Georgia Institute of Technology College of Sciences

This talk explores how to construct meaningful features from noisy, high-dimensional data by leveraging geometric and invariant structures. First, we introduce a geometric framework for dimension reduction using a power-weighted path metric, which effectively de-noises high-dimensional data while preserving its intrinsic geometric structure. This framework is particularly useful for analyzing single-cell RNA data and for multi-manifold clustering, and we provide theoretical guarantees for the convergence of the associated graph Laplacian operators. We then turn to the problem of constructing features invariant to group actions in the multi-reference alignment (MRA) data model. In this setting one has many noisy observation of a hidden signal corrupted by both a group action(s) and additive noise, and one wants to recover the hidden signal from the noisy data. By formulating MRA in function space, we uncover a new connection to deconvolution: the hidden signal can be recovered from second-order Fourier statistics via an approach analogous to Kotlarski’s identity. We extend this identity to general dimensions, analyze recovery in the presence of vanishing Fourier transforms, and validate the resulting deconvolution framework with both theoretical guarantees and numerical experiments.