F\u002Farxiv.org\u002Fabs\u002F2601.05972)\n\n---\n\n### Probability theory\n* [Markov categories](https:\u002F\u002Farxiv.org\u002Fabs\u002F1908.07021)\n* [Markov Categories and Entropy](https:\u002F\u002Farxiv.org\u002Fabs\u002F2212.11719)\n* [Infinite products and zero-one laws in categorical probability](https:\u002F\u002Farxiv.org\u002Fabs\u002F1912.02769)\n* [A Convenient Category for Higher-Order Probability Theory](https:\u002F\u002Farxiv.org\u002Fabs\u002F1701.02547)\n* [Bimonoidal Structure of Probability Monads](https:\u002F\u002Farxiv.org\u002Fabs\u002F1804.03527)\n* [Representable Markov Categories and Comparison of Statistical Experiments in Categorical Probability](https:\u002F\u002Farxiv.org\u002Fabs\u002F2010.07416)\n* [De Finneti's construction as a categorical limit](https:\u002F\u002Farxiv.org\u002Fabs\u002F2003.01964)\n* [A Probability Monad as the Colimit of Spaces of Finite Samples](https:\u002F\u002Farxiv.org\u002Fabs\u002F1712.05363)\n* [A Probabilistic Dependent Type System based on Non-Deterministic Beta Reduction](https:\u002F\u002Farxiv.org\u002Fabs\u002F1602.06420)\n* [Probability, valuations, hyperspace: Three monads on Top and the support as a morphism](https:\u002F\u002Farxiv.org\u002Fabs\u002F1910.03752)\n* [Categorical Probability Theory](https:\u002F\u002Farxiv.org\u002Fabs\u002F1406.6030)\n* [Information structures and their cohomology](https:\u002F\u002Farxiv.org\u002Fabs\u002F1709.07807)\n* [Computable Stochastic Processes](https:\u002F\u002Farxiv.org\u002Fabs\u002F1409.4667)\n* [Compositional Semantics for Probabilistic Programs with Exact Conditioning](https:\u002F\u002Farxiv.org\u002Fabs\u002F2101.11351)\n* [String Diagrams with Factorized Densities](https:\u002F\u002Farxiv.org\u002Fabs\u002F2305.02506)\n* [Partial Markov Categories](https:\u002F\u002Farxiv.org\u002Fabs\u002F2502.03477)\n* [Random Variables, Conditional Independence and Categories of Abstract Sample Spaces](https:\u002F\u002Farxiv.org\u002Fabs\u002F2503.02477)\n* [A categorical account of the Metropolis-Hastings algorithm](https:\u002F\u002Farxiv.org\u002Fabs\u002F2601.22911)\n\n---\n\n### Bayesian\u002FCausal inference\n* [The Compositional Structure of Bayesian Inference](https:\u002F\u002Farxiv.org\u002Fabs\u002F2305.06112)\n* [Compositional Inference for Bayesian Networks and Causality](https:\u002F\u002Farxiv.org\u002Fabs\u002F2512.00209)\n* [Dependent Bayesian Lenses: Categories of Bidirectional Markov Kernels with Canonical Bayesian Inversion](https:\u002F\u002Farxiv.org\u002Fabs\u002F2209.14728)\n* [A category theory framework for Bayesian Learning](https:\u002F\u002Farxiv.org\u002Fabs\u002F2111.14293)\n* [Causal Theories: A Categorical Perspective on Bayesian Networks](https:\u002F\u002Farxiv.org\u002Fabs\u002F1301.6201)\n* [Bayesian machine learning via category theory](https:\u002F\u002Farxiv.org\u002Fabs\u002F1312.1445)\n* [A Categorical Foundation for Bayesian probability](https:\u002F\u002Farxiv.org\u002Fabs\u002F1205.1488)\n* [Bayesian Open Games](https:\u002F\u002Farxiv.org\u002Fabs\u002F1910.03656)\n* [Causal Inference by String Diagram Surgery](https:\u002F\u002Farxiv.org\u002Fabs\u002F1811.08338)\n* [Disintegration and Bayesian Inversion via String Diagrams](https:\u002F\u002Farxiv.org\u002Fabs\u002F1709.00322)\n* [Categorical Stochastic Processes and Likelihood](https:\u002F\u002Farxiv.org\u002Fabs\u002F2005.04735)\n* [Bayesian Updates Compose