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>When ML researchers speak of such a manifold, it's kind of loosey-goosey because a set that includes isolated points that don't have a continuous region around
by whatshisface 5y ago
>When ML researchers speak of such a manifold, it's kind of loosey-goosey because a set that includes isolated points that don't have a continuous region around them aren't actually manifolds.
So they mean "subset" when they say "manifold?"
- JabavuAdams 5y agoI suppose so, but they really are trying to convey that all the points lie close to some lower-dimensional crumpled surface. So a subset that was e.g. just a lattice in the high-dimensional space wouldn't fit this mental image. E.g. Generate 3d points that are on a 2d plane +- some small random offset normal to the plane. If the points are isolated, without each having a local neighbourhood, it's not technically a manifold. But, you could describe the data-set as lying on or near a plane to within some tolerance. So they're trying to describe something that is much more specific / strongly constrained than an arbitrary subset, but it doesn't meet the very stringent (and frankly idealized) requirements of a manifold. I wonder whether there is a math-object to describe this? EDIT> Maybe it's just a matter of saying "close to some lower-dimensional manifold", rather than "on a lower-dimensional manifold."
- whatshisface 5y ago>I wonder whether there is a math-object to describe this? Here is how I would phrase it: there is a probability distribution in the higher-dimensional space that expresses P(this image | given that it's a natural image). The level sets of the probability distribution are manifolds.