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*.
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Now, how do you learn such a curve? That's where deep learning comes in.
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But by this point this thread is LONG and the Keras team sync starts in 30s, so I refer you to DLwP, chapter 5 for how DL models and gradient descent are an awesome way to achieve generalization via interpolation on the latent manifold.https://www.manning.com/books/deep-learning-with-python-second-edition?a_aid=keras&a_bid=76564dff …
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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.
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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.
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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
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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).
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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.
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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*.
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It's only with a sufficiently dense sampling of the latent manifold that it becomes possible to make sense of new inputs by interpolating between past training inputs without having to leverage additional priors.pic.twitter.com/SmRvEN2NXS
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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.
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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).
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In conclusion: the only things you'll find in a DL model is what you put into it: the priors encoded in its architecture and the data it was trained on. DL models are not magic. They're big curves that fit their training samples, with some constraints on their structure.
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End of conversation
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