Optically](https:\u002F\u002Farxiv.org\u002Fabs\u002F2006.01631)\n* [Automatic Backward Filtering Forward Guiding for Markov processes and graphical models](https:\u002F\u002Farxiv.org\u002Fabs\u002F2010.03509)\n* [Compositionality in algorithms for smoothing](https:\u002F\u002Farxiv.org\u002Fabs\u002F2303.13865)\n* [A Channel-Based Perspective on Conjugate Priors](https:\u002F\u002Farxiv.org\u002Fabs\u002F1707.00269)\n* [A Type Theory for Probabilistic and Bayesian Reasoning](https:\u002F\u002Farxiv.org\u002Fabs\u002F1511.09230)\n* [Denotational validation of higher-order Bayesian inference](https:\u002F\u002Farxiv.org\u002Fabs\u002F1711.03219)\n* [The Geometry of Bayesian Programming](https:\u002F\u002Farxiv.org\u002Fabs\u002F1904.07425)\n* [Relational Reasoning for Markov Chains in a Probabilistic Guarded Lambda Calculus](https:\u002F\u002Farxiv.org\u002Fabs\u002F1802.09787)\n* [A Bayesian Interpretation of the Internal Model Principle](https:\u002F\u002Farxiv.org\u002Fabs\u002F2503.00511)\n* [On the Functoriality of Belief Propagation Algorithms on Finite Partially Ordered Sets](https:\u002F\u002Fhal.sorbonne-universite.fr\u002Fhal-04995548\u002Fdocument)\n* [Bayesian Networks, Markov Networks, Moralisation, Triangulation: a Categorical Perspective](https:\u002F\u002Farxiv.org\u002Fabs\u002F2512.09908)\n\n---\n\n### Topological Data Analysis\n* [On Characterizing the Capacity of Neural Networks using Algebraic Topology](https:\u002F\u002Farxiv.org\u002Fabs\u002F1802.04443)\n* [Persistent-Homology-based Machine Learning and its Applications - A Survey](https:\u002F\u002Farxiv.org\u002Fabs\u002F1811.00252)\n* [Topological Expressiveness of Neural Networks](https:\u002F\u002Frun.unl.pt\u002Fbitstream\u002F10362\u002F129615\u002F1\u002FTAA0115.pdf)\n\n---\n\n### Metric space magnitude\n* [Approximating the convex hull via metric space magnitude](https:\u002F\u002Farxiv.org\u002Fabs\u002F1908.02692)\n* [Practical applications of metric space magnitude and weighting vectors](https:\u002F\u002Farxiv.org\u002Fabs\u002F2006.14063)\n* [Weighting vectors for machine learning: numerical harmonic analysis applied to boundary detection](https:\u002F\u002Farxiv.org\u002Fabs\u002F2106.00827)\n* [The magnitude vector of images](https:\u002F\u002Farxiv.org\u002Fabs\u002F2110.15188)\n* [Magnitude of arithmetic scalar and matrix categories](https:\u002F\u002Farxiv.org\u002Fabs\u002F2304.08334)\n* [Metric Space Magnitude for Evaluating Unsupervised Representation Learning](https:\u002F\u002Farxiv.org\u002Fabs\u002F2311.16054)\n* [The Magnitude of Categories of Texts Enriched by Language Models](https:\u002F\u002Farxiv.org\u002Fabs\u002F2501.06662)\n\n---\n\n### Blog posts\n* [Neural Networks, Types, and Functional Programming](https:\u002F\u002Fcolah.github.io\u002Fposts\u002F2015-09-NN-Types-FP\u002F)\n* [Generalized Transformers from Applicative Functors](https:\u002F\u002Fglaive-research.org\u002F2025\u002F02\u002F11\u002FGeneralized-Transformers-from-Applicative-Functors.html)\n* [Towards Categorical Foundations of Learning](https:\u002F\u002Fwww.brunogavranovic.com\u002Fposts\u002F2021-03-03-Towards-Categorical-Foundations-Of-Neural-Networks.html)\n* [Graph Convolutional Neural Networks as Parametric CoKleisli morphisms](https:\u002F\u002Fwww.brunogavranovic.com\u002Fposts\u002F2022-12-05-graph_neural_networks_as_parametric_cokleisli_morphisms.html)\n* [Optics vs Lenses, Operationally](https:\u002F\u002Fwww.brunogavranovic.com\u002Fposts\u002F2022-02-10-optics-vs-lenses-operationally.html)\n* [Meta-learning