More than 2,000 years ago, the Greek mathematician Euclid, known to many as the father of geometry, changed the way we think about shapes.
Building on these ancient foundations and millennia of mathematical progress since then, Justin Solomon uses modern geometric techniques to solve thorny problems that often seem to have nothing to do with shapes.
For example, perhaps a statistician wants to compare two datasets to see how using one for training and one for testing might affect the performance of a machine learning model.
The contents of these datasets may share some geometric structure depending on how the data is arranged in high-dimensional space, explains Solomon, an associate professor in MIT’s Department of Electrical Engineering and Computer Science (EECS) and a member of the Computer Science and Artificial Intelligence Laboratory (CSAIL). Comparing them using geometric tools can yield insight, for example, about whether the same model will work on both datasets.
“The language we use to talk about data often includes distances, similarities, curvature, and shape—exactly the things we’ve been talking about forever in geometry. So geometers have a lot to contribute to abstract problems in data science,” he says.
The sheer range of problems one can solve using geometric techniques is why Solomon gave his Geometric Data Processing Group a “deliberately ambiguous” name.
About half of his team works on problems involving the processing of 2D and 3D geometric data, such as aligning 3D instrument scans in medical imaging or enabling autonomous vehicles to recognize pedestrians in spatial data collected by LiDAR sensors.
The rest conduct high-dimensional statistical research using geometric tools, such as building better AI models. For example, these models learn to generate new images by sampling from certain portions of a data set that are filled with example images. Mapping this image space is, at its core, a geometric problem.
“The algorithms we developed targeting applications in computer animation are almost directly relevant to the artificial intelligence and probability generation tasks that are popular today,” adds Solomon.
Getting into the graphics
An early interest in computer graphics started Solomon on his journey to become an MIT professor.
A math-minded high school student growing up in northern Virginia, he had the opportunity to intern at a research lab outside of Washington, D.C., where he helped develop algorithms for 3D facial recognition.
This experience inspired him to double major in mathematics and computer science at Stanford University, and he arrived on campus eager to dive into more research projects. He remembers walking into the campus career fair as a freshman and talking about a summer internship at Pixar Animation Studios.
“Finally they relented and gave me an interview,” he recalls.
He worked at Pixar every summer throughout college and into graduate school. There, he focused on physically simulating cloth and fluids to enhance the realism of animated films, as well as performing techniques to change the “look” of animated content.
“The graphics are so fun. It is driven by visual content, but beyond that, it presents unique mathematical challenges that set it apart from other parts of computer science,” says Solomon.
After deciding to pursue an academic career, Solomon stayed at Stanford to earn a PhD in computer science. As a graduate student, he eventually focused on a problem known as optimal transportation, where one seeks to move one distribution of some object to another distribution as efficiently as possible.
For example, someone might want to find the cheapest way to ship bags of flour from a collection of manufacturers to a collection of bakeries spread across a city. The farther one sends the flour, the more expensive it is. Optimal transportation seeks the least cost for shipping.
“My focus was initially limited only to optimal transfer computer graphics applications, but the research took off in other directions and applications, which was a surprise to me. But, somehow, that coincidence led to the structure of my research group at MIT,” he says.
Solomon says he was drawn to MIT because of the opportunity to work with outstanding students, postdocs and colleagues on complex but practical problems that could have an impact across multiple disciplines.
Pay it forward
As a faculty member, he is passionate about using his position at MIT to make the field of geometric research accessible to people who might not normally be exposed to it — especially underserved students who often don’t have the opportunity to conduct research in high school or college.
For this purpose, Solomon started the Geometry Summer Initiative, a six-week paid research program for undergraduate students, primarily from underrepresented backgrounds. The program, which provides a hands-on introduction to geometry research, completed its third summer in 2023.
“Not many institutions have someone working in my field, which can lead to imbalances. It means that the typical PhD candidate comes from a limited set of schools. I’m trying to change that and make sure that people who are absolutely brilliant but didn’t have the advantage of being born in the right place still have the opportunity to work in our area,” he says.
The program has real results. Since its inception, Solomon has seen the composition of incoming doctoral student classes change, not only at MIT, but at other institutions as well.
Beyond computer graphics, there is a growing list of problems in machine learning and statistics that can be tackled using geometric techniques, which underscores the need for a more diverse field of researchers bringing new ideas and perspectives, he says.
For his part, Solomon looks forward to applying tools from geometry to improve unsupervised machine learning models. In unsupervised machine learning, models must learn to recognize patterns without having labeled training data.
The vast majority of 3D data is unlabeled, and paying people to label objects in 3D scenes is often prohibitively expensive. But sophisticated models that incorporate geometric insight and inference from data can help computers understand complex, unlabeled 3D scenes so that models can learn from them more effectively.
When Solomon isn’t thinking about this and other odd research problems, he can often be found playing classical music on the piano or cello. He is a fan of the composer Dmitri Shostakovich.
An avid musician, he’s used to joining a symphony in whatever city he moves to, and currently plays cello with the New Philharmonia Orchestra in Newton, Massachusetts.
In a way, it is a harmonious combination of his interests.
“Music is analytical in nature and I have the advantage of being in a research field – computer graphics – that is very closely linked to artistic practice. So both are mutually beneficial,” he says.