Last time we looked at the elementary formulation of an elliptic curve as the solutions to the equation $$y^2 = x^3 + ax + b$$ where $ a,b$ are such that the discriminant is nonzero: $$-16(4a^3 + 27b^2) \neq 0$$ We have yet to explain why we want our equation in this form, and we will get to that, b...

This is a guest post by my friend and colleague Adam Lelkes. Adam’s interests are in algebra and theoretical computer science. This gem came up because Adam gave a talk on probabilistic computation in which he discussed this technique. Problem: simulate a biased coin using a fair coin. Solution: (in...

Finding solutions to systems of polynomial equations is one of the oldest and deepest problems in all of mathematics. This is broadly the domain of algebraic geometry, and mathematicians wield some of the most sophisticated and abstract tools available to attack these problems. The elliptic curve st...

This is a guest post by my friend and colleague Adam Lelkes. Adam’s interests are in algebra and theoretical computer science. This gem came up because Adam gave a talk on probabilistic computation in which he discussed this technique. Problem: Simulate a fair coin given only access to a biased coin...

With all the recent revelations of government spying and backdoors into cryptographic standards, I am starting to disagree with the argument that you should never roll your own cryptography. Of course there are massive pitfalls and very few people actually need home-brewed cryptography, but history ...

A few awesome readers have posted comments in Computing Homology to the effect of, “Your code is not quite correct!” And they’re right! Despite the almost year since that post’s publication, I haven’t bothered to test it for more complicated simplicial complexes, or even the basic edge cases! When I...

This post is intended to be a tutorial on how to access the RealityMining dataset using Python (because who likes Matlab?), and a rant on how annoying the process was to figure out. RealityMining is a dataset of smart-phone data logs from a group of about one hundred MIT students over the course of ...

A professor at Stanford once said, If you really want to impress your friends and confound your enemies, you can invoke tensor products… People run in terror from the $ \otimes$ symbol. He was explaining some aspects of multidimensional Fourier transforms, but this comment is only half in jest; peop...

In tackling machine learning (and computer science in general) we face some deep philosophical questions. Questions like, “What does it mean to learn?” and, “Can a computer learn?” and, “How do you define simplicity?” and, “Why does Occam’s Razor work? (Why do simple hypotheses do well at modelling ...

We’ve studied the Fourier transform quite a bit on this blog: with four primers and the Fast Fourier Transform algorithm under our belt, it’s about time we opened up our eyes to higher dimensions. Indeed, in the decades since Cooley & Tukey’s landmark paper, the most interesting applications of the ...

So far in this series we’ve seen two nontrivial algorithms for bandit learning in two different settings. The first was the UCB1 algorithm, which operated under the assumption that the rewards for the trials were independent and stochastic. That is, each slot machine was essentially a biased coin fl...

For a while I’ve been meaning to do some more advanced posts on optimization problems of all flavors. One technique that comes up over and over again is Lagrange multipliers, so this post is going to be a leisurely reminder of that technique. I often forget how to do these basic calculus-type things...

In the last twenty years there has been a lot of research in a subfield of machine learning called Bandit Learning. The name comes from the problem of being faced with a large sequence of slot machines (once called one-armed bandits) each with a potentially different payout scheme. The problems in t...

startups The software world is always atwitter with predictions on the next big piece of technology. And a lot of chatter focuses on what venture capitalists express interest in. As an investor, how do you pick a good company to invest in? Do you notice quirky names like “Kaggle” and “Meebo,” requir...

A lot of people who like functional programming often give the reason that the functional style is simply more elegant than the imperative style. When compelled or inspired to explain (as I did in my old post, How I Learned to Love Functional Programming), they often point to the three “higher-order...

My First Paper I’m pleased to announce that my first paper, titled “Anti-Coordination Games and Stable Colorings,” has been accepted for publication! The venue is the Symposium on Algorithmic Game Theory, which will take place in Aachen, Germany this October. A professor of mine once told me that ev...

During the 1950’s the famous mathematician Paul Erdős and Alfred Rényi put forth the concept of a random graph and in the subsequent years of study transformed the world of combinatorics. The random graph is the perfect example of a good mathematical definition: it’s simple, has surprisingly intrica...

Machine learning is broadly split into two camps, statistical learning and non-statistical learning. The latter we’ve started to get a good picture of on this blog; we approached Perceptrons, decision trees, and neural networks from a non-statistical perspective. And generally “statistical” learning...

Problem: Prove that for vectors $ v, w$ in an inner product space, the inequality $$\displaystyle |\left \langle v, w \right \rangle | \leq \| v \| \| w \|$$ Solution: There is an elementary proof of the Cauchy-Schwarz inequality (see the Wikipedia article), and this proof is essentially the same. W...

Last time we worked through some basic examples of universal properties, specifically singling out quotients, products, and coproducts. There are many many more universal properties that we will mention as we encounter them, but there is one crucial topic in category theory that we have only hinted ...

Problem: Given a data stream of unknown size $ n$, pick an entry uniformly at random. That is, each entry has a $ 1/n$ chance of being chosen. Solution: (in Python) import random def reservoirSample(stream): for k,x in enumerate(stream, start=1): if random.random() < 1.0 / k: chosen = x return chose...

Guest Post: Torus-Knotted Baklava at Baking And Math, a blog by my friend and colleague, Yen.

Problem: Determine if a number is prime, with an acceptably small error rate. Solution: (in Python) import random def decompose(n): exponentOfTwo = 0 while n % 2 == 0: n = n/2 exponentOfTwo += 1 return exponentOfTwo, n def isWitness(possibleWitness, p, exponent, remainder): possibleWitness = pow(pos...

There has been a lot of news recently on government surveillance of its citizens. The biggest two that have pervaded my news feeds are the protests in Turkey, which in particular have resulted in particular oppression of social media users, and the recent light on the US National Security Agency’s w...

I’ve been spending a little less time on my blog recently then I’d like to, but for good reason: I’ve been attending two weeks of research conferences, I’m getting ready for a summer internship in cybersecurity, and I’ve finally chosen an advisor. Visions, STOC, and CCC I’ve been taking a break from...

Last time we defined and gave some examples of rings. Recapping, a ring is a special kind of group with an additional multiplication operation that “plays nicely” with addition. The important thing to remember is that a ring is intended to remind us arithmetic with integers (though not too much: mul...

Previously in this series we’ve seen the definition of a category and a bunch of examples, basic properties of morphisms, and a first look at how to represent categories as types in ML. In this post we’ll expand these ideas and introduce the notion of a universal property. We’ll see examples from ma...

This post is mainly mathematical. We left it out of our introduction to categories for brevity, but we should lay these definitions down and some examples before continuing on to universal properties and doing more computation. The reader should feel free to skip this post and return to it later whe...

Pablo Picasso Simplicity and the Artist Some of my favorite of Pablo Picasso’s works are his line drawings. He did a number of them about animals: an owl, a camel, a butterfly, etc. This piece called “Dog” is on my wall: Dachshund-Picasso-Sketch (Jump to interactive demo where we recreate “Dog” usin...

In this post we’ll get a quick look at two ways to define a category as a type in ML. The first way will be completely trivial: we’ll just write it as a tuple of functions. The second will involve the terribly-named “functor” expression in ML, which allows one to give a bit more structure on data ty...