François CholletVerified account

However, if you interpolate between two digits *on the latent manifold of the MNIST data*, the mid-point between two digits still lies on the manifold of the data, i.e. it's still a plausible digit.

But wait, what does that really mean? What's a "manifold"? What does "latent" mean? How do you learn to interpolate on a latent manifold?

Ok, so, consider MNIST digits, 28x28 black & white images. You could say the "encoding space" of MNIST has 28 * 28 = 784 dimensions. But does that mean MNIST digits represent "high-dimensional" data?

Not quite! The dimensionality of the encoding space is an entirely artificial measure that only reflects how you choose to encode the data -- it has nothing to do with the intrinsic complexity of the data.

For instance, if you take a set of different numbers between 0 and 1, print them on sheets of paper, then take 2000x2000 RGB pictures of those papers, you end up with a dataset with 12M dimensions. But in reality your data is scalar, i.e. 1-dimensional.

You can always add (and usually remove) encoding dimensions without changing the data, simply by picking a different encoding scheme. That is, until you hit the "intrinsic dimensionality" of the data (past which you would be destroying information).

For MNIST, for instance, very few random grids of 28x28 pixels form a valid digit. Valid digits occupy a tiny, microscopic *subspace* within the encoding space. Like a grain of sand in a stadium. You call it the "latent space". This is broadly true for all forms of data.

Further, the valid digits aren't sprinkled at random within the encoding space. The latent space is *highly structured*. So structured, in fact, that for many problems it is *continuous*.

The fact that this property (latent space = very small subspace + continuous & structured) applies to so many problems is called the *manifold hypothesis*. This concept is central to understanding the nature of generalization in ML.

The manifold hypothesis posits that for many problems, your data samples lie on a low-dimensional manifold embedded in the original encoding space.

A "manifold" is simply a lower-dimensional subspace of some parent space that is locally similar to a linear (Euclidian) space. (i.e. it is continuous and smooth). Like a curved line within a 3D space.

Once you have such a curve, you can walk on the curve to make sense of *samples you've never seen before* (that are interpolated from samples you have seen). This is how a GAN can generate faces that weren't in the training data, or how a MNIST classifier can recognize new digits

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.

The second point -- training data density -- is equally important. You will naturally only be able to train on a very space sampling *of the encoding space*, but you need to *densely cover the latent space*.

The practical implication is that the best way to improve a deep learning model is to get more data or better data (overly noisy / inaccurate data will hurt generalization). A denser coverage of the latent manifold leads a model that generalizes better.

This is why *data augmentation techniques* like exposing a model to variations in image brightness or rotation angle is an extremely effective way to improve test-time performance. Data augmentation is all about densifying your latent space coverage (by leveraging visual priors).