We investigate the impact of multiscale data (large variations in scale across different directions) on machine learning algorithms and reveal multiscale structures in the loss landscape. We also introduce a novel gradient descent algorithm for this scenario adopting a combination of large and small learning rates, and provide convergence guarantees.
2023
Linear Regression on Manifold Structured Data: the Impact of Extrinsic Geometry on Solutions
We study linear regression applied to data structured on a manifold. We reveal the impact of the data manifold’s extrinsic geometry on the regression by deriving explicit solution formulas and numerical experiments. Our findings reveal the role of data manifold geometry in ensuring the stability of regression models for out-of-distribution inferences.
2019
Nearest neighbor sampling of point sets using rays (RaySense)
Liangchen Liu, Louis Ly, Colin Macdonald, and 1 more author
We propose a new framework for the sampling, compression, and analysis of distributions of point sets and other geometric objects. We provide theoretical explanations to its properties along with guarantees for line integral approximation