and Monads](https:\u002F\u002Fwww.brunogavranovic.com\u002Fposts\u002F2021-10-13-meta-learning-and-monads.html)\n\n---\n\n### Automata Learning\n* [Automata Learning: A Categorical Perspective](http:\u002F\u002Fwww.calf-project.org\u002Fpublications\u002Fprakash.pdf)\n* [A Categorical Framework for Learning Generalised Tree Automata](https:\u002F\u002Farxiv.org\u002Fabs\u002F2001.05786)\n* [CALF: Categorical Automata Learning Framework (thesis)](https:\u002F\u002Fdiscovery.ucl.ac.uk\u002Fid\u002Feprint\u002F10110356\u002F1\u002Fthesis_final_ucl.pdf)\n\n---\n\n### Misc\n* [Generalized Convolution and Efficient Language Recognition](https:\u002F\u002Farxiv.org\u002Fabs\u002F1903.10677)\n* [General supervised learning as change propagation with delta lenses](https:\u002F\u002Farxiv.org\u002Fabs\u002F1911.12904)\n* [From Open Learners to Open Games](https:\u002F\u002Farxiv.org\u002Fabs\u002F1902.08666)\n* [Learners Languages](https:\u002F\u002Farxiv.org\u002Fabs\u002F2103.01189)\n* [A Constructive, Type-Theoretic Approach to Regression via Global Optimisation](https:\u002F\u002Farxiv.org\u002Fabs\u002F2006.12868)\n* [Functorial Manifold Learning](https:\u002F\u002Farxiv.org\u002Fabs\u002F2011.07435)\n* [Diegetic representation of feedback in open games](https:\u002F\u002Farxiv.org\u002Fabs\u002F2206.12338)\n* [Assessing the Unitary RNN as an End-to-End Compositional Model of Syntax](https:\u002F\u002Farxiv.org\u002Fabs\u002F2208.05719)\n* [Classifying Clustering Schemes](https:\u002F\u002Farxiv.org\u002Fabs\u002F1011.5270)\n* [Categorical Hopfield Networks](https:\u002F\u002Farxiv.org\u002Fabs\u002F2201.02756)\n* [A Category-theoretical Meta-analysis of Definitions of Disentanglement](https:\u002F\u002Farxiv.org\u002Fabs\u002F2305.06886)\n* [Isomorphism, Normalizing Flows, and Density Estimation: Preserving Relationships Between Data](https:\u002F\u002Fwww.cs.uoregon.edu\u002FReports\u002FAREA-202307-Walton.pdf)\n* [Transport of Algebraic Structure to Latent Embeddings](https:\u002F\u002Fbrendon-anderson.github.io\u002Ffiles\u002Fpublications\u002Fpfrommer2024transport.pdf)\n* [Data for Mathematical Copilots: Better Ways of Presenting Proofs for Machine Learning](https:\u002F\u002Farxiv.org\u002Fabs\u002F2412.15184)\n* [Categorical Diffusion of Weighted Lattices](https:\u002F\u002Farxiv.org\u002Fabs\u002F2501.03890)\n* [Aggregating time-series and image data: functors and double functors](https:\u002F\u002Farxiv.org\u002Fabs\u002F2504.05274)\n* [Logic Explanation of AI Classifiers by Categorical Explaining Functors](https:\u002F\u002Farxiv.org\u002Fabs\u002F2503.16203)\n* [Typing tensor calculus in 2-categories](https:\u002F\u002Farxiv.org\u002Fabs\u002F1908.01212v4)\n* [The Joys of Categorical Conformal Prediction](https:\u002F\u002Farxiv.org\u002Fabs\u002F2507.04441)\n* [The Gauss-Markov Adjunction: Categorical Semantics of Residuals in Supervised Learning](https:\u002F\u002Farxiv.org\u002Fabs\u002F2507.02442)\n* [Modeling Layout Abstractions Using Integer Set Relations](https:\u002F\u002Farxiv.org\u002Fabs\u002F2511.10374)\n","# 范畴论 ∩ 机器学习\n\n范畴论在机器学习中的应用日益广泛。本仓库旨在按领域分类列出所有相关论文。\n\n如需了解范畴论的基本思想，请参阅[此链接](https:\u002F\u002Fgithub.com\u002Fbgavran\u002FCategory_Theory_Resources)。\n\n\u003Cp align=\"center\">\u003Cimg width=\"100%\" src=\"https:\u002F\u002Foss.gittoolsai.com\u002Fimages\u002Fbgavran_Category_Theory_Machine_Learning_readme_121b5441bf8c.png\" \u002F>\u003C\u002Fp>\n\n可能仍有遗漏的论文，且部分论文可能同时属于多个领域。