In reduced-order modeling, complex systems that exhibit high state-space dimensionality are described and evolved using a small number of parameters. These parameters can be obtained in a data-driven way, where a high-dimensional dataset is projected onto a lower-dimensional basis. A complex system is then restricted to states on a low-dimensional manifold where it can be efficiently modeled. While this approach brings computational benefits, obtaining a good quality of the manifold topology becomes a crucial aspect when models, such as nonlinear regression, are built on top of the manifold. Here, we present a quantitative metric for characterizing manifold topologies. Our metric pays attention to non-uniqueness and spatial gradients in physical quantities of interest, and can be applied to manifolds of arbitrary dimensionality. Using the metric as a cost function in optimization algorithms, we show that optimized low-dimensional projections can be found. We delineate a few applications of the cost function to datasets representing argon plasma, reacting flows and atmospheric pollutant dispersion. We demonstrate how the cost function can assess various dimensionality reduction and manifold learning techniques as well as data preprocessing strategies in their capacity to yield quality low-dimensional projections. We show that improved manifold topologies can facilitate building nonlinear regression models.
- Award ID(s):
- 1632738
- NSF-PAR ID:
- 10025813
- Date Published:
- Journal Name:
- arXiv.org
- ISSN:
- 2331-8422
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
Abstract -
Chambers, Erin W. ; Gudmundsson, Joachim (Ed.)Datasets with non-trivial large scale topology can be hard to embed in low-dimensional Euclidean space with existing dimensionality reduction algorithms. We propose to model topologically complex datasets using vector bundles, in such a way that the base space accounts for the large scale topology, while the fibers account for the local geometry. This allows one to reduce the dimensionality of the fibers, while preserving the large scale topology. We formalize this point of view and, as an application, we describe a dimensionality reduction algorithm based on topological inference for vector bundles. The algorithm takes as input a dataset together with an initial representation in Euclidean space, assumed to recover part of its large scale topology, and outputs a new representation that integrates local representations obtained through local linear dimensionality reduction. We demonstrate this algorithm on examples coming from dynamical systems and chemistry. In these examples, our algorithm is able to learn topologically faithful embeddings of the data in lower target dimension than various well known metric-based dimensionality reduction algorithms.more » « less
-
Abstract Visualizing spatial assay data in anatomical images is vital for understanding biological processes in cell, tissue, and organ organizations. Technologies requiring this functionality include traditional one-at-a-time assays, and bulk and single-cell omics experiments, including RNA-seq and proteomics. The spatialHeatmap software provides a series of powerful new methods for these needs, and allows users to work with adequately formatted anatomical images from public collections or custom images. It colors the spatial features (e.g. tissues) annotated in the images according to the measured or predicted abundance levels of biomolecules (e.g. mRNAs) using a color key. This core functionality of the package is called a spatial heatmap plot. Single-cell data can be co-visualized in composite plots that combine spatial heatmaps with embedding plots of high-dimensional data. The resulting spatial context information is essential for gaining insights into the tissue-level organization of single-cell data, or vice versa. Additional core functionalities include the automated identification of biomolecules with spatially selective abundance patterns and clusters of biomolecules sharing similar abundance profiles. To appeal to both non-expert and computational users, spatialHeatmap provides a graphical and a command-line interface, respectively. It is distributed as a free, open-source Bioconductor package (https://bioconductor.org/packages/spatialHeatmap) that users can install on personal computers, shared servers, or cloud systems.
-
null (Ed.)Supervised dimensionality reduction for sequence data learns a transformation that maps the observations in sequences onto a low-dimensional subspace by maximizing the separability of sequences in different classes. It is typically more challenging than conventional dimensionality reduction for static data, because measuring the separability of sequences involves non-linear procedures to manipulate the temporal structures. In this paper, we propose a linear method, called Order-preserving Wasserstein Discriminant Analysis (OWDA), and its deep extension, namely DeepOWDA, to learn linear and non-linear discriminative subspace for sequence data, respectively. We construct novel separability measures between sequence classes based on the order-preserving Wasserstein (OPW) distance to capture the essential differences among their temporal structures. Specifically, for each class, we extract the OPW barycenter and construct the intra-class scatter as the dispersion of the training sequences around the barycenter. The inter-class distance is measured as the OPW distance between the corresponding barycenters. We learn the linear and non-linear transformations by maximizing the inter-class distance and minimizing the intra-class scatter. In this way, the proposed OWDA and DeepOWDA are able to concentrate on the distinctive differences among classes by lifting the geometric relations with temporal constraints. Experiments on four 3D action recognition datasets show the effectiveness of OWDA and DeepOWDA.more » « less
-
Supervised dimensionality reduction for sequence data projects the observations in sequences onto a low-dimensional subspace to better separate different sequence classes. It is typically more challenging than conventional dimensionality reduction for static data, because measuring the separability of sequences involves non-linear procedures to manipulate the temporal structures. This paper presents a linear method, namely Order-preserving Wasserstein Discriminant Analysis (OWDA), which learns the projection by maximizing the inter-class distance and minimizing the intra-class scatter. For each class, OWDA extracts the order-preserving Wasserstein barycenter and constructs the intra-class scatter as the dispersion of the training sequences around the barycenter. The inter-class distance is measured as the order-preserving Wasserstein distance between the corresponding barycenters. OWDA is able to concentrate on the distinctive differences among classes by lifting the geometric relations with temporal constraints. Experiments show that OWDA achieves competitive results on three 3D action recognition datasets.more » « less