# A Variational Autoencoder on the SVHN dataset

In this post, I'm going to share some notes on implementing a variational autoencoder (VAE) on the Street View House Numbers (SVHN) dataset. My last post on variational autoencoders showed a simple example on the MNIST dataset but because it was so simple I thought I might have missed some of the subtler points of VAEs -- boy was I right! The fact that I'm not really a computer vision guy nor a deep learning guy didn't help either. Through this exercise, I picked up some of the basics in the "craft" of computer vision/deep learning area; there are a lot of subtle points that are easy to gloss over if you're just reading someone else's tutorial. I'll share with you some of the details in the math (that I initially got wrong) and also some of the implementation notes along with a notebook that I used to train the VAE. Please check out my previous post on variational autoencoders to get some background.

Update 2017-08-09: I actually found a bug in my original code where I was only using a small subset of the data! I fixed it up in the notebooks and I've added some inline comments below to say what I've changed. For the most part, things have stayed the same but the generated images are a bit blurry because the dataset isn't so easy anymore.

# Variational Autoencoders

This post is going to talk about an incredibly interesting unsupervised learning method in machine learning called variational autoencoders. It's main claim to fame is in building generative models of complex distributions like handwritten digits, faces, and image segments among others. The really cool thing about this topic is that it has firm roots in probability but uses a function approximator (i.e. neural networks) to approximate an otherwise intractable problem. As usual, I'll try to start with some background and motivation, include a healthy does of math, and along the way try to convey some of the intuition of why it works. I've also annotated a basic example so you can see how the math relates to an actual implementation. I based much of this post on Carl Doersch's tutorial, which has a great explanation on this whole topic, so make sure you check that out too.

# Variational Bayes and The Mean-Field Approximation

This post is going to cover Variational Bayesian methods and, in particular, the most common one, the mean-field approximation. This is a topic that I've been trying to understand for a while now but didn't quite have all the background that I needed. After picking up the main ideas from variational calculus and getting more fluent in manipulating probability statements like in my EM post, this variational Bayes stuff seems a lot easier.

Variational Bayesian methods are a set of techniques to approximate posterior distributions in Bayesian Inference. If this sounds a bit terse, keep reading! I hope to provide some intuition so that the big ideas are easy to understand (which they are), but of course we can't do that well unless we have a healthy dose of mathematics. For some of the background concepts, I'll try to refer you to good sources (including my own), which I find is the main blocker to understanding this subject (admittedly, the math can sometimes be a bit cryptic too). Enjoy!

# The Calculus of Variations

This post is going to describe a specialized type of calculus called variational calculus. Analogous to the usual methods of calculus that we learn in university, this one deals with functions of functions and how to minimize or maximize them. It's used extensively in physics problems such as finding the minimum energy path a particle takes under certain conditions. As you can also imagine, it's also used in machine learning/statistics where you want to find a density that optimizes an objective 1. The explanation I'm going to use (at least for the first part) is heavily based upon Svetitsky's Notes on Functionals, which so far is the most intuitive explanation I've read. I'll try to follow Svetitsky's notes to give some intuition on how we arrive at variational calculus from regular calculus with a bunch of examples along the way. Eventually we'll get to an application that relates back to probability. I think with the right intuition and explanation, it's actually not too difficult, enjoy!

# Maximum Entropy Distributions

This post will talk about a method to find the probability distribution that best fits your given state of knowledge. Using the principle of maximum entropy and some testable information (e.g. the mean), you can find the distribution that makes the fewest assumptions about your data (the one with maximal information entropy). As you may have guessed, this is used often in Bayesian inference to determine prior distributions and also (at least implicitly) in natural language processing applications with maximum entropy (MaxEnt) classifiers (i.e. a multinomial logistic regression). As usual, I'll go through some intuition, some math, and some examples. Hope you find this topic as interesting as I do!