欢迎通过创建拉取请求或提交议题来为本列表贡献内容或提出改进建议。\n\n### 学位论文\n\n* [深度学习的基本组件：范畴论视角](https:\u002F\u002Farxiv.org\u002Fabs\u002F2403.13001)\n* [深度学习架构的鲁棒图示：应用与理论](https:\u002F\u002Fwww.vtabbott.io\u002Fcontent\u002Ffiles\u002F2023\u002F11\u002FRobust-Diagrams-for-Deep-Learning-Architectures.pdf)\n* [用于学习多项式电路的范畴论数据结构与算法](https:\u002F\u002Feprints.soton.ac.uk\u002F483757\u002F1\u002Fpaul_wilson_thesis_acrobat_fixup.pdf)\n* [量子自然语言处理中的范畴论](https:\u002F\u002Farxiv.org\u002Fabs\u002F2212.06615)\n* [迈向深度学习的范畴论基础：综述](https:\u002F\u002Farxiv.org\u002Fabs\u002F2410.05353)\n  \n---\n\n### 通用深度学习\n* [基于梯度学习的范畴论基础](https:\u002F\u002Farxiv.org\u002Fabs\u002F2103.01931)\n* [观点：范畴论深度学习是所有架构的代数理论](https:\u002F\u002Farxiv.org\u002Fabs\u002F2402.15332)\n* [用于范畴论深度学习的余代数：可表示性与通用逼近](https:\u002F\u002Farxiv.org\u002Fabs\u002F2603.03227)\n* [观点：拓扑深度学习是关系学习的新前沿](https:\u002F\u002Farxiv.org\u002Fabs\u002F2402.08871)\n* [作为函子的反向传播](https:\u002F\u002Farxiv.org\u002Fabs\u002F1711.10455)\n* [透镜与学习者](https:\u002F\u002Farxiv.org\u002Fabs\u002F1903.03671)\n* [逆导数上升法](https:\u002F\u002Farxiv.org\u002Fabs\u002F2101.10488)\n* [双目透镜](http:\u002F\u002Fevents.cs.bham.ac.uk\u002Fsyco\u002Fstrings3-syco5\u002Fpapers\u002Fdalrymple.pdf)\n* [使用梯度下降学习函子](http:\u002F\u002Fwww.cs.ox.ac.uk\u002FACT2019\u002Fpreproceedings\u002FBruno%20Gavranovic.pdf)（更长版本见[这里](https:\u002F\u002Farxiv.org\u002Fabs\u002F1907.08292)）\n* [递归神经网络的组合性](https:\u002F\u002Farxiv.org\u002Fabs\u002F1901.10723)\n* [深度神经网络作为嵌套的动力系统](https:\u002F\u002Farxiv.org\u002Fabs\u002F2111.01297)\n* [机器学习中的范畴论](https:\u002F\u002Farxiv.org\u002Fabs\u002F2106.07032)\n* [神经网络层作为参数化跨度](https:\u002F\u002Farxiv.org\u002Fabs\u002F2208.00809)\n* [用于机器学习的可微多项式电路范畴](https:\u002F\u002Farxiv.org\u002Fabs\u002F2203.06430)\n* [学习者几乎是自由紧闭的](https:\u002F\u002Foxford24.github.io\u002Fassets\u002Fact-papers\u002F39_learners_are_almost_free_compa.pdf)\n* [超越神经网络特征相似性：网络特征复杂度及其范畴论解释](https:\u002F\u002Farxiv.org\u002Fabs\u002F2310.06756)\n* [关注拓扑空间：细胞变换器](https:\u002F\u002Farxiv.org\u002Fabs\u002F2405.14094)\n* [关于注意力机制的解剖学研究](https:\u002F\u002Farxiv.org\u002Fabs\u002F2407.02423)\n* [机器学习中的代数动力系统](https:\u002F\u002Farxiv.org\u002Fabs\u002F2311.03118)\n* [神经算子是否总是可以连续离散化？](https:\u002F\u002Farxiv.org\u002Fabs\u002F2412.03393)\n* [机器学习背景下的序理论：一项应用](https:\u002F\u002Farxiv.org\u002Fabs\u002F2412.06097)\n* [神经代数几何邀请](https:\u002F\u002Farxiv.org\u002Fabs\u002F2501.18915)\n* [代数位置编码](https:\u002F\u002Farxiv.org\u002Fabs\u002F2312.16045)\n* [学习依赖类型的结构感知表示](https:\u002F\u002Farxiv.org\u002Fabs\u002F2402.02104)\n* [通过范畴论加速机器学习系统：应用于基因调控网络的球面注意力](https:\u002F\u002Farxiv.org\u002Fabs\u002F2505.09326)\n* [上层集拓扑神经网络：一种广义深度学习框架](https:\u002F\u002Farxiv.org\u002Fabs\u002F2505.21251)\n* [学习动态的范畴不变量](https:\u002F\u002Farxiv.org\u002Fabs\u002F2510.04376)\n* [乘积交互：深度学习架构的代数形式化](https:\u002F\u002Farxiv.org\u002Fabs\u002F2602.02573)\n\n\n#### 等变性\n* [等变神经网络与分段线性表示理论](https:\u002F\u002Farxiv.org\u002Fabs\u002F2408.00949)\n* [图神经网络的局部置换等变性](https:\u002F\u002Farxiv.org\u002Fabs\u002F2111.11840)\n* [马尔可夫范畴中的随机神经网络对称化](https:\u002F\u002Farxiv.org\u002Fabs\u002F2406.11814)\n* [面向克利福德群等变神经网络的度量学习](https:\u002F\u002Farxiv.org\u002Fabs\u002F2407.09926)\n* [群等变神经网络的范畴化](https:\u002F\u002Farxiv.org\u002Fabs\u002F2304.14144)\n* [通过诱导和限制表示进行等变单视图姿态预测](https:\u002F\u002Farxiv.org\u002Fabs\u002F2307.03704)\n* [利用范畴论刻画学习算法的不变性](https:\u002F\u002Farxiv.org\u002Fabs\u002F1905.02072)\n* [可解释等变代理模型的数学基础](https:\u002F\u002Farxiv.org\u002Fabs\u002F2503.01942)\n* [滤波等变函数：关于列表长度泛化外推的对称性解释](https:\u002F\u002Farxiv.org\u002Fabs\u002F2507.08796)\n* [注意力机制上的关系归纳偏置](https:\u002F\u002Farxiv.org\u002Fabs\u002F2507.04117)\n* [可识别等变网络是逐层等变的](https:\u002F\u002Farxiv.org\u002Fabs\u002F2601.21645)\n\n#### 