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Another great question! Interpolative generalization works as long as the manifold hypothesis applies. This works independently of whether your latent data space is entirely connected, locally connected, or fully discrete.https://twitter.com/etsiJnaJ/status/1450558699714535428 …
You only need the existence of a manifold (entirely connected) that intercepts your locally-connected areas or discrete points. For instance, predicting y = a * x ** 2 where x is an integer is a discrete problem... that is interpolative in nature.pic.twitter.com/whpII1KkMN
In such a case your learned manifold will feature lots of areas that aren't actually meaningful as per the original domain, but that could be interpreted as a continuous extension of the domain (according to the model). Like reals are to integers.
This enables you to solve seemingly-discrete problems with deep learning -- as long as they are so structured as to be interpolative, and as long as you can gather a dense sampling of the manifold. The manifold hypothesis must apply.
However, when faced with a problem where the manifold hypothesis no longer applies, fitting a differentiable curve via gradient is no longer a good strategy. This is actually *most* of the space of all interesting/useful problems! E.g. most programs that software engineers write.
Again, if this is your jam, then this is your book:https://www.manning.com/books/deep-learning-with-python-second-edition?a_aid=keras&a_bid=76564dff …
Ah, but this is incorrect! Virtually all perception problems *do* feature entirely-connected latent manifolds. Natural images are no exception. This is very easy to demonstrate, and something that cinematographers are intimately familiar with...https://twitter.com/david_picard/status/1450697372930088963 …
Take a picture of your dog in your backyard, and a picture of your coffee mug on your kitchen table. Unconnected, right? You can't morph one into the other in a continuous fashion, right? Of course you can!
There's an infinity of ways you could do it, but here's a few trivial ones.
1. The handheld camera: just keep the camera rolling as you walk from your kitchen to your backward. Boom, a smooth transition where *every single frame* is extracted from the real world.
2. Fade to black: this trick exploits the fact that changing the brightness of an image still gives you a "valid" image. Just fade one frame to black then fade in to the new frame. A bit trivial, I know
3. The shared focus: zoom in on one object shared between the two scenes (or background color/texture, e.g. the wood of the table) then zoom out into the new frame.
4. A variant of this would be to, say, turn the camera towards the sky you can see from the window, then move it down, and boom, it's your backyard
So anyways, the visual world is in fact an entirely connected space (you can go from any valid image to any other valid image via a continuous path where every frame is a valid image). This means that there exists a natural, intrinsic manifold of the visual world.
This is true for every perception problem, remarkably. The *physical world* lies on a manifold. And that's what makes deep learning so effective -- its assumptions are a good match to reality.
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