This document provides an overview of autoencoders and variational autoencoders. It discusses how principal component analysis (PCA) is related to linear autoencoders and can be performed using backpropagation. Deep and nonlinear autoencoders are also covered. The document then introduces variational autoencoders, which combine variational inference with autoencoders to allow for probabilistic latent space modeling. It explains how variational autoencoders are trained using backpropagation through reparameterization to maximize the evidence lower bound.

1. Autoencoders
NN club, March 21 2018
Jonathan Ronen

2. Agenda
● PCA and linear autoencoders
● Deep and nonlinear autoencoders
● Variational autoencoders

3. PCA for dimensionality reduction

4. PCA for dimensionality reduction
● The U that maximizes the variance of PC1
● also minimizes the reconstruction error
○ Note: this is not the same as OLS,
which minimizes
There are efficient solvers for this, but we could also use backpropagation

5. PCA through backpropagation
●
● This is an autoencoder
● If the neurons are linear, it is similar to PCA
○ Caveat: PCs are orthogonal, autoencoded
components are not - but they will span the
same space

6. PCA vs linear autoencoders for MNIST

7. PCA vs linear autoencoders for MNIST

8. Autoencoders can be nonlinear

9. Nonlinear autoencoder with 32 hidden neurons

10. Autoencoders can be deep
ReLu
ReLu
ReLu
ReLu
ReLu
ReLu
ReLu
ReLu
ReLu
ReLu
Sigmoid
Sigmoid
Sigmoid
Sigmoid
Sigmoid

11. Deep autoencoder (bottleneck of 2)
Guess which one is deep (has intermediate layer)?

12. Many variations of autoencoders
● Sparse autoencoders
● Denoising autoencoders
● Convolutional autoencoders
○ UNet is a sort of autoencoder
● And more…
● I’d like to introduce Variational Autoencoders

13. Variational autoencoders
Variational Bayesian Inference
Variational Inference
+
autoencoders
z
x observation
latent variable

14. Variational Inference (quick overview)
z
x observation
latent variable

15. Variational Inference (quick overview)
z
x observation
latent variable
problematic...

16. Variational Inference (quick overview)
z
x observation
latent variable
problematic...
Variational Inference Solution:
Chosen to be a
distribution we can work
with

17. Side note on
● Information
○ “How many bits do we need to represent event x if we optimized for p(x)?”
● Entropy
○ “What is the expected amount of information in each event drawn from p(x)?” (how many bits?)
● Cross-entropy
○ “What is the expected amount of information in p(x) if we optimized for q(x)?” (how many bits?)
● Kullback-Leibler divergence: “cross-entropy - entropy”
○ “How many more bits will we need to represent events from p(x) if we optimized for q(x)?

18. Variational Inference (quick overview)
skipping the math...
Maximizing the Evidence LOwer Bound (ELBO)

19. Variational inference is methods to maximize ELBO
How does it fit in with autoencoders?

20. What if autoencoders were probabilistic?

21. What if autoencoders were probabilistic?
Regular autoencoder
Variational autoencoder

22. Variational Autoencoder loss - negative ELBO
reconstruction error divergence from prior

23. Backpropagation through VAEs
sampling

24. Backpropagation through VAEs - reparameterizing

25. VAE 2d embedding

26. VAEs are a generative model

27. Regular autoencoder as a generative model?

28. Jupyter Notebook with all analysis in this talk
https://nbviewer.jupyter.org/gist/jonathanronen/69902c1a97149ab4aae42e099d1d1367

29. Further reading
● https://arxiv.org/abs/1312.6114
● https://www.youtube.com/watch?v=uaaqyVS9-rM
● https://www.jeremyjordan.me/variational-autoencoders/
● https://blog.keras.io/building-autoencoders-in-keras.html