François CholletVerified account

If you're in a high-dimensional encoding space, this curve is, of course, a high-dimensional curve. But that's because it needs to deal with the encoding space, not because the problem is intrinsically high-dimensional (as mentioned earlier).

Now, how do you learn such a curve? That's where deep learning comes in.

I'm back, just wanted to add one important note to conclude the thread: deep learning models are basically big curves fitted via gradient, that approximate the latent manifold of a dataset. The *quality of this approximation* determines how well the model will generalize.

The ideal model literally just encodes the latent space -- it would be able to perfectly generalize to *any* new sample. An imperfect model will partially deviate from the latent space, leading to possible errors.

Being able to fit a curve that approximates the latent space relies critically on two factors: 1. The structure of the latent space itself! (a property of the data, not of your model) 2. The availability of a "sufficiently dense" sampling of the latent manifold, i.e. enough data

You *cannot* generalize in this way to a problem where the manifold hypothesis does not apply (i.e. a true discrete problem, like finding prime numbers).

In this case, there is no latent manifold to fit to, which means that your curve (i.e. deep learning model) will simply memorize the data -- interpolated points on the curve will be meaningless. Your model will be a very inefficient hashtable that embeds your discrete space.