Thoughts on Machine Learning, Representation Theory, and RoShamBo – Part I

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.

RPS25 is a generalized form of Rock Paper Scissors.  I'll leave generalizations of this post to the RPS25 case as an exercise to the reader...
RPS25 is a generalized form of Rock Paper Scissors. I’ll leave generalizations of this post to the RPS25 case as an exercise to the reader…

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Know Your Sensor

Recently I’ve been playing with building a regression model for the brightness of images produced with the Raspberry Pi’s camera board. Essentially, I want to quickly figure out – hopefully from a single image – what shutter speed and ISO to choose to get an image of a given brightness.

This is a pretty standard regression problem: We some data, extract some information from it, and use that information to make a prediction. To get a better handle on the algorithms involved, I wrote my own code to perform the regression, using NumPy for fast linear algebra operations. You always learn something from re-inventing the wheel, after all.

Data from images of my back yard.  Each point is one image.  The y-axis is the brightness of the image, the x-axis is the ISO the image was taken at, and the color indicates the shutter speed the image was taken with.
Data from images of my back yard. Each point is one image. The y-axis is the brightness of the image, the x-axis is the ISO the image was taken at, and the color indicates the shutter speed the image was taken with.  The shutter speeds and ISO’s chosen were evenly spaced (except for a few rogue images).  We can see there’s a kind of ‘sweet area’ for SS/ISO, with some upper threshold where things get too bright/dark very quickly.  We also see that the increase in brightness as ISO increases is non-linear.

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Pi-Lapse 3: Toronto Skyline

A still from a test-reel of timelapse taken at Owen Sound, in North-ish Ontario last week.
A still from a test-reel of timelapse taken at Owen Sound, in North-ish Ontario last week.

I’ve done quite a bit to improve my Raspberry Pi-based timelapser.  I have a pretty good system together now, thanks in part to recent firmware updates to the Pi which allow me to directly control the shutter speed and ISO of the camera sensor instead of relying on auto-exposure settings.  The auto-exposure is pretty unreliable from one shot to the next: the camera makes a different decision each time it snaps a picture, which leads to quite a lot of flicker.  Previously, I was dealing with this flicker by manipulating the images in post-production, but I’ve now written some code to get the camera to try to maintain a constant image brightness across a long shoot.

The code now consists of a ‘timelapser’ class, which keeps track of its current shutterspeed and ISO (SS/ISO henceforth), and the brightness of the last few images taken.  It then adjusts SS/ISO to try to get the image brightness to 100.  By keeping track of the last few images, it is a bit less susceptible to being upset by one strange image (like, say, if I put my hand over the lens for one shot, producing a black image), or more standard movement within the frame.  On the other hand, it takes a while longer to settle down to the ‘right’ SS/ISO.  So it’s currently set up with an initialization step, where it finds a good SS/ISO pretty quickly, and then transitions to actually taking pictures.  The result is very little flicker as the timelapse goes on, and a pretty constant level of image brightness when light levels gradually change: like when we watch dawn or dusk.  (If you’re interested in playing around with the code, I’ve set up a github repository here.)

As an example, this is a video that we shot over about three days on my friends Ketan and Ananya’s balcony.  They have a great view over Toronto, from the CN Tower to Honest Ed’s.

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Pi-Lapse 2: Wychwood Barns Farmer’s Market

Aerial raspberry pi installation. The boxy-head is the pi(s), attached to a tripod held up by a jib arm.  A ten-port powered usb hub is duct taped to the tripod, providing lots of data storage and internet to the Pi's.
Aerial raspberry pi installation. The boxy-head is the pi(s), attached to a tripod held up by a jib arm. A ten-port powered usb hub is duct taped to the tripod, providing lots of data storage and internet to the Pi’s.

Almost immediately after my first foray with the Raspberry Pi timelapse I was contacted by Cookie Roscoe, who coordinates a weekly farmer’s market for The Stop.  The Stop is a community food center here in Toronto, providing lots of programming and emergency food access for lower-income people.  My partner, Elizabeth, worked with them for about three years, so I was super happy to try doing a time-lapse of their market.

Here’s the results!  Lots of technical stuff about what went into the video after the break!