图神经网络\n* [图神经网络是动态编程者](https:\u002F\u002Farxiv.org\u002Fabs\u002F2203.15544)\n* [自然图网络](https:\u002F\u002Farxiv.org\u002Fabs\u002F2007.08349)\n* [层析表示学习](https:\u002F\u002Fwww.proquest.com\u002Fopenview\u002F5ab4860c57313d5f1a02fbe90a505a7f\u002F1?pq-origsite=gscholar&cbl=18750&diss=y)\n* [层析神经网络](https:\u002F\u002Farxiv.org\u002Fabs\u002F2012.06333)\n* [带有连接拉普拉斯算子的层析神经网络](https:\u002F\u002Farxiv.org\u002Fabs\u002F2206.08702)\n* [神经层析扩散：从拓扑视角看图神经网络中的异质性和过度平滑问题](https:\u002F\u002Farxiv.org\u002Fabs\u002F2202.04579)\n* [图神经网络中的非线性层析扩散](https:\u002F\u002Farxiv.org\u002Fabs\u002F2403.00337)\n* [图卷积神经网络作为参数化柯莱斯里态射](https:\u002F\u002Farxiv.org\u002Fabs\u002F2212.00542)\n* [图神经网络的可学习交换幺半群](https:\u002F\u002Farxiv.org\u002Fabs\u002F2212.08541)\n* [基于图的推荐系统的层析神经网络](https:\u002F\u002Farxiv.org\u002Fabs\u002F2304.09097)\n* [层析理论：从深度几何到深度学习](https:\u002F\u002Farxiv.org\u002Fabs\u002F2502.15476)\n* [利用上循环进行异步算法对齐](https:\u002F\u002Farxiv.org\u002Fabs\u002F2306.15632)\n* [用于形状判别的拓扑赋值图](https:\u002F\u002Farxiv.org\u002Fabs\u002F2306.17805)\n* [格罗滕迪克图神经网络框架：构建拓扑感知图神经网络的代数平台](https:\u002F\u002Farxiv.org\u002Fabs\u002F2412.08835)\n* [别害怕细胞复形！从应用角度的介绍](https:\u002F\u002Farxiv.org\u002Fabs\u002F2506.09726)\n* [图谱系与骨架图乘积](https:\u002F\u002Farxiv.org\u002Fabs\u002F2508.00197)\n* [关于高阶消息传递的层析化](https:\u002F\u002Farxiv.org\u002Fabs\u002F2509.23020)\n* [通过范畴论的视角解开超边](https:\u002F\u002Farxiv.org\u002Fabs\u002F2510.16289)\n* [魏斯费勒—莱曼方法进入范畴论](https:\u002F\u002Farxiv.org\u002Fabs\u002F2602.06787)\n* [建模赋值图中拓扑对节点属性分布的影响](https:\u002F\u002Farxiv.org\u002Fabs\u002F2602.01454)\n\n---\n\n### 可微编程 \u002F 自动微分\n* [CHAD：组合同态自动微分](https:\u002F\u002Farxiv.org\u002Fabs\u002F2103.15776)\n* [适用于表达性全函数语言的CHAD](https:\u002F\u002Farxiv.org\u002Fabs\u002F2110.00446)\n* [可微的Curry语言](https:\u002F\u002Fdimitriv.github.io\u002Fpapers\u002Fhoad-workshop.pdf)\n* [用于反向模式自动微分的函子型字符串图](https:\u002F\u002Farxiv.org\u002Fabs\u002F2107.13433)\n* [通过延迟迹实现可微的因果计算](https:\u002F\u002Farxiv.org\u002Fabs\u002F1903.01093)\n* [自动微分的简单本质](https:\u002F\u002Farxiv.org\u002Fabs\u002F1804.00746)\n* [反向导数范畴](https:\u002F\u002Farxiv.org\u002Fabs\u002F1910.07065)\n* [笛卡尔反向微分范畴中的反向Faà di Bruno公式](https:\u002F\u002Farxiv.org\u002Fabs\u002F2509.20931)\n* [利用切触范畴形式化并扩展微分编程](http:\u002F\u002Fwww.cs.ox.ac.uk\u002FACT2019\u002Fpreproceedings\u002FJonathan%20Gallagher,%20Geoff%20Cruttwell%20and%20Ben%20MacAdam.pdf)\n* [基于微分流形与范畴粘合的自动微分正确性](https:\u002F\u002Farxiv.org\u002Fabs\u002F2001.02209)\n* [语义正确、纯函数式、高效的反向模式自动微分](https:\u002F\u002Farxiv.org\u002Fabs\u002F2212.09801)\n* [高阶函数的高阶自动微分](https:\u002F\u002Farxiv.org\u002Fabs\u002F2101.06757)\n* [通过高阶范畴论探讨透镜与光学的空间—时间权衡](https:\u002F\u002Farxiv.org\u002Fabs\u002F2209.09351)\n* [利用重写策略实现高效的函数式自动微分](https:\u002F\u002Farxiv.org\u002Fabs\u002F2307.02447)\n\n### CuTe 布局\n\n* [CuTe 布局的表示与代数](https:\u002F\u002Farxiv.org\u002Fabs\u002F2603.02298v1)\n* [CuTe 布局的范畴论基础](https:\u002F\u002Farxiv.org\u002Fabs\u002F2601.05972)\n\n---\n\n### 