Deanna Needell (left) and Rachel Ward
The 2016 IMA Prize in Mathematics and its Applications is shared by Deanna Needell, an associate professor in the Department of Mathematics at Claremont McKenna College, and Rachel Ward, an associate professor in the Department of Mathematics at the University of Texas at Austin.
While Needell is recognized for her contributions to sparse approximation, signal processing, and stochastic optimization, and Ward is recognized for her contributions to the mathematics of machine learning and signal processing, much of their research overlaps.
It is not surprising that Needell and Ward are frequent collaborators, as both are lauded by their peers as being among the most talented applied analysts in the country. Their most oft mentioned work in respect to this award was their 2013 paper on “Stable image reconstruction using total variation minimization,” published in the SIAM Journal on Imaging Sciences.
“This gave theoretical guarantees for total variation minimization in compressed sensing, which prove that given undersampled measurements of some signal (e.g. an image), one can search for the signal with the same measurements that has the lowest variation – that is, smallest gradient – and that this reconstruction yields near-optimal error,” Needell explained.
A particularly striking application of these principles arises in medical imaging. Consider magnetic resonance imaging (MRI), where the underlying image to be recovered is, say, a horizontal section of a brain or neck. Like most natural images of interest, this section can be thought of as a two-dimensional function that will be constant or slowly varying over most of the domain, interrupted only by sharp changes across a low-dimensional set of values corresponding to edges.
At the same time, each measurement in an MRI scan corresponds to a Fourier transform component, representing the response of the image to a particular frequency. Each measurement takes time and costs money, and thus it is desirable to obtain high-quality MRI reconstructions using as few measurements as possible.
“My work has answered questions like: what is a good subset of frequencies to take if the total scan time is limited to one hour, or alternatively, if one has a fixed budget of frequencies? And how should one reconstruct the underlying image from these frequencies?” Ward said. “The joint work with Deanna gave theoretical guarantees for a popular reconstruction method used in practice, total variation minimization, and suggested a stochastic sampling strategy for selecting frequencies which achieves these guarantees.”
Needell says that “rather than measuring in every ‘direction’ as in a typical MRI, compressed sensing promotes measuring in a small number of random directions, and it turns out that is enough to still ensure accurate image reconstruction.”
Other applications for this type of data acquisition and analysis include sensor and distributed networks, statistical problems, compression, and image processing problems.
Ward credits her Ph.D. advisor, Ingrid Daubechies, for getting her interested in these types of problems.
“Her construction of compactly supported smooth wavelets. The combination of practicality and mathematical beauty blew my mind,” Ward said.
Needell’s Ph.D. advisor, Roman Vershynin, had a large influence on her career as well, having introduced her to the field and to her first experience in real mathematical research. It also helped that she enjoyed classes in probability and analysis, and it turned out that these topics were used in a lot of results of compressed sensing.
“Even though compressed sensing is an applied field, it uses many tools from ‘pure math.’ So for me, it is an area that is the best of both worlds,” Needell noted.
Needell is also expanding her research horizons into methods for stochastic and combinatorial optimization.
“Often, these methods involve intriguing geometric and probabilistic problems which are fun to solve, and also have a wide array of important applications,” she said. “I'm also interested in using these kinds of techniques along with other compressive methods to analyze large-scale medical data, a personal passion of mine.”
Ward plans to spend a sabbatical year working in industry doing machine learning research.
“It will be good for me to get out of the ivory tower for a while and face the applications head on,” she added.
The IMA Prize in Mathematics and its Applications is awarded annually to a mathematical scientist who is within 10 years of having received his or her Ph.D. degree. The award recognizes an individual who has made a transformative impact on the mathematical sciences and their applications. The prize can recognize either a single notable achievement or acknowledge a body of work. The prize consists of a certificate and a cash award of $3,000. Funding for the IMA Prize in Mathematics and its Applications is made possible by generous donations of friends of the IMA.