More than 2,000 years in the past, the Greek mathematician Euclid, recognized to many as the father of geometry, modified the method we take into consideration shapes.
Building off these historic foundations and millennia of mathematical progress since, Justin Solomon is utilizing trendy geometric strategies to unravel thorny issues that always appear to have nothing to do with shapes.
For occasion, maybe a statistician needs to check two datasets to see how utilizing one for coaching and the different for testing would possibly influence the efficiency of a machine-learning mannequin.
The contents of these datasets would possibly share some geometric construction relying on how the knowledge are organized in high-dimensional area, explains Solomon, an affiliate professor in the MIT Department of Electrical Engineering and Computer Science (EECS) and a member of the Computer Science and Artificial Intelligence Laboratory (CSAIL). Comparing them utilizing geometric instruments can carry perception, for instance, into whether or not the similar mannequin will work on each datasets.
“The language we use to talk about data often involves distances, similarities, curvature, and shape — exactly the kinds of things that we’ve been talking about in geometry forever. So, geometers have a lot to contribute to abstract problems in data science,” he says.
The sheer breadth of issues one can remedy utilizing geometric strategies is the cause Solomon gave his Geometric Data Processing Group a “purposefully ambiguous” title.
About half of his workforce works on issues that contain processing two- and three-dimensional geometric knowledge, like aligning 3D organ scans in medical imaging or enabling autonomous autos to establish pedestrians in spatial knowledge gathered by LiDAR sensors.
The relaxation conduct high-dimensional statistical analysis utilizing geometric instruments, corresponding to to assemble higher generative AI fashions. For instance, these fashions be taught to create new pictures by sampling from sure components of a dataset crammed with instance pictures. Mapping that area of pictures is, at its core, a geometrical drawback.
“The algorithms we developed targeting applications in computer animation are almost directly relevant to generative AI and probability tasks that are popular today,” Solomon provides.
Getting into graphics
An early curiosity in computer graphics began Solomon on his journey to change into an MIT professor.
As a math-minded highschool scholar rising up in northern Virginia, he had the alternative to intern at a analysis lab outdoors Washington, the place he helped to develop algorithms for 3D face recognition.
That expertise impressed him to double-major in math and computer science at Stanford University, and he arrived on campus eager to dive into extra analysis initiatives. He remembers charging into the campus profession truthful as a first-year and speaking his method right into a summer season internship at Pixar Animation Studios.
“They finally relented and granted me an interview,” he remembers.
He labored at Pixar each summer season all through faculty and into graduate faculty. There, he centered on bodily simulation of material and fluids to enhance the realism of animated movies, in addition to rendering strategies to vary the “look” of animated content material.
“Graphics is so much 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,” Solomon says.
After deciding to launch an instructional profession, Solomon stayed at Stanford to earn a computer science PhD. As a graduate scholar, he ultimately centered on an issue generally known as optimum transport, the place one seeks to maneuver a distribution of some merchandise to a different distribution as effectively as doable.
For occasion, maybe somebody needs to search out the most cost-effective strategy to ship baggage of flour from a group of producers to a group of bakeries unfold throughout a metropolis. The farther one ships the flour, the dearer it’s; optimum transport seeks the minimal price for cargo.
“My focus was originally narrowed to only computer graphics applications of optimal transport, but 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 drawn to MIT as a result of of the alternative to work with good college students, postdocs, and colleagues on complicated, but sensible issues that might have an effect on many disciplines.
Paying it ahead
As a college member, he’s captivated with utilizing his place at MIT to make the discipline of geometric analysis accessible to individuals who aren’t normally uncovered to it — particularly underserved college students who usually don’t have the alternative to conduct analysis in highschool or faculty.
To that finish, Solomon launched the Summer Geometry Initiative, a six-week paid analysis program for undergraduates, largely drawn from underrepresented backgrounds. The program, which supplies a hands-on introduction to geometry analysis, accomplished its third summer season in 2023.
“There aren’t many institutions that have someone who works in my field, which can lead to imbalances. It means the typical PhD applicant comes from a restricted set of schools. I’m trying to change that, and to make sure folks 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 gotten actual outcomes. Since its launch, Solomon has seen the composition of the incoming courses of PhD college students change, not simply at MIT, however at different establishments, as effectively.
Beyond computer graphics, there’s a rising record of issues in machine studying and statistics that may be tackled utilizing geometric strategies, which underscores the want for a extra various discipline of researchers who carry new concepts and views, he says.
For his half, Solomon is trying ahead to making use of instruments from geometry to enhance unsupervised machine studying fashions. In unsupervised machine studying, fashions should be taught to acknowledge patterns with out having labeled coaching knowledge.
The overwhelming majority of 3D knowledge should not labeled, and paying people to hand-label objects in 3D scenes is commonly prohibitively costly. But refined fashions incorporating geometric perception and inference from knowledge may also help computer systems determine complicated, unlabeled 3D scenes, so fashions can be taught from them extra successfully.
When Solomon isn’t pondering this and different knotty analysis quandaries, he can usually be discovered taking part in classical music on the piano or cello. He’s a fan of composer Dmitri Shostakovich.
An avid musician, he’s made a behavior of becoming a member of a symphony in no matter metropolis he strikes to, and at the moment performs cello with the New Philharmonia Orchestra in Newton, Massachusetts.
In a method, it’s a harmonious mixture of his pursuits.
“Music is analytical in nature, and I have the advantage of being in a research field — computer graphics — that is very closely connected to artistic practice. So the two are mutually beneficial,” he says.