A pc scientist pushes the boundaries of geometry

More than 2,000 years ago, the Greek mathematician Euclid, known to many as the daddy of geometry, modified the best way we take into consideration shapes.

Building on these ancient foundations and millennia of mathematical progress, Justin Solomon uses modern geometric techniques to unravel thorny problems that usually seem unrelated to shapes.

For example, a statistician might want to match two data sets to see how using one data set for training and the opposite for testing might affect the performance of a machine learning model.

The contents of those datasets could have some geometric structure depending on how the information is arranged in high-dimensional space, explains Solomon, an associate professor within the MIT Department of Electrical Engineering and Computer Science (EECS) and a member of the Computer Science Department and Computer Science Laboratory artificial intelligence (CSAIL). For example, comparison with geometric tools can reveal whether the identical model works for each data sets.

“The language we use to discuss data often involves distances, similarities, curvatures, and shapes—precisely the sorts of things we have at all times talked about in geometry. So geometers can contribute lots to abstract problems in data science,” he says.

The sheer range of problems that will be solved using geometric techniques is why Solomon gave his geometric computing group an “intentionally ambiguous” name.

About half of his team works on problems that involve processing two- and three-dimensional geometric data, comparable to the alignment of 3D organ scans in medical imaging or the power of autonomous vehicles to discover pedestrians in spatial data detected by LiDAR sensors.

The rest conduct high-dimensional statistical research using geometric tools, for instance to construct higher generative AI models. For example, these models learn to create recent images by sampling specific parts of a knowledge set crammed with example images. Mapping this image space is actually a geometrical problem.

“The algorithms we developed for applications in computer animation are almost directly relevant to generative AI and probabilistic tasks which can be popular today,” adds Solomon.

Getting began with graphics

His early interest in computer graphics initiated Solomon’s path to becoming an MIT professor.

As a math-minded highschool student growing up in Northern Virginia, he had the chance to intern at a research lab outside of Washington, where he helped develop algorithms for 3D facial recognition.

This experience inspired him to pursue a double major in mathematics and computer science at Stanford University, and when he arrived on campus he was wanting to dive into more research projects. He remembers rushing into the campus profession fair as a freshman and convincing himself to take a summer internship at Pixar Animation Studios.

“They finally gave in and gave me an interview,” he remembers.

He worked at Pixar every summer throughout college and graduate school. There he focused on the physical simulation of gear and fluids to enhance the realism of animated movies, in addition to rendering techniques to vary the “look” of animated content.

“Graphics are a lot fun. It relies on visual content, but in addition presents unique mathematical challenges that differentiate it from other areas of computer science,” says Solomon.

After selecting an educational profession, Solomon stayed at Stanford to earn a doctorate in computer science. As a graduate student, he ended up specializing in an issue often called optimal transportation, through which one tries to maneuver one distribution of an item to a different distribution as efficiently as possible.

For example, perhaps someone wants to search out the most affordable approach to ship bags of flour from a group of manufacturers to a group of bakeries across the town. The further you ship the flour, the costlier it’s; Optimal transport strives for minimal transport costs.

“Originally I only focused on optimal transport computer graphics applications, however the research shifted to other directions and applications, which was a surprise to me. But in a way, that coincidence led to the structure of my research group at MIT,” he says.

Solomon says he was drawn to MIT by the chance to work with sensible students, postdocs and colleagues on complex but practical problems that might have implications for a lot of disciplines.

I’ll pay it forward

As a school member, he’s captivated with using his position at MIT to make the sector of geometric research accessible to individuals who may not normally be exposed to it—particularly underserved students who often lack opportunities in highschool must do research or college.

For this purpose Solomon began the Summer Geometry Initiative, a six-week paid research program for college students who largely come from underrepresented backgrounds. The program, which provides a practical introduction to geometry research, accomplished its third summer in 2023.

“There aren’t many institutions which have someone working in my field, which might result in imbalances. This signifies that the everyday graduate student comes from a limited number of colleges. “I’m trying to vary that and be certain that that people who find themselves absolutely sensible but haven’t had the advantage of being born in the precise place still have the chance to work in our area,” he says.

The program has achieved real results. Since its launch, Solomon has seen the composition of incoming graduate student classes change not only at MIT but at other institutions as well.

Beyond computer graphics, there may be a growing list of problems in machine learning and statistics that will be solved using geometric techniques, highlighting the necessity for a more diverse field of researchers who bring recent ideas and perspectives, he says.

For his part, Solomon is looking forward to applying tools from geometry to enhance unsupervised machine learning models. Unsupervised machine learning requires models to learn to acknowledge patterns without labeling training data.

The overwhelming majority of 3D data is unlabeled, and paying people to manually label objects in 3D scenes is usually prohibitively expensive. But sophisticated models that incorporate geometric insights and inferences from data may also help computers determine complex, unlabeled 3D scenes in order that models can learn from them more effectively.

When Solomon is just not serious about this and other difficult research problems, he is usually found playing classical music on the piano or cello. He is a fan of the composer Dmitri Shostakovich.

An avid musician, he has made it a habit to hitch a symphony orchestra in every city he moves to and currently plays the cello there New Philharmonia Orchestra in Newton, Massachusetts.

In a way, it’s a harmonious combination of his interests.

“Music has an analytical character and I even have the advantage of working in a research area – computer graphics – that could be very closely linked to artistic practice. So each are mutually helpful,” he says.