Variational Auto Encoder and the Math Behind
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The Math and internal workings of Variational Auto Encoders. It talks about Variational Inference and how VAEs take advantage of them to bypass having to calculate Marginal Probability distribution.
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- 2. Generative Models ? They take in data as input and learn to generate new data points from the same data distribution. ? They learn the hidden representations using unsupervised learning techniques
- 3. Variational Autoencoder ? As the name suggests, it is an auto-encoder, which learns attributes from the data points and represents them in terms of latent variables.
- 4. Problems ? How can we make use of the auto-encoder architecture to generate new data points ? ? Assuming we can pass a vector from that learnt latent space to the decoder, how can guarantee it¡¯s not going to result in a garbage output. ? VAEs address the above problems
- 5. 1. How do we use the Auto Encoder Architecture for generation ? So, if we train the Auto encoder network and somehow learn the data distribution of the latent space, we can then sample from that latent space and pass it down to decoder and generate data points But, there is a problem.
- 7. 2. How can we make sure that if we sample from our latent space, we are going to get new and meaningful output ? ? VAEs achieve this by constricting the latent space. ? The encoder and decoder parameters are tuned to accommodate for this setup
- 8. But,
- 9. Calculating Marginal Probability ? If X = (x1, x2, x3) and Z = (z1, z2) then, ? ? ? = ?(?|?)??(?) ?(?) Here, the P(X) is very difficult to calculate, especially in higher dimensions. It takes the form of ?1 ?2 ? ?1, ?2, ?3, ?1, ?2 ??1 ? ??2 and is intractable. There are ways to solving this by using, 1. Using Monte Carlo Integration techniques 2. Variational Inference
- 11. As x is already given, log p(x) is a constant. And KL(q(z) || p(z|x)) is what we wanted to minimize and it is always >= 0 0 <= p(x) <= 1 and KL >= 0 So, it is equivalent to maximizing the 3rd term. It is called Variational Lower bound
- 12. This is nothing but, Expectation of p(x|z) w.r.t q(z|x) + KL(q(z) || p(z|x)) So, Maximizing lower bound means, Minimizing KL (as >=0), and For a given q and z, maximize the the likelihood of observing the x
- 18. VAE Architecture
- 20. References ? Lecture by Ali Ghodsi https://www.youtube.com/watch?v=uaaqyVS9-rM ? Lecture by Pascal Poupart https://www.youtube.com/watch?v=DWVlEw0D3gA
- 21. Hierarchy of Generative Models Figure from Ian Goodfellow¡¯s tutorial on GANs, NIPS 2016
- 22. Internals of a VAE¡¯s learning algorithm KL Divergence