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Pi-Lapse Test Footage

A camera made of legos!  I've built two of these; the photo was taken with the _other_ raspberyy pi camera.
A camera made of legos! I’ve built two of these; this photo was taken with the other raspberry pi camera, during a three-week time-lapse shot.

My recent DIY electronics project has been putting together a Raspberry Pi-based camera.  The Pi foundation sells a camera board which plugs into the Pi; it’s sold as ‘a bit better than the average camera in a mobile phone.’ But the Pi’s default Raspbian Linux installation comes with a couple programs for controlling the camera, and lets you take still pictures and videos easily.

In the interest of using the camera for something you can’t usually do with a store-bought digital camera, I wrote a short python script which takes a photo assigns it a file name based on the date and time taken.  It then does some sampling of the picture, and only keeps pictures which aren’t ‘too dark.’  And then cron runs the photo script once every five minutes.  In other words, the Pi is set up for long-form time-lapse photography.  The resulting pictures are then easy to compile into a movie.

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An Evening at Hacklab.to

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A 3D printer at HackLab printing a model of a jet engine turbine.

On Tuesday evening I finally made it to the Toronto Hacklab, after meaning to make it over for weeks!

It’s a really interesting space.  There are currently five 3D printers, lots of tools for playing with electronics, and a giant computer-driven laser in the bathroom for etching and cutting plastic or acrylic.  (Actually, I should try to make an acrylic LakeHub logo to send back to Kenya….)  According to Eric, who gave me a tour of the space, about a two thirds of the members come through to work on these sorts of projects, and a third are mainly software people who use it as a common work space.

A couple of my favorite toys that I saw on the tour were a giant LED pixel board and a ‘flip dot board’, both of which had been salvaged from the Toronto Transit Commission, which runs all the buses and subways.  Members of the Hacklab built electronic interfaces to both of the devices: the LED board is run by an arduion and can be sent messages to display, and otherwise acts a clock.  The flip-dot board looks like it’s run by some custom microcontrollers, and is hooked up to a joystick for playing ‘snake.’

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LakeHub is Hiring!

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The awesome LakeHub logo,
designed by Allan from All Day Tuesday

I’m happy to say that LakeHub will soon be hiring its first employee!

During my early involvement in Kenya, I had the pleasure to chat with Prof. Haynes Miller, an MIT mathematician, about the potential for e-learning in Africa. He related the story of a relative who had gone to work in a small developing country, creating an interesting technological component in a library. After working for a year to get everything set up, the project leader went back to North America. Two years later, the leader went back to visit and found the library in terrible disrepair. Computers busted, the e-library unusable. The lesson here is that a successful project needs to have a good mix of leadership and capacity on the ground to continue after the mzungu founders have departed. A good project needs to have long-term participants involved in the design and deployment, and respond directly to the needs and capacities of the local community. And as one unravels these ideas, the whole notion of the ‘mzungu founders’ is itself quite problematic; our success is really measured by our ability to promote quality leadership, build capacity, and facilitate the vision of people who are committed to live and work in Africa for the long term.

All of which is to say that I think LakeHub hits a number of the right boxes. The leadership is almost entirely Kenyan, and the project is about harnessing and amplifying the abilities of the tech community in Western Kenya. I’m extremely excited to see it continue and thrive now that I’ve wandered back to North America. The Kenyans who are taking the lead on the project so fa are fantastic, and have the skills and vision to help take it forward.  These are people like James Odede, Google’s student ambassador to Maseno University and head of the Maseno ICT guild, and Simeon Obwego, who works for Innovations in Poverty Action, and is a truly fantastic self-starter.  Evan Green-Lowe, also from IPA, has also been a great contributor, and is largely responsible for the current push to get a full-time employee.

But we’ve run into the limits of volunteer organizing: there’s a lot of vital-but-time-consuming work that needs to get done. Like scouting for potential spaces, building partnerships with local businesses and community organizations, and, yeah, locking in funding sources. The plan is to search the budding Kenyan tech community for someone with a great combination of drive and vision to help build LakeHub up to meet its fantastic potential.

