Machine learning works spectacularly well, but mathematicians aren’t quite sure why

At a dinner I attended some years ago, the distinguished differential geometer Eugenio Calabi volunteered to me his tongue-in-cheek distinction between pure and applied mathematicians. A pure mathematician, when stuck on the problem under study, often decides to narrow the problem further and so avoid the obstruction. An applied mathematician interprets being stuck as an indication that it is time to learn more mathematics and find better tools.

I have always loved this point of view; it explains how applied mathematicians will always need to make use of the new concepts and structures that are constantly being developed in more foundational mathematics. This is particularly evident today in the on

Our current mathematical understanding of many techniques that are central to the on

In mathematical terms, these supervised-learning systems are given a large set of inputs and the corresponding outputs; the goal is for a computer to learn the function that will reliably transform a new input into the correct output. To do this, the computer breaks down the mystery function into a number of layers of unknown functions called sigmoid functions. These S-shaped functions look like a street-to-curb transition: a smoothened step from on

Inputs enter the first layer of sigmoid functions, which spits out results that can be combined before being fed into a second layer of sigmoid functions, and so on. This web of resulting functions constitutes the “network” in a neural network. A “deep” on

*Olena Shmahalo/Quanta Magazine*

Decades ago, researchers proved that these networks are universal, meaning that they can generate all possible functions. Other researchers later proved a number of theoretical results about the unique correspondence between a network and the function it generates. But these results assume networks that can have extremely large numbers of layers and of function nodes within each layer. In practice, neural networks use anywhere between two and two dozen layers.* Because of this limitation, none of the classical results come close to explaining why neural networks and deep learning work as spectacularly well as they do.

It is the guiding principle of many applied mathematicians that if something mathematical works really well, there must be a good underlying mathematical reason for it, and we ought to be able to understand it. In this particular case, it may be that we don’t even have the appropriate mathematical framework to figure it out yet. (Or, if we do, it may have been developed within an area of “pure” mathematics from which it hasn’t yet spread to other mathematical disciplines.)

Another technique used in machine learning is unsupervised learning, which is used to discover hidden connections in large da

What you would like to do, as a researcher, is identify this lower-dimensional surface, thereby reducing the personality portraits of the 200,000 subjects to their essential properties — a task that is similar to finding that two variables suffice to identify any point in the mountain-range surface. Perhaps the personality-test surface can also be described with a simple function, a connection between a number of variables that is significantly smaller than 500. This function is likely to reflect a hidden structure in the da

In the last 15 years or so, researchers have created a number of tools to probe the geometry of these hidden structures. For example, you might build a model of the surface by first zooming in at many different points. At each point, you would place a drop of virtual ink on the surface and watch how it spread out. Depending on how the surface is curved at each point, the ink would diffuse in some directions but not in others. If you were to connect all the drops of ink, you would get a pretty good picture of what the surface looks like as a whole. And with this information in hand, you would no longer have just a collection of da

These methods are already leading to interesting and useful results, but many more techniques will be needed. Applied mathematicians have plenty of work to do. And in the face of such challenges, they trust that many of their “purer” colleagues will keep an open mind, follow what is going on, and help discover connections with other existing mathematical frameworks. Or perhaps even build new on

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