科学与数学johnhw.github.io
前一百万个整数的质因数可视化
A visualization of the prime factors of the first million integers
2018 — 2022，它一次次回到首页
黑点是上首页（≥ 50 分）的那次投稿，空心点是没冲上去的投稿，点越大分数越高。
导读
作者用 UMAP 降维算法将前一百万个整数的质因数分解（视为高维稀疏向量）投影到二维平面上，生成了令人惊叹的可视化图像。图中呈现出明显的簇状结构，展示了数字质因数构成在低维空间中的分布形态。
为什么经久不衰：它把抽象的数论和现代降维算法结合起来，图像既美又引发对“结构是数字本身的还是算法伪影”的思考，多年来反复引发讨论。
HN 评论精选
质疑图中结构可能来自算法而非数字本身，建议用 PCA 对照验证
浮现出的结构非常漂亮；但这可能是虚假的——它更多反映的是布局算法的特性，而非数字本身的“真实”结构。如果用比 UMAP 更简单、更不“随意”的 PCA 来做，图像还相似的话，那我们看到的才是关于数字的东西，而不是关于方法的东西。
OscarCunningham查看英文原文
> A very pretty structure emerges; this might be spurious in that it captures more about the layout algorithm than any "true" structure of numbers. What would it look like if we used PCA rather than UMAP? PCA is simpler than UMAP, so it's in some sense "less arbitrary". If the image is similar then we know we're seeing something about numbers rather than about our methods.
认为 UMAP 是高维分析的强大通用工具，并解释其相对 t-SNE 的优势
质因数分解是终极的高维空间。我越看 UMAP 越觉得它会成为高维分析的核心通用工具。目前我的理解是：它是“加强版 t-SNE”——既保留局部邻近性又不撕裂全局结构，速度更快；而且 UMAP 学到的是变换模型而非移动点本身，因此能对未见数据生成嵌入，还支持监督式降维和自定义距离度量。
BenoitP查看英文原文
Reposting my comment from here: https://news.ycombinator.com/item?id=17816981 ---- Prime factors of a number is the ultimate high-dimensional space. Damn. The more I see UMAP, the more I think it is going to be a central and generic tool for high-dimensional analysis. I haven't taken the time to go in depth into it yet, though :/ So far, my understanding of it is: t-SNE on steroids * t-SNE is great for local proximity, but it 'rips' high dimensional global structures too early. UMAP solves both scales by using transformations to map overlapping points of the different lower dimensional spaces that are locally relevant. * It is faster than t-SNE, and has a better scale factor. * t-SNE is about moving the points when UMAP is about finding the transformations that move the points.. which means: a) it yields a model that you can use to create embeddings for unseen data. This means sharing your work by contributing to the public model zoos. b) And you can also do supervised dimension reduction as you create your embedding. Ie You can judge if the shape looks good for unseen data (aka it generalizes well), and then correct the embedding by choosing which unseen instances to add to the training set. This means you control the cost of labeling data. You can see where your errors are, and back-propagate them to the collection process in a cost effective manner. For high dimensional data. * You can choose your metric! Specific a distance function and you're good to go. Haversine for a great orange peeling, Levenshtein for visualizing word spelling (and maybe provide an embedding for ML-based spell checking?) * You can choose the output space to be greater than 2 or 3, in order to stop the compression at a specified level. I believe it will replace t-SNE in the long term. Here is a great video of the author presenting his work: https://www.youtube.com/watch?v=nq6iPZVUxZU
指出图中部分细节是旧版 UMAP 的收敛伪影
据 Dmitry Kobak 说，这些图中的某些细节只是算法收敛的伪影，使用更新版本的 UMAP 时就不再出现了。
superbatfish查看英文原文
According to Dmitry Kobak, some details in these figures are merely convergence artifacts, and no longer produced when using more recent versions of UMAP. https://twitter.com/hippopedoid/status/1318917878364672001?lang=en
困惑于图中簇结构究竟是数字的深层性质还是算法伪影
我没搞明白：这些簇到底有没有意义？感觉它应该在告诉我们关于数字的某种重要东西。虽然我不太懂 UMAP，但如果它降维真有那么好，簇就是簇——这结构太清晰了，不太可能只是 UMAP 自身的怪异伪影。还是说真是？
krick查看英文原文
I don't get it. So, do all these clusters mean anything at all? It feels like that should be telling us something eerily important about numbers. Granted, I don't really understand UMAP, but still, if it's any good for dimensionality reduction at all (and it appears to be), then clusters are clusters. This seems to have way too distinct structure to be essentially just a weird artifact of UMAP itself. Or is it?
历次讨论
同类还值得读
如果月球只有 1 像素：一个极其精确的太阳系模型（2014）
If the moon were only 1 pixel: A tediously accurate solar system model (2014)
一个网页版太阳系模型，以“如果月亮只有 1 像素”的真实比例呈现：你必须一路横向滚动，才能体会行星之间空旷到何等程度。
科学与数学上首页 7 次 · 2014–2025joshworth.com编辑精选沉浸式线性代数（2016）
Immersive Linear Algebra (2016)
《Immersive Linear Algebra》是一本完全免费、开放在线的线性代数“超文本教科书”，其最大特色是大量可交互的图形：读者可以直接拖动、操作图中的向量、矩阵和变换…
科学与数学上首页 5 次 · 2015–2024immersivemath.com编辑精选如何将贝果切成两个相连的半环（2009）
How to Slice a Bagel into Two Linked Halves (2009)
数学家 George Hart 的经典教程，演示如何用一刀把贝果切成两个相互扣住的环，并附上切割步骤与数学原理。
科学与数学上首页 5 次 · 2009–2022georgehart.com“犹太问题”：一组用于歧视的数学题
Jewish problems
一篇 arXiv 论文，整理并讨论苏联时期犹太学生在申请大学数学系时被问到的特殊数学难题：这些题往往有简单解法却极难想到，被用作面试中拒绝犹太考生的手段。
科学与数学上首页 5 次 · 2011–2021arxiv.org你的身体并非为长久而造：人类死亡率数据的一课
Your body wasn’t built to last: a lesson from human mortality rates
文章从人类死亡率随年龄变化的统计规律出发，解释身体并非为长久使用而设计：年轻时死亡风险较低，之后随年龄近似指数上升，像复杂系统在不断累积损伤。
科学与数学上首页 5 次 · 2009–2018gravityandlevity.wordpress.com