Greater 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.
Constructing off those ancient foundations and millennia of mathematical progress since, Justin Solomon is using modern geometric techniques to unravel thorny problems that always appear to don’t have anything to do with shapes.
As an example, perhaps a statistician wants to check two datasets to see how using one for training and the opposite for testing might impact the performance of a machine-learning model.
The contents of those datasets might share some geometric structure depending on how the information are 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 and Artificial Intelligence Laboratory (CSAIL). Comparing them using geometric tools can bring insight, for instance, into whether the identical model will work on each datasets.
“The language we use to speak about data often involves distances, similarities, curvature, and shape — precisely the sorts of things that we’ve been talking about in geometry ceaselessly. So, geometers have quite a bit to contribute to abstract problems in data science,” he says.
The sheer breadth of problems one can solve using geometric techniques is the rationale Solomon gave his Geometric Data Processing Group a “purposefully ambiguous” name.
About half of his team works on problems that involve processing two- and three-dimensional geometric data, like aligning 3D organ scans in medical imaging or enabling autonomous vehicles to discover pedestrians in spatial data gathered by LiDAR sensors.
The remaining conduct high-dimensional statistical research using geometric tools, similar to to construct higher generative AI models. For instance, these models learn to create recent images by sampling from certain parts of a dataset stuffed with example images. Mapping that space of images is, at its core, a geometrical problem.
“The algorithms we developed targeting applications in computer animation are almost directly relevant to generative AI and probability tasks which are popular today,” Solomon adds.
Entering into graphics
An early interest in computer graphics began Solomon on his journey to grow to be 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 Washington, where he helped to develop algorithms for 3D face recognition.
That have inspired him to double-major in math and computer science at Stanford University, and he arrived on campus keen to dive into more research projects. He remembers charging into the campus profession fair as a first-year and talking his way right into a summer internship at Pixar Animation Studios.
“They finally relented and granted me an interview,” he recalls.
He worked at Pixar every summer throughout college and into graduate school. There, he focused on physical simulation of material and fluids to enhance the realism of animated movies, in addition to rendering techniques to alter the “look” of animated content.
“Graphics is a lot fun. It’s driven by visual content, but beyond that, it presents unique mathematical challenges that set it other than other parts of computer science,” Solomon says.
After deciding to launch an educational profession, Solomon stayed at Stanford to earn a pc science PhD. As a graduate student, he eventually focused on an issue referred to as optimal transport, where one seeks to maneuver a distribution of some item to a different distribution as efficiently as possible.
As an example, perhaps someone wants to seek out the most affordable solution to ship bags of flour from a set of manufacturers to a set of bakeries spread across a city. The farther one ships the flour, the dearer it’s; optimal transport seeks the minimum cost for shipment.
“My focus was originally narrowed to only computer graphics applications of optimal transport, however the research took off in other directions and applications, which was a surprise to me. But, in a way, this coincidence led to the structure of my research group at MIT,” he says.
Solomon says he was interested in MIT due to the opportunity to work with good students, postdocs, and colleagues on complex, yet practical problems that would have an effect on many disciplines.
Paying it forward
As a college member, he’s keen about using his position at MIT to make the sector of geometric research accessible to individuals who aren’t normally exposed to it — especially underserved students who often don’t have the chance to conduct research in highschool or college.
To that end, Solomon launched the Summer Geometry Initiative, a six-week paid research program for undergraduates, mostly drawn from underrepresented backgrounds. This system, which provides a hands-on introduction to geometry research, accomplished its third summer in 2023.
“There aren’t many institutions which have someone who works in my field, which might result in imbalances. It means the standard PhD applicant comes from a restricted set of colleges. I’m trying to alter that, and to make sure that folks who’re absolutely good but didn’t have the advantage of being born in the proper place still have the chance to work in our area,” he says.
This system has gotten real results. Since its launch, Solomon has seen the composition of the incoming classes of PhD students change, not only at MIT, but at other institutions, as well.
Beyond computer graphics, there’s a growing list of problems in machine learning and statistics that may be tackled using geometric techniques, which underscores 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. In unsupervised machine learning, models must learn to acknowledge patterns without having labeled training data.
The overwhelming majority of 3D data are usually not labeled, and paying humans to hand-label objects in 3D scenes is usually prohibitively expensive. But sophisticated models incorporating geometric insight and inference from data will help computers determine complex, unlabeled 3D scenes, so models can learn from them more effectively.
When Solomon isn’t pondering this and other knotty research quandaries, he can often be found playing classical music on the piano or cello. He’s a fan of composer Dmitri Shostakovich.
An avid musician, he’s made a habit of joining a symphony in whatever city he moves to, and currently plays cello with the Recent Philharmonia Orchestra in Newton, Massachusetts.
In a way, it’s a harmonious combination of his interests.
“Music is analytical in nature, and I actually have the advantage of being in a research field — computer graphics — that could be very closely connected to artistic practice. So the 2 are mutually useful,” he says.