Scenes From a Mall

The Art Caffe at Junction, where I drank many Americanos and wrote many blog entries.
The Art Caffe at Junction, where I drank many Americanos and wrote many blog entries. (Image Source)

I’ve never been a hang-out-at-the-mall kind of person.  The prices are higher than things I can find in the internets or strange alleys, the food courts unappealing, and the general level of “shiny” doesn’t really match my self-concept.

And yet, during my year in Kenya, I spent a lot of time at various malls.  When staying in Nairobi, I would find myself in a mall every other day, it seemed.  And it makes sense in retrospect: The specialist shops and on-line stores where I usually use to get my obscure electronics aren’t available in Kenya, and food safety seems a bit more reliable in the malls.  The biggest factor, though, are the coffee shops.  I spend a lot of time in coffee shops wherever I travel, unwinding thoughts and sketching proofs and sipping on a simple black Americano.  And most of the coffee shops in Kenya are in malls.

The mall I spent the most time at in Nairobi was The Junction, and most of that time was spent at Art Caffe.  It’s a space with a great slice of the mix of modern Nairobi.  Yeah, there are a lot of jackass travellers like myself plugged into their laptops, but just as many people stopping through for a business lunch, mommas chatting about their kids, young people hanging out with friends, and men from unidentifiable Eastern European countries discussing possibly shady business in an unidentifiable language.  It’s a great cross-section.

It’s also a bit of an escape.

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Third Maseno Math Camp

3rd Maseno Maths Camp group photo (silly version).
3rd Maseno Maths Camp group photo (silly version).

The third annual Maseno math camp ran in the third week of August; it’s a bit hard to believe that we’re up to three already!

The week started with a great talk from Rejoyce Gavhi, a South African mathematician who just finished a postdoc in Canada, and who is now starting a new job with AIMS:Sec.  She talked about the challenges she overcame in pursuing mathematics as a woman from Africa, and was quite inspirational for everyone involved.

As usual, we divided this year’s camp up into five ‘themes.’  The themes this year were programming, modelling, geometry, combinatorics, and code-breaking.  I mainly helped put together the combinatorics section with Ingrid Mostert (from AIMS:Sec) and Santiago Borio, a Geogebra virtuoso who teaches school in London; the sessions were about the bijection between subsets and lattice paths, and seeing the binomial coefficients from different perspectives.  Chris Clarke put together some great sessions in the modelling section (for example, using massively multi-player dice-games to model the spread of a disease in a population).  The programming section focused on building flow charts to describe algorithms, which was a pretty different tack than we’d previously considered, and I think a good one.  I never really think of flow charts when I program, but breaking a process down into some ‘decision points’ and considering all possible outcomes is quite useful as a programmer.  Approaching the process via flow charts is a great way to organize that process in a visual way.

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Back in Canada with a new domain

Return to the Fields Institute!
Return to the Fields Institute!

And I’m back in Canada now and still somewhat jet-lagged, which means going to sleep and waking up at hours that would be normal for anyone else.  I’m doing the second year of a postdoc with Nantel Bergeron and Mike Zabrocki, keeping up the research while looking for the next big thing, or at least a job. I have a great big pile of projects to work on this year, too, which have been on back-burners during the end of my time in Kenya.  I decided to do more people-oriented things while in Africa, and come back to some things (like programming projects) once back in a place where the location matters a little less to the project.

I’m really looking forward to a lot of these projects: thinking more about getting Sage into an online homework system, q-counting simultaneous core partitions, inventing a new type-free definition of the k-Schur functions, experimenting with the Raspberry Pi camera board, building a Pi-based Beowulf cluster, and moving my website to a new domain.  Oh wait, I guess that last one’s done now….  I also need to catch up on my writing!  There’s a recap of the third Maseno math camp waiting to be written, and an article for the Notices of the AMS.

So off to work!

Speaking of writing…  I’ve finally decided to buck up and get my own domain; partly so people can find the site more easily, and partly because I think the African Maths Initiative site where I’ve been writing is probably going to undergo some big changes over the next few months: the site should be much more focused on showing what AMI does, and there’s a separate plan to build a kind of registry/network for math initiatives across Africa.  This is also my 50th post on the blog, so we can think of it as a nice upgrade to commemorate the occasion.