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 ongoing effort to understand “big data”—data sets that are too large or complex to be understood using traditional data-processing techniques.

Our current mathematical understanding of many techniques that are central to the ongoing big-data revolution is inadequate, at best. Consider the simplest case, that of supervised learning, which has been used by companies such as Google, Facebook and Apple to create voice- or image-recognition technologies with a near-human level of accuracy. These systems start with a massive corpus of training samples—millions or billions of images or voice recordings—which are used to train a deep neural network to spot statistical regularities. As in other areas of machine learning, the hope is that computers can churn through enough data to “learn” the task: Instead of being programmed with the detailed steps necessary for the decision process, the computers follow algorithms that gradually lead them to focus on the relevant patterns.

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 one level to another, where the starting level, the height of the step and the width of the transition region are not determined ahead of time.

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” one has many layers.

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.)