A new study by MIT researchers shows the first method for machine learning with symmetry that is provably efficient in terms of both the amount of computation and data needed. These results clarify a foundational question, and they could aid researchers in the development of more powerful machine-learning models that are designed to handle symmetry. Such models would be useful in a variety of applications, from discovering new materials to identifying astronomical anomalies to unraveling complex climate patterns. “These symmetries are important because they are some sort of information that nature is telling us about the data, and we should take it into account in our machine-learning models. We’ve now shown that it is possible to do machine-learning with symmetric data in an efficient way,” says Behrooz Tahmasebi, an MIT graduate student and co-lead author of this study. They explored the statistical-computational tradeoff in machine learning with symmetric data. This tradeoff means methods that require fewer data can be more computationally expensive, so researchers need to find the right balance. Building on this theoretical evaluation, the researchers designed an efficient algorithm for machine learning with symmetric data. To do this, they borrowed ideas from algebra to shrink and simplify the problem. Then, they reformulated the problem using ideas from geometry that effectively capture symmetry. Finally, they combined the algebra and the geometry into an optimization problem that can be solved efficiently, resulting in their new algorithm. “Most of the theory and applications were focusing on either algebra or geometry. Here we just combined them,” Tahmasebi says.