PCAfold 2.0-Novel tools and algorithms for low-dimensional manifold assessment and optimization

We describe an update to our open-source Python package, PCAfold, designed to help researchers generate, analyze and improve low-dimensional data manifolds. In the current version, PCAfold 2.0, we introduce novel tools and algorithms for assessing and optimizing low-dimensional manifolds. This includes a method that generates a mapof local feature sizes that can help pinpoint researchers to problematic regions on a manifold. We introduce a novel cost function that characterizes the quality of a manifold topology with a single number. We develop two algorithms for feature selection based on principal component analysis (PCA) that use the cost function as an objective function to minimize. We introduce a quantity of interest (QoI)-aware dimensionality reduction strategy where data projections are computed using an artificial neural network and are directly optimized towards representing various projection-independent and projection-dependent QoIs. We also introduce an implementation of partition of unity networks (POUnets) for efficient reconstruction of QoIs from low-dimensional manifolds based on combining neural network classification with localized polynomial regression. Our software can be broadly applicable in all domains of science and engineering that aim to reduce data dimensionality, as well as in the fundamental research on representation learning.& COPY; 2023 The Author(s). Published by Elsevier B.V. All rights reserved.

Automated summary

We present an update to our open-source Python package, PCAfold, which assists researchers in generating, analyzing, and improving low-dimensional data manifolds. The new version, PCAfold 2.0, introduces innovative tools and algorithms for evaluating and optimizing low-dimensional manifolds. These include a method for generating a map of local feature sizes to identify problematic regions, a novel cost function for characterizing manifold topology, and two feature selection algorithms based on principal component analysis. Additionally, we propose a dimensionality reduction strategy that considers the quantity of interest (QoI) and an implementation of partition of unity networks (POUnets) for efficient reconstruction of QoIs from low-dimensional manifolds.