AI 与机器学习2024arxiv.org
几何、拓扑与代数结构机器学习指南
Guide to Machine Learning with Geometric, Topological, and Algebraic Structures
进阶 · 约 120 分钟读完前置知识：机器学习基础、微分几何、代数拓扑
导读
这篇 arXiv 论文是一份指南，介绍如何将几何、拓扑和代数结构（如群论、代数拓扑、微分几何等）作为先验融入机器学习，涵盖几何深度学习、拓扑数据分析等方向，并讨论了在蛋白质折叠等任务中的应用。它试图为“数据具有非欧几里得结构时如何选择合适的模型结构”提供系统性的方法论。
HN 关注点：这类方法能否真正突破当前 ML 依赖海量算力、泛化性差的现状，以及这些漂亮的数学在工业界是否有实际落地价值。
HN 评论精选
这几条评论里：支持 1 质疑 1 补充 2
质疑质疑拓扑数据分析等方向在工业界的实际落地情况
几何、拓扑、代数方法在 ML/数据分析中真的被工业界使用吗？数学上确实很美。读研时我遇到几个纯数学博士，说毕业后要去工业界做拓扑数据分析（那是十年前，ML 还没这么火），但我从没听说过有人靠这个计划真正成功。
funnygiraffe查看英文原文
Is geometric, topological, and algebraic ML/data analysis actually used in the industry? It is certainly beautiful math. However, during grad school I met a few pure math PhD students who were saying that after finishing their PhD they will just go into industry to do topological data analysis (this was about 10 years ago and ML wasn't yet as hyped up). However, I have never heard of anybody actually having success on that plan.
支持认为几何/拓扑方法有望解决 ML 的根本性数据-空间错配问题
我百分之百确信这类方法能把 ML 研究从当前资源饥渴、泛化性差的状态中解救出来。低维欧氏几何是特例，高维欧氏空间则不然。大多数真实数据是高维、不光滑、结构上根本不能厚着脸皮称为欧氏的。看看表格数据（从业者实际处理的大部分数据）上什么有效——是梯度提升树，不是神经网络。我们处理的数据和我们硬塞进去的空间之间存在根本性错配。代数拓扑和几何工具在物理学里早已是老一套，在 ML 中理应更有用。
mjhay查看英文原文
I am 100% convinced that these kind of approaches will be what delivers ML research from the current resource-hungry and ungeneralizable status quo. Low-dimensional Euclidean geometry is special. Higher-dimensional Euclidean spaces are less special. Most real-life data is high-dimensional, not at all smooth, and possessing a structure you cannot call Euclidean with a straight face. Look at what works with tabular data (which is probably most of what practitioners work with in the wild). It's gradient boosted trees, not neural networks. There is a fundamental mismatch between the data we usually work with and the spaces we shove it into. Tools from algebraic topology and geometry are old hat in physics. If anything, they should be even more useful in ML.
补充推荐几何深度学习的延伸阅读资料
论文的参考文献里有不少适合入门的好资料，比如这篇内容密集的：Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges（https://geometricdeeplearning.com/）。
dpflan查看英文原文
The paper's references have some good ones for getting more acquainted with these subjects; this one being a nice dense one to start with: - Geometric Deep Learning Grids, Groups, Graphs, Geodesics, and Gauges: https://geometricdeeplearning.com/
补充指出选择合适结构先验很难，并设想多模态数据的应用场景
我在论文中（如蛋白质折叠部分）看到一个共同主题：先识别哪些性质（几何、代数、拓扑）重要、哪个是有用的先验，然后用这份指南选择初始结构。这可能比听起来难——类似贝叶斯先验，都需要特殊假设，但不像后者那样宽容。我好奇：能否用它来整合多模态数据和任务网络？比如把传感器数据、地图拓扑、城市拓扑、图像结合起来，用这份指南得出“街道数据可以与传感器数据嵌入，用 XYZ 网络结构做 ABC 推理”这样的结论？
itissid查看英文原文
One common theme I see in the paper(e.g. in protein folding) is: "Identify what properties are important (geometry, algebra, topo) and which one is an useful prior and then "use" the guide to select an initial struct. This is probably harder than it sounds(unlike bayesian priors which are more forgiving for one to select, but quite like them in that they both require special assumptions)." I wonder: could one use it to bring together certain multimodal data and a proposed network for a task? Like could one bring in sensor, map topology, urban topology, pictures which have certain properties and that help me use this guide to make a statement like : "Street data could be embedded with Sensor data to do ABC kind of inference using XYZ NNetwork structure because this paper suggests that is a reasonable thing to do"?
相关资源
从这篇论文所有 HN 讨论帖的全部评论里整理出来的链接。
- 评论区提到的论文
历次讨论
- 2024Hacker News 讨论帖176 分 · 27 评
同领域还值得读
DeepSeek-R1：通过强化学习激励 LLM 的推理能力
DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via RL
研究如何用强化学习激励大语言模型的推理能力，提出 DeepSeek-R1，并探讨 RL 训练对模型推理表现的影响。
AI 与机器学习2025 · 1,351 分 · 1,056 评 · 引用 30进阶 · 约 80 分钟编辑精选1-bit LLM 时代：面向低成本计算的三值参数
The Era of 1-bit LLMs: ternary parameters for cost-effective computing
提出用三值参数 {-1, 0, 1} 替代 LLM 中的浮点权重，即所谓“1-bit LLM”。
AI 与机器学习2024 · 1,040 分 · 447 评 · 引用 19进阶 · 约 45 分钟编辑精选DeepSeek-V3.2：推进开放大语言模型的前沿
DeepSeek-v3.2: Pushing the frontier of open large language models
这是 DeepSeek-V3.2 的技术报告，延续了 DeepSeek 开源权重模型的路线，目标是让开放模型继续逼近甚至追平前沿闭源模型。
AI 与机器学习2025 · 982 分 · 465 评进阶 · 约 90 分钟编辑精选通过自我对弈与通用强化学习掌握国际象棋和将棋（AlphaZero）
Mastering Chess and Shogi by Self-Play with General Reinforcement Learning
DeepMind 的 AlphaZero 论文：用通用强化学习算法（自我博弈 + MCTS + 神经网络），从零开始仅数小时训练便在国际象棋和将棋上击败了世界顶级引擎 Stock…
AI 与机器学习2017 · 539 分 · 270 评 · 引用 13进阶 · 约 90 分钟编辑精选用强化学习发现更快的矩阵乘法算法
Discovering faster matrix multiplication algorithms with reinforcement learning
论文提出 AlphaTensor，用强化学习搜索矩阵乘法算法。
AI 与机器学习2022 · 518 分 · 111 评专业 · 约 90 分钟编辑精选