概率论\n* [马尔可夫范畴](https:\u002F\u002Farxiv.org\u002Fabs\u002F1908.07021)\n* [马尔可夫范畴与熵](https:\u002F\u002Farxiv.org\u002Fabs\u002F2212.11719)\n* [范畴概率中的无穷乘积与零一律](https:\u002F\u002Farxiv.org\u002Fabs\u002F1912.02769)\n* [高阶概率论的便利范畴](https:\u002F\u002Farxiv.org\u002Fabs\u002F1701.02547)\n* [概率单子的双幺半群结构](https:\u002F\u002Farxiv.org\u002Fabs\u002F1804.03527)\n* [可表的马尔可夫范畴及范畴概率中统计实验的比较](https:\u002F\u002Farxiv.org\u002Fabs\u002F2010.07416)\n* [德芬内蒂构造作为范畴极限](https:\u002F\u002Farxiv.org\u002Fabs\u002F2003.01964)\n* [作为有限样本空间余极限的概率单子](https:\u002F\u002Farxiv.org\u002Fabs\u002F1712.05363)\n* [基于非确定性β约简的概率依赖类型系统](https:\u002F\u002Farxiv.org\u002Fabs\u002F1602.06420)\n* [概率、赋值、超空间：Top上的三个单子以及支集作为态射](https:\u002F\u002Farxiv.org\u002Fabs\u002F1910.03752)\n* [范畴概率论](https:\u002F\u002Farxiv.org\u002Fabs\u002F1406.6030)\n* [信息结构及其上同调](https:\u002F\u002Farxiv.org\u002Fabs\u002F1709.07807)\n* [可计算的随机过程](https:\u002F\u002Farxiv.org\u002Fabs\u002F1409.4667)\n* [具有精确条件化的概率程序的组合语义](https:\u002F\u002Farxiv.org\u002Fabs\u002F2101.11351)\n* [带因式分解密度的字符串图](https:\u002F\u002Farxiv.org\u002Fabs\u002F2305.02506)\n* [部分马尔可夫范畴](https:\u002F\u002Farxiv.org\u002Fabs\u002F2502.03477)\n* [随机变量、条件独立性与抽象样本空间范畴](https:\u002F\u002Farxiv.org\u002Fabs\u002F2503.02477)\n* [梅特罗波利斯—哈斯廷斯算法的范畴论解释](https:\u002F\u002Farxiv.org\u002Fabs\u002F2601.22911)\n\n---\n\n### 贝叶斯\u002F因果推断\n* [贝叶斯推断的组合结构](https:\u002F\u002Farxiv.org\u002Fabs\u002F2305.06112)\n* [贝叶斯网络与因果关系的组合推理](https:\u002F\u002Farxiv.org\u002Fabs\u002F2512.00209)\n* [依赖性贝叶斯透镜：带有规范贝叶斯逆的双向马尔可夫核范畴](https:\u002F\u002Farxiv.org\u002Fabs\u002F2209.14728)\n* [贝叶斯学习的范畴论框架](https:\u002F\u002Farxiv.org\u002Fabs\u002F2111.14293)\n* [因果理论：贝叶斯网络的范畴视角](https:\u002F\u002Farxiv.org\u002Fabs\u002F1301.6201)\n* [基于范畴论的贝叶斯机器学习](https:\u002F\u002Farxiv.org\u002Fabs\u002F1312.1445)\n* [贝叶斯概率的范畴论基础](https:\u002F\u002Farxiv.org\u002Fabs\u002F1205.1488)\n* [贝叶斯开放博弈](https:\u002F\u002Farxiv.org\u002Fabs\u002F1910.03656)\n* [通过字符串图手术进行因果推断](https:\u002F\u002Farxiv.org\u002Fabs\u002F1811.08338)\n* [基于字符串图的分解与贝叶斯逆](https:\u002F\u002Farxiv.org\u002Fabs\u002F1709.00322)\n* [范畴随机过程与似然度](https:\u002F\u002Farxiv.org\u002Fabs\u002F2005.04735)\n* [贝叶斯更新以光学方式组合](https:\u002F\u002Farxiv.org\u002Fabs\u002F2006.01631)\n* [马尔可夫过程与图模型的自动后向滤波—前向引导](https:\u002F\u002Farxiv.org\u002Fabs\u002F2010.03509)\n* [平滑算法中的组合性](https:\u002F\u002Farxiv.org\u002Fabs\u002F2303.13865)\n* [共轭先验的信道视角](https:\u002F\u002Farxiv.org\u002Fabs\u002F1707.00269)\n* [用于概率与贝叶斯推理的类型论](https:\u002F\u002Farxiv.org\u002Fabs\u002F1511.09230)\n* [高阶贝叶斯推断的指称验证](https:\u002F\u002Farxiv.org\u002Fabs\u002F1711.03219)\n* [贝叶斯编程的几何学](https:\u002F\u002Farxiv.org\u002Fabs\u002F1904.07425)\n* [在概率守卫型λ演算中对马尔可夫链的关系推理](https:\u002F\u002Farxiv.org\u002Fabs\u002F1802.09787)\n* [内部模型原理的贝叶斯解释](https:\u002F\u002Farxiv.org\u002Fabs\u002F2503.00511)\n* [信念传播算法在有限偏序集上的函子性研究](https:\u002F\u002Fhal.sorbonne-universite.fr\u002Fhal-04995548\u002Fdocument)\n* [贝叶斯网络、马尔可夫网络、道德化、三角化：范畴视角](https:\u002F\u002Farxiv.org\u002Fabs\u002F2512.09908)\n\n---\n\n### 拓扑数据分析\n* [利用代数拓扑刻画神经网络的容量](https:\u002F\u002Farxiv.org\u002Fabs\u002F1802.04443)\n* [基于持久同调的机器学习及其应用——综述](https:\u002F\u002Farxiv.org\u002Fabs\u002F1811.00252)\n* [神经网络的拓扑表达能力](https:\u002F\u002Frun.unl.pt\u002Fbitstream\u002F10362\u002F129615\u002F1\u002FTAA0115.pdf)\n\n---\n\n### 度量空间数量\n* [通过度量空间数量近似凸包](https:\u002F\u002Farxiv.org\u002Fabs\u002F1908.02692)\n* [度量空间数量与加权向量的实际应用](https:\u002F\u002Farxiv.org\u002Fabs\u002F2006.14063)\n* [用于机器学习的加权向量：数值谐波分析在边界检测中的应用](https:\u002F\u002Farxiv.org\u002Fabs\u002F2106.00827)\n* [图像的数量向量](https:\u002F\u002Farxiv.org\u002Fabs\u002F2110.15188)\n* [算术标量与矩阵范畴的数量](https:\u002F\u002Farxiv.org\u002Fabs\u002F2304.08334)\n* [度量空间数量用于评估无监督表征学习](https:\u002F\u002Farxiv.org\u002Fabs\u002F2311.16054)\n* [经语言模型增强的文本范畴的数量](https:\u002F\u002Farxiv.org\u002Fabs\u002F2501.06662)\n\n---\n\n### 