AI 与机器学习2019arxiv.org
深度学习的几何理解
Geometric Understanding of Deep Learning (2018)
专业 · 约 90 分钟读完前置知识：微分几何与流形基础、最优传输理论、深度学习基础
导读
这篇论文从几何视角解释深度学习为何有效：作者（包括菲尔兹奖得主丘成桐）论证深度学习成功的根本原理是数据中的流形结构——高维自然数据集中在低维流形附近，深度网络学习的是该流形及其上的概率分布。论文还提出“修正线性复杂度”来度量网络的学习能力与流形的学习难度，并证明固定结构的网络总存在无法学习的流形，同时用最优质量传输理论控制隐空间中的概率分布。
HN 关注点：是因为这是首次有声誉的团队声称证明了长期被相信的“流形假设”是深度学习有效的根本原理，且作者阵容（丘成桐）引人注目。
HN 评论精选
这几条评论里：支持 1 补充 3
支持认为这是首次有声誉地证明流形假设是深度学习根本原理的工作，并期待与 normalizing flows 的比较
据我所知，这是第一次有声誉的研究者声称证明了“流形假设”是深度学习有效的根本原理：自然高维数据集中在低维流形附近，深度学习学习的是流形及其上的概率分布。作者还提出用“修正线性复杂度”度量网络的学习能力和流形的学习难度，并证明固定结构的网络总存在学不了的流形。他们还提出用最优质量传输理论控制隐空间概率分布，我很好奇这与近期的 normalizing flows 工作相比如何。
cs702查看英文原文
Wow. As far as I know, this is the first time anyone reputable[a] has claimed to show (!) that the "manifold hypothesis" is the fundamental principle that makes deep learning work, as has long been believed: "In this work, we give a geometric view to understand deep learning: we show that the fundamental principle attributing to the success is the manifold structure in data, namely natural high dimensional data concentrates close to a low-dimensional manifold, deep learning learns the manifold and the probability distribution on it." Moreover, the authors also claim to have come up with a way of measuring how hard it is for any deep neural net (of fixed size) to learn a parametric representation of a particular lower-dimensional manifold embedded in some higher-dimensional space: "We further introduce the concepts of rectified linear complexity for deep neural network measuring its learning capability, rectified linear complexity of an embedding manifold describing the difficulty to be learned. Then we show for any deep neural network with fixed architecture, there exists a manifold that cannot be learned by the network." Finally, the authors also propose a novel way to control the probability distribution in the latent space. I'm curious to see how their method compares and relates to recent work, e.g., with discrete and continuous normalizing flows: "...we propose to apply optimal mass transportation theory to control the probability distribution in the latent space." This is not going to be a light read... -- [a] One of the authors, Shing-Tung Yau, is a Fields medalist: https://news.ycombinator.com/item?id=18987219
补充好奇跨洲学术合作的运作方式
（一个学术圈外的问题）两位作者在中国高校，两位在美国不同院系。这种跨洲合作一般是怎么开始的、如何推进？多人合作的理论论文中角色如何划分？有没有帮助协作写论文的工具？
lovelearning查看英文原文
[An OT question as somebody not familiar with the academic world] Two of the authors are in a Chinese university. Two of them in different departments in a US university. In general, how does this kind of intercontinental collaboration start, and how do they progress? How are roles defined when multiple people are involved in a theoretical paper like this? Are there some tools that help with collaborative paper writing?
补充从群论/几何角度提出对网络结构的个人理解与开放问题
或许没那么精妙，我把深度网络看作仿射映射与非线性各向同性的三明治结构。仿射映射可以用 GL(n+1) 中的齐次坐标技巧（与神经元偏置相关）很好地表示。一个问题是：仿射与各向同性之间的相互作用在群论意义上是否有值得说的东西。
jesuslop查看英文原文
Possibly less sophisticatedly I think of them as a sandwich of affine maps and nonlinear isotropies (as those giving irregular rings in tree trunks). The affinities are represented nicely in GL(n+1) with a homogenous coordinates trick related to neuron biases. A question would be if there's something interesting to say about the interactions of the affinities and isotropies in group theoretic terms (which I dunno).
补充指出作者中有菲尔兹奖得主丘成桐
读着读着才意识到其中一位作者是著名的菲尔兹奖得主丘成桐 :-)
yoquan查看英文原文
I'm reading this and just realized one author is the famous Field medalist, Shing-Tung Yau :-)
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