December 2013

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So I have been wanting to understand Category Theory (see [1], [2], [3]) mainly because I thought it would help me understand advanced functional programming in Haskell and because of the book “Physics, topology, logic and computation: a Rosetta Stone” by Baez and Stay (2011).  A number of the constructs in Haskell were derived from Category Theory so that was enough motivation for me to learn it.  But also, monoidal categories are useful in several fields: physics, computer science, pure mathematics, and statistics. The biggest application for me is the idea of generalizing probabilistic graphical models through either Ologs or monoidal categories / tensor networks (see Brendan Fong’s thesis “Causal Theories: A Categorical Perspective on Bayesian Networks“).

To start my investigation of Category Theory, I began with the $20 thin book “Basic Category Theory for Computer Scientists” by Benjamin Pierce (see also the free online version “A taste of category theory for computer scientists” (1988)).

I really enjoyed the book.  Dr. Pierce’s style is a little informal compared to pure math books like Mac Lane’s “Categories for the Working Mathematician” (pdf / Amazon), but I enjoy that more relaxed style of writing when I am first learning a field.

The biggest obstacle for learning category theory is the fact that category theory generalizes a lot of areas of pure mathematics like topology, abstract algebra, and geometry.  It’s hard to generalize before you have examples to generalize, but the examples being generalized in category theory are mostly from higher level mathematics found in senior level undergraduate and graduate level courses. Pierce ameliorates this problem by introducing some of the most basic categories first: sets, ordered sets, partially ordered sets, groups, monoids, vector spaces, measure spaces, topological spaces, proofs, and a simple functional computer language.  He takes the time to explicitly define most of these ideas, so, in theory, you could read this book without a background in theoretical mathematics, but it would be hard.

After defining categories and introducing the most basic categories, Pierce describes and defines the most basic ideas in category theory: subcategories, commutative diagrams, monomorphisms, epimorphisms, isomorphisms, initial/terminal objects, products, coproducts, universal constructions, equalizers, pullbacks, pushouts, limits, cones, colimits, cocones, exponentiation, and closed Cartesian categories. These ideas are spelled out over the thirty pages of chapter one including illuminating homework exercises. The homework exercises varied significantly in difficulty.  Many of the exercises were trivial and there are two or three that I am still working on despite investing several hours of thought.  Generally, I found the exercises to be a bit harder than those in Mac Lane’s book, but Pierce’s book required less of a background in mathematics.  A couple of the exercises were incorrectly stated or impossible.

Chapter two introduced functors, natural transformations, adjoints, and F-algebras. After reading this chapter, I was finally able to understand the definition of monads which are an important part of the computer language Haskell!  Pierce provides many examples of each of these ideas and enjoyable homework exercises to increase understanding.  Pierce’s definition of adjoints is much easier to understand than the standard definitions using counit adjunction or Hom Sets.

The last major chapter concerns applications of category theory to computer science–specifically lambda-calculus and programming language design.  

The first two chapters of the book give a reasonable condensed introduction to category theory for students that have taken a course in abstract algebra.  A course in topology or linear algebra would be another useful prerequisite.  I carried around the light 100 page book for a few months so that I could learn something whenever I had some extra time.  I had hoped that when I had proven that several functors where monads, I would then really understand monads, but a full understanding still eludes me.  Similarly, I have proven that several functor pairs are adjoint, but I still don’t feel as though I understand adjoint functors.  I guess more work and learning are needed.

Have a great year.  –  Cheers, Hein

 

Check out Markus Beissinger’s blog post “Deep Learning 101″.  Markus reviews a lot of deep learning basics derived from the papers “Representation Learning: A Review and New Perspectives” (Bengio, Courville, Vincen 2012) and “Deep Learning of Representations: Looking Forward” (Bengio 2013). Beissinger covers the following topics:

  • An easy intro to Deep Learing
  • The Current State of Deep Learing
  • Probabilistic Graphical Models
  • Principal Component Analysis
  • Restricted Boltzman Machines
  • Auto-Encoders
  • “Challenges Looking Ahead”

This is a great intro and I highly recommend it.

If you want more information, check out Ng’s lecture notesHonglak Lee’s 2010 NIPS slides, and Hinton’s Videos ([2009] [2013]).

Zygmunt Zając at fastml.com has a great post titled “13 NIPS Papers that caught our eye.” Zajac provides a short readable summary of his favorite NIPS 2013 papers.  (NIPS 2013 which just ended last week.) The papers are:

  1. Understanding Dropout  by Baldi and Sadowski
  2. Training and Analysing Deep Recurrent Neural Networks by Hermans and Schrauwen
  3. RNADE: The real-valued neural autoregressive density-estimator by Uria, Murray, and Larochelle
  4. Predicting Parameters in Deep Learning by Denil, Shakibi, Dinh,  Ranzato, and Freitas
  5. Pass-efficient unsupervised feature selection by Maung and Schweitzer
  6. Multi-Prediction Deep Boltzmann Machines by Goodfellow, Mirza, Courville, and Bengio
  7. Memoized Online Variational Inference for Dirichlet Process Mixture Models by Hughes, and Sudderth
  8. Learning word embeddings efficiently with noise-contrastive estimation by Andriy , and Kavukcuoglu
  9. Learning Stochastic Feedforward Neural Networks by Tang and Salakhutdinov
  10. Distributed Representations of Words and Phrases and their Compositionality by Mikolov, Sutskever, Chen, Corrado, and Dean
  11. Correlated random features for fast semi-supervised learning by McWilliams, Balduzzi, and Buhmann
  12. Convex Two-Layer Modeling by Aslan, Cheng, Zhang, and Schuurmans, and
  13. Approximate inference in latent Gaussian-Markov models from continuous time observations by Cseke, Opper, and Sanguinetti

I suggest reading Zajac’s summaries before diving in.

