Reflections on: On Uncertainty Calibration for Equivariant Functions

> Code, arxiv, Workshop Version (NeurReps)

Most project pages for academic papers serve as the most consolidated version of the research project. They contain all of the relevant links to the code and paper and give a high level overview of what is contained in the paper. Instead, I want to give a little bit more of a high level intuition as to how this project came to be, and what my thoughts are going forward.

This project was conceived from a real world problem I was trying to solve. With my collaborators at AstroAI, I was working on an emulator for chemical spectra that included uncertainty estimates. This uncertainty estimates are useful for downstream tasks such as retrieval using Hamiltonian Monte Carlo. I was at a stage where I was happy with my model's qualitative output, but I wanted to quantify this. I had recently read Dian's paper on approximation error bounds for equivariant functions, and I began wondering if the bounds presented in his paper could similarly be applied to the Calibration objective I was trying to meet. The calibration objective just means that my uncertainty is representative of my actual error. For example, if I am 70% confident, then I should be 70% accurate. Calibration errors compute these discrepancies between accuracy and confidence or error and variance.

The key idea in this paper is that we can use approximation error bounds for equivariant functions from Dian's paper on different confidence fibers. This allows us to say things such as "the model can be no less than X% accurate when it is Y% confident due to equivariance constraints." The total calibration bounds follow by averaging the calibration errors over all fibers. We show how symmetry mismatch can tighten both lower and upper calibration error bounds in different settings. Additionally, we note how models that are overconstrained by symmetry are unable to distinguish between the origin of their uncertain (aleatoric / epistemic).

There is more to say about how equivariant models can become more or less uncertain when the equivariance is extrinsic, or there is some covariate shift. In fact, I think there can be made a precise analogy between extrinsic equivariance and covariate shift, though it is currently escaping me. If you are interested in working on these topics, feel free to reach out.

I am a fairly sentimental person! While we have yet to reach the final goal (paper is still under review at TMLR at time of writing 11/05/25), now seems as good a time as ever to share some gratitudes. My first thank you goes to Boston, and the great community within it. My work with my advisor and the co-authors on this paper started from a conversation at the Symposium on Geometry Processesing, which was serrindipitously hosted at MIT while I was a student at Northeastern.

A lot of what allowed me to dedicate significant time to this project was taking the Research Capstone option at my university. My classmates are wonderfully talented, and I truly enjoyed and appreciated them sharing their passions with me. I hope they were similarly inspired by EquiUQ :). Special shout out to my classmate and co-author Jake, who worked on this project with me despite being from an entirely different field of computer science (security / PL).

To that end, an enormous thanks is also owed to my awesome co-authors, as well as Shubhendu Trivedi for offering his insights down the stretch of submission.