博文\n* [神经网络、类型与函数式编程](https:\u002F\u002Fcolah.github.io\u002Fposts\u002F2015-09-NN-Types-FP\u002F)\n* [由应用函子导出的广义 Transformer](https:\u002F\u002Fglaive-research.org\u002F2025\u002F02\u002F11\u002FGeneralized-Transformers-from-Applicative-Functors.html)\n* [迈向学习的范畴论基础](https:\u002F\u002Fwww.brunogavranovic.com\u002Fposts\u002F2021-03-03-Towards-Categorical-Foundations-Of-Neural-Networks.html)\n* [图卷积神经网络作为参数化柯克莱斯利态射](https:\u002F\u002Fwww.brunogavranovic.com\u002Fposts\u002F2022-12-05-graph_neural_networks_as_parametric_cokleisli_morphisms.html)\n* [从操作角度看光学与透镜](https:\u002F\u002Fwww.brunogavranovic.com\u002Fposts\u002F2022-02-10-optics-vs-lenses-operationally.html)\n* [元学习与单子](https:\u002F\u002Fwww.brunogavranovic.com\u002Fposts\u002F2021-10-13-meta-learning-and-monads.html)\n\n---\n\n### 自动机学习\n* [自动机学习：范畴论视角](http:\u002F\u002Fwww.calf-project.org\u002Fpublications\u002Fprakash.pdf)\n* [用于学习广义树自动机的范畴论框架](https:\u002F\u002Farxiv.org\u002Fabs\u002F2001.05786)\n* [CALF：范畴论自动机学习框架（学位论文）](https:\u002F\u002Fdiscovery.ucl.ac.uk\u002Fid\u002Feprint\u002F10110356\u002F1\u002Fthesis_final_ucl.pdf)\n\n---\n\n### 其他\n* [广义卷积与高效语言识别](https:\u002F\u002Farxiv.org\u002Fabs\u002F1903.10677)\n* [广义监督学习视为使用 delta 透镜进行变化传播](https:\u002F\u002Farxiv.org\u002Fabs\u002F1911.12904)\n* [从开放学习者到开放博弈](https:\u002F\u002Farxiv.org\u002Fabs\u002F1902.08666)\n* [学习者语言](https:\u002F\u002Farxiv.org\u002Fabs\u002F2103.01189)\n* [基于全局优化的构造性类型论回归方法](https:\u002F\u002Farxiv.org\u002Fabs\u002F2006.12868)\n* [函子式的流形学习](https:\u002F\u002Farxiv.org\u002Fabs\u002F2011.07435)\n* [开放博弈中反馈的叙事性表示](https:\u002F\u002Farxiv.org\u002Fabs\u002F2206.12338)\n* [评估单位 RNN 作为句法的端到端组合模型](https:\u002F\u002Farxiv.org\u002Fabs\u002F2208.05719)\n* [聚类方案的分类](https:\u002F\u002Farxiv.org\u002Fabs\u002F1011.5270)\n* [范畴论霍普菲尔德网络](https:\u002F\u002Farxiv.org\u002Fabs\u002F2201.02756)\n* [去耦合定义的范畴论元分析](https:\u002F\u002Farxiv.org\u002Fabs\u002F2305.06886)\n* [同构、归一化流与密度估计：保持数据间的关系](https:\u002F\u002Fwww.cs.uoregon.edu\u002FReports\u002FAREA-202307-Walton.pdf)\n* [代数结构向潜在嵌入的传输](https:\u002F\u002Fbrendon-anderson.github.io\u002Ffiles\u002Fpublications\u002Fpfrommer2024transport.pdf)\n* [面向数学协作助手的数据：为机器学习更好地呈现证明的方式](https:\u002F\u002Farxiv.org\u002Fabs\u002F2412.15184)\n* [加权格的范畴论扩散](https:\u002F\u002Farxiv.org\u002Fabs\u002F2501.03890)\n* [时间序列与图像数据的聚合：函子与双函子](https:\u002F\u002Farxiv.org\u002Fabs\u002F2504.05274)\n* [通过范畴解释函子对 AI 分类器的逻辑解释](https:\u002F\u002Farxiv.org\u002Fabs\u002F2503.16203)\n* [在 2-范畴中对张量微积分进行类型标注](https:\u002F\u002Farxiv.org\u002Fabs\u002F1908.01212v4)\n* [范畴论一致预测的乐趣](https:\u002F\u002Farxiv.org\u002Fabs\u002F2507.04441)\n* [高斯–马尔可夫伴随：监督学习中残差的范畴语义](https:\u002F\u002Farxiv.org\u002Fabs\u002F2507.02442)\n* [利用整数集关系建模布局抽象](https:\u002F\u002Farxiv.org\u002Fabs\u002F2511.10374)","# Category_Theory_Machine_Learning 快速上手指南\n\n**工具简介**\n`Category_Theory_Machine_Learning` 并非一个可直接安装运行的软件库或框架，而是一个**学术资源汇总仓库**。它致力于收集并分类整理将“范畴论（Category Theory）”应用于“机器学习”领域的相关论文、学位论文及技术报告。本指南旨在帮助开发者高效利用该仓库进行理论研究与技术探索。\n\n## 1. 环境准备\n\n由于本项目主要为文档和链接集合，无需复杂的运行时环境，仅需具备基础的代码托管平台访问能力。\n\n*   **系统要求**：Windows \u002F macOS \u002F Linux (任意支持现代浏览器的操作系统)\n*   **前置依赖**：\n    *   Git (用于克隆仓库)\n    *   现代 Web 浏览器 (Chrome, Edge, Firefox 等)\n    *   学术文献阅读工具 (如 Zotero, Mendeley 或 PDF 阅读器)\n*   **网络建议**：\n    *   项目资源主要托管于 GitHub 和 arXiv。