Data Science at the 2019 AAAS Annual Meeting: Illustrating Opportunities for Undergraduate Mathematics Education
By Michael Pearson, Executive Director of the Mathematical Association of America
This year’s annual meeting of the AAAS was held in Washington, DC, in mid February. Since it was only a few blocks from MAA’s offices, I decided it was a great opportunity to listen to colleagues from outside the mathematics community and, in particular, look for ways that mathematics is relevant to current issues that affect all of us.
Not surprisingly, there were a number of sessions that, broadly speaking, fall into the “big data” category. The release by the White House of the American AI Initiative on the same week as the AAAS meeting makes this topic even more important for us to address. The AI Initiative calls in a broad sense for the U.S. to maintain leadership in the development and deployment of AI across all sectors of business, industry, and government. Relevant for those of us concerned about post secondary mathematics education, the report calls for us to “train current and future generations of American workers with the skills to develop and apply AI technologies to prepare them for today’s economy and jobs of the future.”
Of course that is extremely broad, and most MAA members will intuit that undergraduate mathematics must be central to the training of future AI professionals (even if that designation is not well-defined). However, at the AAAS meeting, I was struck by two basic ideas that relate active and deep research to undergraduate mathematics. Now I’d have to do a lot of hand-waving in any discussion of the underlying mathematics, but I’ll leave it to others to do the necessary work to make deeper sense of my observations.
First, there were sessions on deep learning and neural networks. As I understand it, a neural network is a realization of the Universal Approximation Theorem. Basically any function (think “input-output” in the broadest sense) can be approximated arbitrarily closely as a (finite) sum of just about any non-constant function, with appropriately chosen parameters as coefficients in the sum, and both shift and scaling factors.
Approximation of data is something we routinely study in undergraduate courses, from polynomial approximations (think, e.g., of Taylor polynomials), to more complex approximation using trigonometric series and orthogonal polynomials, and moving on to interpolation and splines on data sets, to least squares approximations of noisy data.
In fact, the basic study of polynomials that we usually begin in high school algebra classes seems to me to be justified (if it is -- another topic) almost solely because they serve as such a convenient class of basic approximators. I wish someone had explained this to me when I was in high school. I had to intuit this in much later studies of PDEs and harmonic analysis (though yes, I should have made the connection the first time I saw Taylor series and the results around convergence of the same!).
A second topic that really intrigued me is differential privacy. The session I attended first told the unfortunate tale of the 2010 census data being reconstructed at the record level by running the publicly-released data against other publicly available databases (e.g., of names and addresses). This is definitely something the Census Bureau wants to avoid repeating, but at the same time it’s critically important that these large public datasets are made available for appropriate analysis and research. This work drives public policy decisions that affect all of us.
In undergraduate statistics courses, it’s fairly common to deal with messy datasets, and, assuming one knows something about the kind of noise that has corrupted the data (e.g., gaussian or biased because of the underlying mechanism used to record the data), there are methods for analyzing the data and measuring the confidence in the conclusions drawn from that analysis.
Differential privacy is in some sense the reverse of this. Starting with the clean (census) data, noise with known properties is used to perturb the dataset in such a way that large-scale analysis can effectively be carried out, and conclusions confidently drawn, while also safeguarding against reconstruction of record-level data.
It’s an exciting time to be in mathematics. Through MAA’s Preparing for Industrial Careers in Mathematics project and our participation in the National Academies Roundtable on Data Science Postsecondary Education, as well as other initiatives and partnerships, I expect MAA to continue to serve as a source for our community to find effective ways to engage in these important issues.