Abhiti Mishra | Searcharxiv

arXiv2026

funFMC: Overlapping Clustering for Functional Data

In applications such as neuroscience and environmental science, data are naturally modeled as multivariate functional data and often exhibit overlapping cluster structure. Existing clustering methods for functional data typically impose mutually exclusive memberships and therefore fail to capture such structure. We propose a latent factor model based approach with functional factors and a real-valued loading matrix that encodes potentially overlapping cluster memberships. Under mild conditions, we establish identifiability of the loading matrix up to permutation, ensuring that the overlapping cluster structure is recoverable up to label switching. We develop a procedure for estimating both the number of clusters and the associated cluster memberships. This involves solving an infinite-dimensional regression problem in operators, whose solution is characterized using the inner product on the space of Hilbert-Schmidt operators and expressed in terms of real-valued matrices. This formulation enables rigorous asymptotic analysis, and we establish a central limit theorem to facilitate statistical inference on overlapping cluster memberships. We demonstrate the performance of our method using numerical studies and an application to functional magnetic resonance imaging data.

stat.ME↗

arXiv2025

Continuum Transformers Perform In-Context Learning by Operator Gradient Descent

Transformers robustly exhibit the ability to perform in-context learning, whereby their predictive accuracy on a task can increase not by parameter updates but merely with the placement of training samples in their context windows. Recent works have shown that transformers achieve this by implementing gradient descent in their forward passes. Such results, however, are restricted to standard transformer architectures, which handle finite-dimensional inputs. In the space of PDE surrogate modeling, a generalization of transformers to handle infinite-dimensional function inputs, known as "continuum transformers," has been proposed and similarly observed to exhibit in-context learning. Despite impressive empirical performance, such in-context learning has yet to be theoretically characterized. We herein demonstrate that continuum transformers perform in-context operator learning by performing gradient descent in an operator RKHS. We demonstrate this using novel proof strategies that leverage a generalized representer theorem for Hilbert spaces and gradient flows over the space of functionals of a Hilbert space. We additionally show the operator learned in context is the Bayes Optimal Predictor in the infinite depth limit of the transformer. We then provide empirical validations of this optimality result and demonstrate that the parameters under which such gradient descent is performed are recovered through the continuum transformer training.

stat.ML↗