To explain the notion of a manifold, Izenman (2008, page 615) begins with a story:"Imagine an ant at a picnic, where there are all sorts of items from cups to doughnuts. The ant crawls all over the picnic items, but because of its small size, the ant sees everything on a very small scale as flat and featureless. A manifold . . . can be thought of in similar terms, as a topological space that locally looks flat and featureless."What looks like two dimensions, could actually be a world of three or more dimensions.

Consider the famous data problem referred to as the Swiss roll, also illustrated in Izenman (2008, page 615). Linear dimension reduction techniques fail us as a way of understanding or visualizing the Swiss roll. But t-SNE may not be the right tool either, as documented at https://jlmelville.github.io/smallvis/swisssne.html Links to an external site. What about autoencoders?

What do you think about these kinds of problems? As practicing data scientists, should we care about manifolds? Should we can about nonlinear dimension reduction?

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