Nonparametric estimation of high-dimensional shape spaces with applications to structural biology

Over the last twenty years, there have been major advances in non-linear dimensionality reduction, or manifold learning, and nonparametric regression of high-dimensional datasets with low intrinsic dimensionality.  A key idea in this field is the use of data-dependent Fourier-like basis vectors given by the eigenvectors of a graph Laplacian.  These eigenvectors provide a natural basis for representing and estimating smooth signals. Their use for estimation over arbitrary domains generalizes the classical notion of regression using orthogonal function series. In this talk, I will discuss the application of such methods for mapping spaces of volumetric shapes with continuous motion. Three lines of research will be presented:

(i) High-dimensional nonparametric estimation of distributions of volumetric signals from noisy linear measurements.

(ii) Leveraging the Wasserstein optimal transport metric for manifold learning and clustering.

(iii) Non-linear independent component analysis for analyzing independent motions.

A key motivation for this work comes from structural biology, where breakthrough advances in cryo-electron microscopy have led to thousands of atomic-resolution reconstructions of various proteins in their native states.  However, the success of this field has been mostly limited to the estimation of rigid structures, while many important macromolecules contain several parts that can move in a continuous fashion, thus forming a manifold of conformations which cannot be estimated using existing tools.  The methods described in this talk present progress towards the solution of this grand challenge, namely the extension of point-estimation methods which output a single 3D conformation to estimators of entire manifolds of conformations.