38,370 research outputs found
On the state space geometry of the Kuramoto-Sivashinsky flow in a periodic domain
The continuous and discrete symmetries of the Kuramoto-Sivashinsky system
restricted to a spatially periodic domain play a prominent role in shaping the
invariant sets of its chaotic dynamics. The continuous spatial translation
symmetry leads to relative equilibrium (traveling wave) and relative periodic
orbit (modulated traveling wave) solutions. The discrete symmetries lead to
existence of equilibrium and periodic orbit solutions, induce decomposition of
state space into invariant subspaces, and enforce certain structurally stable
heteroclinic connections between equilibria. We show, on the example of a
particular small-cell Kuramoto-Sivashinsky system, how the geometry of its
dynamical state space is organized by a rigid `cage' built by heteroclinic
connections between equilibria, and demonstrate the preponderance of unstable
relative periodic orbits and their likely role as the skeleton underpinning
spatiotemporal turbulence in systems with continuous symmetries. We also offer
novel visualizations of the high-dimensional Kuramoto-Sivashinsky state space
flow through projections onto low-dimensional, PDE representation independent,
dynamically invariant intrinsic coordinate frames, as well as in terms of the
physical, symmetry invariant energy transfer rates.Comment: 31 pages, 17 figures; added references, corrected typos. Due to file
size restrictions some figures in this preprint are of low quality. A high
quality copy may be obtained from
http://www.cns.gatech.edu/~predrag/papers/preprints.html#rp
Path finding on a spherical self-organizing map using distance transformations
Spatialization methods create visualizations that allow users to analyze high-dimensional data in an intuitive manner and facilitates the extraction of meaningful information. Just as geographic maps are simpli ed representations of geographic spaces, these visualizations are esssentially maps of abstract data spaces that are created through dimensionality reduction. While we are familiar with geographic maps for path planning/ nding applications, research into using maps of high-dimensional spaces for such purposes has been largely ignored. However, literature has shown that it is possible to use these maps to track temporal and state changes within a high-dimensional space. A popular dimensionality reduction method that produces a mapping for these purposes is the Self-Organizing Map. By using its topology preserving capabilities with a colour-based visualization method known as the U-Matrix, state transitions can be visualized as trajectories on the resulting mapping. Through these trajectories, one can gather information on the transition path between two points in the original high-dimensional state space. This raises the interesting question of whether or not the Self-Organizing Map can be used to discover the transition path between two points in an n-dimensional space. In this thesis, we use a spherically structured Self-Organizing Map called the Geodesic Self-Organizing Map for dimensionality reduction and the creation of a topological mapping that approximates the n-dimensional space. We rst present an intuitive method for a user to navigate the surface of the Geodesic SOM. A new application of the distance transformation algorithm is then proposed to compute the path between two points on the surface of the SOM, which corresponds to two points in the data space. Discussions will then follow on how this application could be improved using some form of surface shape analysis. The new approach presented in this thesis would then be evaluated by analyzing the results of using the Geodesic SOM for manifold embedding and by carrying out data analyses using carbon dioxide emissions data
Path finding on a spherical self-organizing map using distance transformations
Spatialization methods create visualizations that allow users to analyze high-dimensional data in an intuitive manner and facilitates the extraction of meaningful information. Just as geographic maps are simpli ed representations of geographic spaces, these visualizations are esssentially maps of abstract data spaces that are created through dimensionality reduction. While we are familiar with geographic maps for path planning/ nding applications, research into using maps of high-dimensional spaces for such purposes has been largely ignored. However, literature has shown that it is possible to use these maps to track temporal and state changes within a high-dimensional space. A popular dimensionality reduction method that produces a mapping for these purposes is the Self-Organizing Map. By using its topology preserving capabilities with a colour-based visualization method known as the U-Matrix, state transitions can be visualized as trajectories on the resulting mapping. Through these trajectories, one can gather information on the transition path between two points in the original high-dimensional state space. This raises the interesting question of whether or not the Self-Organizing Map can be used to discover the transition path between two points in an n-dimensional space. In this thesis, we use a spherically structured Self-Organizing Map called the Geodesic Self-Organizing Map for dimensionality reduction and the creation of a topological mapping that approximates the n-dimensional space. We rst present an intuitive method for a user to navigate the surface of the Geodesic SOM. A new application of the distance transformation algorithm is then proposed to compute the path between two points on the surface of the SOM, which corresponds to two points in the data space. Discussions will then follow on how this application could be improved using some form of surface shape analysis. The new approach presented in this thesis would then be evaluated by analyzing the results of using the Geodesic SOM for manifold embedding and by carrying out data analyses using carbon dioxide emissions data
Image-Based Visualization of Classifier Decision Boundaries
Understanding how a classifier partitions a high-dimensional input space and assigns labels to the parts is an important task in machine learning. Current methods for this task mainly use color-coded sample scatterplots, which do not explicitly show the actual decision boundaries or confusion zones. We propose an image-based technique to improve such visualizations. The method samples the 2D space of a dimensionality-reduction projection and color-code relevant classifier outputs, such as the majority class label, the confusion, and the sample density, to render a dense depiction of the high-dimensional decision boundaries. Our technique is simple to implement, handles any classifier, and has only two simple-to-control free parameters. We demonstrate our proposal on several real-world high-dimensional datasets, classifiers, and two different dimensionality reduction methods
Unsupervised visualization of image datasets using contrastive learning
Visualization methods based on the nearest neighbor graph, such as t-SNE or
UMAP, are widely used for visualizing high-dimensional data. Yet, these
approaches only produce meaningful results if the nearest neighbors themselves
are meaningful. For images represented in pixel space this is not the case, as
distances in pixel space are often not capturing our sense of similarity and
therefore neighbors are not semantically close. This problem can be
circumvented by self-supervised approaches based on contrastive learning, such
as SimCLR, relying on data augmentation to generate implicit neighbors, but
these methods do not produce two-dimensional embeddings suitable for
visualization. Here, we present a new method, called t-SimCNE, for unsupervised
visualization of image data. T-SimCNE combines ideas from contrastive learning
and neighbor embeddings, and trains a parametric mapping from the
high-dimensional pixel space into two dimensions. We show that the resulting 2D
embeddings achieve classification accuracy comparable to the state-of-the-art
high-dimensional SimCLR representations, thus faithfully capturing semantic
relationships. Using t-SimCNE, we obtain informative visualizations of the
CIFAR-10 and CIFAR-100 datasets, showing rich cluster structure and
highlighting artifacts and outliers
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