Reddit has this pretty cool subreddit ELI5 = “Explain like I’m 5″ where people try to explain topics as if they are talking to a five year old.  It’s an interesting exercise to see how much of a difficult topic like nuclear energy or solar physics you could explain to a five year old.  Here is my attempt to explain Entropy and Mutual Information to a bright five year old that isn’t afraid of big words or long sentences.

(This post is going to be amazingly long and very simplistic, so I suggest you don’t bother reading it unless you think it might be fun to consider how to explain a complex concept like mutual information to a five year old.)

 

Random Variables

In order to explain (information theoretic) entropy we need to talk about random variables.  A random variable is a number that you get from a repeatable experiment.  For example, if you roll a die, the outcome is 1, 2, 3, 4, 5, or 6.  We say that the result of each die roll is a random variable.

 

Entropy

So random variables have something called entropy.  In order to talk about entropy it’s helpful to first talk about the doubling numbers.  The doubling numbers are 1, 2, 4, 8, 16, ….  I call them doubling numbers because 1+1 = 2, 2+2 =4, 4+4 =8, and so on.  (The official name of doubling numbers is “the powers of two” but let’s just call them doubling numbers for now.)

 

BITS

If a random variable has two equally likely outcomes, we say that the variable has an entropy of 1 bit.  If it has four equally likely outcomes, then the variable has 2 bits, eight 3 bits, and 16 outcome gives 4 bits.  Of course, some random variables have like 6 outcomes and that is not a doubling number, so we could say that it has about two and a half bits because 6 outcomes is half way between 4 outcomes which is 2 bits and 8 outcomes which is 3 bits.

How did we get the name bits?  Well, in a computer bits are little electronic parts that are either on or off.  We use the code 0 for off and 1 for on.  (This code is called the binary code.)  We could represent the result of an experiment (a random variable) with these bits.  Suppose that our experiment is flipping a coin and we call tails 1 and heads 0.  Then we would need one bit to record the results of each coin flip.

 

Three bits for an eight sided DIE

The number of bits needed to record the roll of an eight sided die would be larger. It can be done with three bits as follows.   If you roll a 1, set the bits to off-off-on (001).  If you roll a two, set the bits to 010.  A 3 set them to 011, 4 ->  100, 5 -> 101, 6 -> 110, 7 -> 111, and 8 -> 000.  Notice that we were able to represent every possible outcome with only 3 bits.  That’s why we say the entropy of an eight sided die roll is three bits.

 

A multi-stage experiment

More complicated experiments would require some more difficult computations. Sometimes, the outcomes are not all equally likely.  Suppose our experiment has two stages.  In stage one we flip a nickel.  If the result is tails, we quit.  If the result is heads, we flip a quarter.  There are three possible results for this experiment:

  1. The nickel is tails
  2. The nickel is heads and the quarter is tails.
  3. The nickel is heads and the quarter is heads.

The nickel is tails about half the time and heads half the time.  So we only have to record the quarter half of the time.  The average number of bits needed to record this experiment is one and a half.  One bit for the nickel and one half bit for the quarter because we only need to write down the quarter flip outcome (1 bit) half of the time.

 

MUTUAL INFORMATiON

Sometimes random variables share bits.  When that happens, we say that they have mutual information. Most random variables do not share information. In that case, we say the variables are independent and the mutual information (shared information) is zero.

 

A COMPLICATED EXAMPLE

Suppose that Harry and Sally are doing an experiment.  They flip a penny, a nickel, a dime, and a quarter.  After the experiment, Harry writes down the results of the penny flip and the nickel flip.  Sally writes down the results of the nickel, the dime, and the quarter.  There are four possible outcomes for Harry: Head-Head, Head-Tail, Tail-Head, and Tail-Tail.  We say that Harry’s recording uses 2 bits, so his entropy is 2 bits.  Sally could have 8 different results, but she would only need 3 bits, one for each coin, to record her results, so her entropy is 3 bits.  The mutual information between Harry and Sally is the nickel flip which is just 1 bit (1 for heads, 0 for tails).  The overall entropy of Harry and Sally together is 4 bits because there are four coins.  The Joint entropy of Harry and Sally together is the number of bits needed to record both Harry’s result and Sally’s result.  Harry uses 2 bits and Sally 3 bits, but only 4 bits are needed because the nickel is shared.

 

THE FORMULA

There is a general formula for the mutual information between Harry and Sally.

Mutual Information = Harry’s Entropy + Sally’s Entropy – The Joint Entropy.

For this experiment the mutual information is  2 + 3 – 4  which is 1 bit.

 

 End Note

Well, that’s it.  Maybe I gave an explanation appropriate for twelve year olds. Maybe not.  I had my eighth grade son read it and then I gave him a 6 question quiz. After I corrected two of his answers, he was able to calculate the mutual information of two, fairly simple random variables.