Two years after a friend gave me a copy, I finally got around to reading Flann O’Brien’s ‘The Third Policeman.’ I liked the part with the bicycles. (Which is to say, pretty much the whole book.) Lots of spoilers below, if you’re afraid of that sort of thing, for a book written around 1940 and published in 1966.
Google unveiled it’s newest iteration of the driverless car, this time without a steering wheel or brake pedals. It looks a bit like a smart car crossed with a Beetle and a koala, but I’m still pretty excited about it.
I have never bothered to get a driver’s license, and it seems now like I was a (relatively) early-adopter of car-free life instead of a life-long freak: the proportion of young people with licenses has been steadily declining over the last ten years. When I was 16, I looked around and saw lots of teenagers getting crappy cars to drive to crappy jobs to pay for their crappy cars, and decided that I didn’t really want to be involved in that cycle. Spending 15-30 hours working at Dairy Queen didn’t seem like a terribly valuable experience.
Back in late December, I planted a Raspberry Pi camera at a cottage on Georgian Bay, in Northern Ontario, set to take a picture once every two minutes. I had been planning the shoot for a couple months prior to the deployment: There were two Raspberry Pi’s involved, in case one failed somewhere during the winter. One of the Pi’s was set to reboot once a week, just in case the software crashed but the Pi was still awake. I had also written some software for the time-lapse to ensure that pictures were only taken during the day time, and to try to maintain a balance of well-lit, consistent images over the course of each day.
In spite of all the planning, I had a sense that something would go horribly wrong, and, indeed, when we showed up to the cottage, the windows were completely frosted over. The cameras had to be placed inside, so we figured we would mainly see the back-side of an icy window when we retrieved the cameras. Or that the camera boards would literally freeze after about a week of sub-zero temperatures in the unheated cottage. Or that a raccoon would find it way in and gnaw through the shiny Lego cases. Or something else entirely unplanned for.
So it was a bit of a surprise when it turned out that the shoot went perfectly. We retrieved the cameras about a week ago, on May 7th, and found over 42,000 photos waiting for us on one of the cameras and somewhat fewer on the other. Both cameras had survived the winter just fine!
All told, I think the result was really cool! The video at the top is the ‘highlights’ reel, with all of the best days. It comes to 13 minutes at 18 frames per second. Turns out it was a fantastic winter for doing a time-lapse, with lots of snow storms and ice. There’s even the occasional bit of wildlife, if you watch closely. I’ll post the full 40-minute time-lapse on Youtube sometime next week.
The talk was about using Fourier transforms to get polynomial-time encodings of permutation statistics, as well as a look at the Kondor-Borgwardt approach to graph invariants via the Fourier transform over . The talk was given at a representation theory conference, and I was making the point that we can get new research ideas by taking trips into the world of applications – in my case, by looking at machine learning problems. The opening joke was that I asked my computer for the best possible title for the talk, and received the click-bait title as a response. It was admittedly a pretty funny moment watching the chair of the session trying to decide whether to read the title originally submitted for the talk (‘Compressed Combinatorial Statistics’) or the ridiculousness on the screen (he went with the original).
I attended a really nice talk by Arash Amini yesterday about detecting ‘communities’ in sparse graphs. The basic problem is: In a big graph (like the Facebook graph, or the graph of scientific papers citations) you have clusters of friends/associates, and you want to pick out those clusters. Dr. Amini and his collaborators have been working on methods to solve this problem in particularly noisy cases. The methods presented were developed for cases where you know the specific number of clusters you are seeking to find. It was a fascinating talk, and I came away wondering if there are good methods to use when you aren’t sure of how many clusters there are.
I’ve finally had a bit of time to finish up the code for the Rock-Paper-Scissors bot discussed in the previous posts; I’ve put the code in a GitHub repository here. Feel free to file issues if you would like it to be a bit more user friendly.
The bot which uses the Fourier transform on move probabilities to search for profitable patterns is named `diaconis`, after Persi Diaconis. It’s currently working just fine, but is a bit slow computationally. On startup, it generates all possible move patterns that it will investigate during the course of each game, as well as some character tables. After each play, it tests some of these patterns, and tries to determine if their predictive power is better than any of the patterns seen thus far. If so, it begins using that pattern to choose move probabilities.