\n    *   **国内加速方案**：若访问 GitHub 或 arXiv 速度较慢，建议使用国内镜像站（如 `https:\u002F\u002Farxiv.org.cn` 查阅论文）或通过 Gitee 搜索是否有同步镜像仓库。\n\n## 2. 安装步骤（获取资源）\n\n你可以通过克隆仓库到本地或直接在线浏览来获取资源列表。\n\n### 方式一：克隆到本地（推荐）\n\n在终端中执行以下命令：\n\n```bash\ngit clone https:\u002F\u002Fgithub.com\u002Fbrendanfong\u002FCategory_Theory_Machine_Learning.git\ncd Category_Theory_Machine_Learning\n```\n\n*注：若原仓库地址变更，请以上述 README 中提供的实际仓库地址为准。*\n\n### 方式二：在线浏览\n\n直接访问 GitHub 仓库页面查看按领域分类的论文列表：\n*   **通用深度学习** (General Deep Learning)\n*   **等变性** (Equivariance)\n*   **图神经网络** (Graph Neural Networks)\n*   **可微编程\u002F自动微分** (Differentiable programming)\n*   **概率论** (Probability theory)\n*   **贝叶斯\u002F因果推断** (Bayesian\u002FCausal inference)\n\n## 3. 基本使用\n\n本仓库的核心价值在于**按需检索**与**理论溯源**。以下是典型的使用流程：\n\n### 步骤 1：确定研究方向\n打开本地的 `README.md` 文件或在线页面，根据你关注的机器学习子领域查找对应章节。例如，如果你关注**图神经网络 (GNN)** 中的拓扑结构，请定位到 `#### Graph Neural Networks` 部分。\n\n### 步骤 2：获取论文资源\n点击列表中对应的论文标题链接（通常指向 arXiv 或学术会议页面）。\n\n**示例：研究 Sheaf Neural Networks**\n在 `Graph Neural Networks` 章节下，找到如下条目：\n*   [Sheaf Neural Networks](https:\u002F\u002Farxiv.org\u002Fabs\u002F2012.06333)\n*   [Neural Sheaf Diffusion: A Topological Perspective on Heterophily and Oversmoothing in GNNs](https:\u002F\u002Farxiv.org\u002Fabs\u002F2202.04579)\n\n### 步骤 3：理论与实践结合\n阅读论文摘要与核心公式。由于这是理论型资源库，若要复现论文中的算法：\n1.  检查论文是否附带了官方代码链接（通常在论文首页或 GitHub 描述中）。\n2.  若无官方实现，需依据论文中的范畴论定义（如函子、自然变换、伴随性等）自行构建数学模型。\n\n### 进阶：贡献与更新\n该列表欢迎社区补充。若你发现遗漏的重要论文，可通过以下方式贡献：\n```bash\n# 创建分支并修改 README.md 添加新论文链接\ngit checkout -b add-new-paper\n# 编辑文件后提交\ngit commit -m \"Add paper: [Paper Title]\"\n# 推送并发起 Pull Request\ngit push origin add-new-paper\n```\n\n---\n**提示**：范畴论门槛较高，建议初学者先阅读仓库顶部推荐的入门资源 [Category_Theory_Resources](https:\u002F\u002Fgithub.com\u002Fbgavran\u002FCategory_Theory_Resources) 建立基础概念，再深入阅读具体领域的论文。","某生物科技公司算法团队正试图为复杂的基因调控网络设计一种新型注意力机制，以突破传统模型在捕捉长程依赖关系上的瓶颈。\n\n### 没有 Category_Theory_Machine_Learning 时\n- **理论检索如大海捞针**：团队需在 arXiv 海量论文中手动筛选，难以发现“球面注意力”或“拓扑神经网络”等跨学科前沿成果，导致研发方向局限。\n- **架构创新缺乏数学根基**：尝试修改网络结构时仅凭经验试错，缺乏范畴论中“函子”或“参量跨度”等形式化工具指导，新架构的可解释性与收敛性无法保证。\n- **复现与迁移成本高昂**：由于缺少统一的代数理论框架，不同论文间的概念（如反向传播作为函子）难以互通，导致代码复用率低，重复造轮子现象严重。\n\n### 使用 Category_Theory_Machine_Learning 后\n- **精准定位交叉领域文献**：通过该工具分类索引，团队迅速锁定了《Accelerating Machine Learning Systems via Category Theory》等关键论文，直接获取了针对基因网络的球面注意力方案。\n- **基于代数形式化设计架构**：借鉴列表中关于“微分多项式电路”和“余层拓扑神经网络”的理论，团队用严谨的数学语言构建了新模型，显著提升了训练稳定性和泛化能力。\n- **统一视角加速系统迭代**：利用工具提供的统一范畴论视角，团队将梯度下降、反向传播等模块抽象为标准“学习器”，实现了模块化开发，大幅缩短了从理论到原型的周期。\n\nCategory_Theory_Machine_Learning 通过将深奥的范畴论论文系统化整理，为开发者架起了一座连接抽象数学理论与落地 AI 架构创新的坚实桥梁。","https:\u002F\u002Foss.gittoolsai.com\u002Fimages\u002Fbgavran_Category_Theory_Machine_Learning_121b5441.png","bgavran","Bruno Gavranović","https:\u002F\u002Foss.gittoolsai.com\u002Favatars\u002Fbgavran_5de4ccf1.jpg","Building neural networks that generate provably correct code, and the software infrastructure for training them.",null,"London, United Kingdom","bruno@brunogavranovic.com","www.brunogavranovic.com","https:\u002F\u002Fgithub.com\u002Fbgavran",[82],{"name":83,"color":84,"percentage":85},"Python","#3572A5",100,1499,"2026-04-13T09:45:11",1,"","未说明",{"notes":92,"python":90,"dependencies":93},"该仓库并非一个可执行的软件工具或代码库，而是一个关于“范畴论与机器学习”交叉领域的学术论文列表（Awesome List）。README 内容仅包含论文标题和链接，分类涵盖深度学习、图神经网络、概率论、贝叶斯推断等方向。因此，该项目没有操作系统、GPU、内存、Python 版本或依赖库等运行环境需求。用户只需通过浏览器访问链接阅读论文，或使用 Git 克隆该仓库以获取列表文件。",[],[14],[96,97,98,99],"neural-networks","lenses","machine-learning","category-theory","2026-03-27T02:49:30.150509","2026-04-13T23:52:53.116768",[],[]]