Low-Dimensional Manifold 低维流形

低维流形，一个被滥用的概念。最近在 Reddit 上看到了一个 10 年前非常通俗易懂的解释：

Imagine your patterned bedsheets. They’ve got a nice plaid grid look to them, very easy to predict what the next few centimeters of material look like.

Now imagine someone tied them in a knot, tore them a little, balled them up, and made them a crumpled mess, then handed them to you. Now if you were to look at the plaid pattern, it would be very difficult to discern how the pattern is, or predict what will be the color of any point of the arbitrary 3D ball of mess you’ve got. But if you untangled it, you could clearly see the pattern again.

The data is a bunch of colors in an arbitrary 3D ball of mess with very difficult to discern patterns. But the data lies on a low dimensional manifold (the 2D bedsheet). If you could figure out the manifold (flatten the bedsheet), the data will be easy to model.

翻译：

想象一下你的带图案床单。它们有着漂亮的格子网格外观，很容易预测接下来几厘米的面料会是什么样子。

现在想象有人把它们打了个结，撕破了一点，揉成一团，弄成了一团皱巴巴的乱糟糟的东西，然后递给你。这时如果你去看那个格子图案，将会非常难以辨清图案是如何分布的，也无法预测你那团任意的三维乱糟糟球体上任意一点的颜色会是什么。但如果你把它解开摊平，就能再次清晰地看到图案。

数据就像是一个任意的三维乱糟糟球体中的一堆颜色，其中的模式极难辨认。但这些数据实际上位于一个低维流形上（即那张二维床单）。如果你能找出这个流形（把床单摊平），数据就会变得易于建模。

Source: https://www.reddit.com/r/MachineLearning/comments/4huo36/what_is_low_dimensional_manifold