From the Lab: Kavita Ramanan puts randomness to work

A professor of applied mathematics at Brown, Ramanan uses randomness as a tool to make precise predictions about complex systems ranging from statistical physics to communication networks.

PROVIDENCE R.I. [Brown University] — Mathematics is a field of proofs and axioms, iron-clad truths that reveal the order underlying the world. In that context, randomness — a state of apparent patternlessness and unpredictability — would seem to exist largely as something to be stamped out or explained away.

But to Kavita Ramanan, a professor of applied mathematics at Brown and an expert in probability theory, randomness is no pariah. In fact, it’s a powerful tool — one that can be used to create useful models of everything from communications networks to the spread of infectious diseases. 

“I think most people are more comfortable with things that are deterministic and predictable,” Ramanan said. “But the world around us is unpredictable in many ways. Probability theory and randomness can help to make sense of that.”

Randomness can manifest itself in a variety of ways, Ramanan says. In one sense, it may be embedded deep in the workings of the universe. Quantum mechanics — the rules governing the world at the scale of individual elementary particles — appears to contain purely random elements. But in the everyday world, she says, randomness often manifests itself as an incomplete understanding of complex systems. In systems with lots of moving parts, the forces driving the activity of each individual part may be unknown. And even if all the forces are known, it may be computationally impossible to keep track of all the information.

It's in these cases where randomness becomes a powerful mathematical tool, Ramanan says. Rather than trying to calculate everything that’s happening within a complex system — which might be computationally impossible — treating some parts of it as random can enable useful predictions about the system as a whole.

“In a big group of people, for example, you may not be able to predict what any one person will do, but you can use probability theory to predict what the larger group might do,” Ramanan said. “Basically, if you aggregate a lot of independent randomness, you can start to see patterns that become predictable.”

One particular area of interest for Ramanan is modeling what are known as high-dimensional stochastic interacting systems. In these models, randomly evolving individuals or particles are labelled by the nodes on a graph that captures the nature of the interaction: a change in the random state of one node directly influences the evolution of the states of only the nodes in its neighborhood. Ramanan has developed new mathematical frameworks for analyzing and approximating large random interacting particle systems that model phenomena as disparate as spiking in neuronal networks and scheduling in communication networks.

In other work, Ramanan has developed models that help to predict the appropriate allocation of power and frequency to accommodate users that may pop up randomly on wireless communication networks. She and her colleagues hold four patents related to optimization of wireless communications.

In more recent work, Ramanan is using randomness to address a persistent problem in data science: the curse of dimensionality. The kinds of datasets involved in the “big data” revolution are often high-dimensional, meaning they record lots of attributes or features. Health care data, for example, can contain dozens of attributes for each patient — height, weight, blood pressure, heart rate, cholesterol and so on. The high dimensionality of these datasets can make them difficult to analyze and visualize.

“When we visualize data, we use graph plots or sometimes 3D pictures — how else would you visualize things?” Ramanan said. “But for these high-dimensional datasets, you need some reduction of dimensionality to project it down into two- or three-dimensional visualizations. It turns out that if you do these projections randomly instead of in a fully deterministic way, you get projections that have all sorts of interesting properties.”

Those interesting properties can lead to new insights that can be gleaned from the data. And because high-dimensional data is ubiquitous in data science, the random projection technique could be useful in diverse data-analysis problems. For Ramanan, that’s the real appeal of this research area.

“What I love about probability theory is that one can develop abstract fundamental results that are both mathematically elegant and at the same time useful for understanding concrete phenomena and real applications,” she said.

"I don't think people understand what a social profession mathematics is. The process of explaining your work to others, either as a teacher or as a researcher, gives rise to many wonderful interactions."

Kavita Ramanan Professor of Applied Mathematics
Photo of Kavita Ramanan

Ramanan’s work in probability theory has earned her an abundance of awards and honors. Last May, she received the Vannevar Bush Faculty Fellowship, the most prestigious award issued by the U.S. Department of Defense. She’s a member of the American Academy of Arts and Sciences, a fellow of several societies, including the American Mathematical Society, the American Association for the Advancement of Science, and the Society for Industrial and Applied Mathematics, and has also won fellowships from the Simons Foundation and the Guggenheim Foundation.

In addition to her research, Ramanan also treasures the opportunity to work with students at Brown and colleagues around the world.

“I don't think people understand what a social profession mathematics is,” she said. “The process of explaining your work to others, either as a teacher or as a researcher, gives rise to many wonderful interactions. And I really enjoy interacting with young students — advising them and seeing them grow.”

In fact, Ramanan said, it was an interaction with a Brown student that led to her work on random projections of high-dimensional data. The student, an enthusiastic participant in Ramanan’s probability course, asked Ramanan for some additional material to read outside of class. She chose a paper she thought might be of interest to the student, and then read it herself to prepare for discussing it with him. Reading that paper, which she may not have found otherwise, led Ramanan to develop her new ideas on the subject.  

Meeting that particular student who had an interest in that particular paper was a random encounter. And as Ramanan’s work proves, randomness can be a powerful thing.

Photo of Kavita Ramanan talking with a student

What made you want to dedicate your professional life to math?

My enjoyment of math is rooted in its power of abstraction and its "unreasonable effectiveness" in helping us understand and control the world around us. Another particularly appealing feature is that math caters to both an artistic and scientific temperament — it requires creativity and intuitive leaps of faith that often lead to conjectures, but is also unrelenting in its demand for precision and clarity of thought.

What's one thing you hope people understand about what you do?

There is never a dull moment. My job combines constant intellectual stimulation, each day bringing unpredictable and interesting new challenges, with endless opportunities to try to have a small impact on the broader community, be it through dissemination and applications of one's research, or education, mentorship, advising and advocacy.

What's your favorite thing about your job?

The life of a research mathematician is one of continual exploration, learning and growth, with the additional bonus of having the chance to travel around the world, and meet and collaborate with diverse people of all ages.