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Now that’s interesting.. what exactly distinguishes latent representations and the manifold? IMHO, those are the same, and you’re constructing a piecewise function of the manifold itself. Decoders also produce manifolds much in the same way, with the distinction being that the encoder isn’t learned but static after initialisation. So fundamentally it is still DOING the same operation.

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The latent representations of the data are like points on a surface. That surface is the manifold. We don't typically have the full manifold and can only sample points from it by embedding data into it.

Worth noting a different manifold "exists" after each transformation (e.g. layer). You only sample from the same manifold when you apply the same transformation(s).

Also worth noting that in reality manifolds will be "spiky" in very high dimension, so the idea of a "surface" is best understood through patterns of distance between samples in embedding space and way they collapse in low D.

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