This works fine as a proof-of-concept of the basic ideas. Additional improvements could be had by doing some code optimization to speed things up a bit, and keeping a list of good patterns and allowing a bit more dexterity in switching between the patterns used for prediction.
I’m teaching an intro calculus class this year (specifically, ‘Math for Life and Social Science’), and came a while ago to the section on optimization. It’s a really important subject, and yet the optimization problems one finds in Calculus books (even good ones) tend to be contrived examples which I refer to as ‘box factory problems.’ Things along the lines of ‘minimize the surface area of a rectangular box with volume ‘.
These are fine for a problem or two: There’s a useful skill in taking a real-sounding problem and translating it into the mathematics you’re learning. We use the constraints (in this case, on the volume) to reduce the number of dimensions, turn the problem into a one-variable calculus problem, and then solve. All well and good, but these problems somehow completely miss the impact of optimization on society at large. Largely because the optimization problems that occur most commonly in the wild have a slightly different flavour.
Problem: In Boston, we observe that the monthly rent for three one-bedroom apartments are $1300, $1150, and $950. Rent on three two-bedroom apartments were $1500, $1700, and $1200. Assuming that the cost of a 0-bedroom apartment is $500, find the best possible line describing the rent as a function of the number of bedrooms. Continue reading
In the last two posts, we’ve looked at using machine learning for playing iterated Roshambo. Specifically, we saw how to use Bayes’ theorem to try to detect and exploit patterns, and then saw how Fourier transforms can give us a concrete measurement of the randomness (and non-randomness) in our opponent’s play. Today’s post is about how we can use representation theory to improve our chances of finding interesting patterns.
In the last post, we looked at using an algorithm suggested by Bayes’ Theorem to learn patterns in an opponent’s play and exploit them. The game we’re playing is iterated rock-paper-scissors, with 1000 rounds of play per game. The opponent’s moves are a string of choices, ‘r’, ‘p’, or ‘s’, and if we can predict what they will play, we’ll be able to beat them. In trying to discover patterns automatically we’ll gain some general knowledge about detecting patterns in streams of characters, which has interesting applications ranging from biology (imagine ‘GATC’ instead of ‘rps’) to cryptography.
Fourier analysis is helpful in a wide variety of domains, ranging from music to image encoding. A great example suggested by ‘Building Machine Learning Algorithms with Python‘ is classifying pieces of music by genre. If we’re given a wave-form of a piece of music, automatically detecting its genre is difficult. But applying the Fourier transform breaks the music up into its component frequencies, which turn out to be quite useful in determining whether a song is (say) classical or metal.
I’ve recently been doing some reading on machine learning with a mind towards applying some of my prior knowledge of representation theory. The initial realization that representation theory might have some interesting applications in machine learning came from discussions with Chris Olah at the Toronto HackLab a couple months ago; you can get interesting new insights by exploring new spaces! Over winter break I’ve been reading Bishop’s ‘Pattern Recognition and Machine Learning‘ (slowly), alongside faster reads like ‘Building Machine Learning Systems with Python.‘ As I’ve read, I’ve realized that there is plenty of room for introducing group theory into machine learning in interesting ways. (Note: This is the first of a few posts on this topic.)
There’s a strong tradition of statisticians using group theory, perhaps most famously Persi Diaconis, who used representation theory of the symmetric group to find the mixing time for card shuffling. His notes ‘Group Representations in Probability and Statistics‘ are an excellent place to pick up the background material with a strong eye towards applications. Over the next few posts I’ll make a case for the use of representation theory in machine learning, emphasizing automatic factor selection and Bayesian methods.
First, an extremely brief overview of what machine learning is about, and an introduction to using a Bayesian approach to play RoShamBo, or rock paper scissors. In the second post, I’ll motivate the viewpoint of representation by exploring the Fourier transform and how to use it to beat repetitive opponents. Finally, in the third post I’ll look at how we can use representations to select factors for our Bayesian algorithm by examining likelihood functions as functions on a group.