Will lecture on
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. In this talk, I will discuss the application of these 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 solution of the inverse problem aimed at recovering a single rigid structure, while many important macromolecules contain several parts that can move in a continuous fashion, thus forming a manifold of conformations. The methods described in this talk present progress towards the solution of this grand challenge, namely the extension of reconstruction methods which estimate a single 3D structure to methods that recover entire manifolds of conformations.