Pietraho took time out of his teaching schedule during the spring semester to entertain faculty colleagues with a talk about the huge strides that have been made recently in artificial intelligence (AI), and what this means for him as a mathematician, or more precisely, an algebraist. The talk was titled “Birds, Books, and Matrices: A Brief Adventure in Artificial Intelligence and Neural Networks.”
The development, over the last five or six years, of artificial neural networks – computing systems inspired by the biology of the human brain – means there are computers out there that can learn to identify a wide range of objects and patterns, says Pietraho. Before discussing neural networks, however, let’s start with a really basic question.
What is a neuron?
A mathematical neuron is just an abstraction. It is a part of a computer program that takes in a numerical input, performs a very simple computation, and returns a numerical output – the artificial equivalent of a biological neuron that receives an electrical impulse, and based on its intensity, sends a signal to other neurons. So a mathematical neuron is a loose attempt to replicate these complex structures as part of a computer program.
So a neural network is….?
It’s a system of interconnected mathematical neurons that interact to perform a complex computation. We understand the role of each individual neuron, but when you put a few thousand of them together (neural networks often contain thousands of neurons working together), the network exhibits behavior that is complex and sometimes unpredictable. It’s man-made, computer-driven, and it works, but we don’t completely understand how.
Here’s an analogy: Neurons in a neural network behave like fish swimming in a school. Each fish follows a simple set of rules and wants to stay near other fish without bumping into them. When you observe entire schools of fish, however, you see some very complex swarming behavior which you wouldn’t really understand by studying individual fish. a neural network has similar properties: An individual neuron is a very simple machine following a very simple rule, but together, networks of them make wonderful and complex things happen.
Where do “birds and books” fit into all this?
Computer programs using neural networks can learn how to identify and classify objects and patterns. They’re especially good at recognizing images. I looked at two families of examples: bird images and book covers. Using about 300,000 images of birds, covering 300 species, I designed a neural network that learned to identify certain features and effectively decide for itself the species of the bird in each image. A similar model developed by the Cornell Lab of Ornithology has a 90 percent success rate.
Working with Parikshit Sharma ’17 (now working for IndieBio), I tried to see whether a neural network could identify genres of different types of books solely by examining the images of their covers. Maybe romance novels have bolder colors, and science books have more writing on them? There are features that are common to some books and uncommon in others, and it turned out that a computer was able detect those and make an accurate classification. So, it seems you can judge a book by its cover!
AI programs can make spectacular mistakes as well, though. Fo example, one common misidentification is that every so often a picture of a cat will be classified as a burrito! People who worry about self-driving cars have a point – sometimes neural networks fail and we don’t know completely understand why.
How good are neural networks at doing your job, algebra?
They’re pretty good, but only up to a point. They can only approximate, and a lot of mathematics requires exact answers, which a neural network will never be able to provide. Neural networks can suggest what a correct answer might be, but it’s still left to the mathematician to take this approximation and make it exact. So neural networks can help guide humans to the right answer.
Neural networks are not perfect. Sometimes they make unexpected mistakes, such as confusing cats and burritos.
So your job as an algebraist is safe?
Essentially yes, and hopefully the tedious part of my job can be replaced by a computer and I can just do the fun stuff.
But given how embryonic neural networks were only six years ago, how can you be sure they won’t overtake you?
It’s easily to extrapolate like that, but science is a punctuated equilibrium. It lies in a steady state for a long time, then there’s a massive and rapid improvement, after which it returns to an equilibrium. So it would be irrational to project that the rate of the last few years’ innovations will continue. If it did, we would all have to up our games.