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Optimal Bounds for Johnson-Lindenstrauss Transformations
In 1984, Johnson and Lindenstrauss proved that any finite set of data in...
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Burning sage: Reversing the curse of dimensionality in the visualization of high-dimensional data
In high-dimensional data analysis the curse of dimensionality reasons th...
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Dimension Reduction with Non-degrading Generalization
Visualizing high dimensional data by projecting them into two or three d...
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Capacity Preserving Mapping for High-dimensional Data Visualization
We provide a rigorous mathematical treatment to the crowding issue in da...
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Nonlinear Quality of Life Index
We present details of the analysis of the nonlinear quality of life inde...
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A Distance-preserving Matrix Sketch
Visualizing very large matrices involves many formidable problems. Vario...
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Improving the Projection of Global Structures in Data through Spanning Trees
The connection of edges in a graph generates a structure that is indepen...
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TopoMap: A 0-dimensional Homology Preserving Projection of High-Dimensional Data
Multidimensional Projection is a fundamental tool for high-dimensional data analytics and visualization. With very few exceptions, projection techniques are designed to map data from a high-dimensional space to a visual space so as to preserve some dissimilarity (similarity) measure, such as the Euclidean distance for example. In fact, although adopting distinct mathematical formulations designed to favor different aspects of the data, most multidimensional projection methods strive to preserve dissimilarity measures that encapsulate geometric properties such as distances or the proximity relation between data objects. However, geometric relations are not the only interesting property to be preserved in a projection. For instance, the analysis of particular structures such as clusters and outliers could be more reliably performed if the mapping process gives some guarantee as to topological invariants such as connected components and loops. This paper introduces TopoMap, a novel projection technique which provides topological guarantees during the mapping process. In particular, the proposed method performs the mapping from a high-dimensional space to a visual space, while preserving the 0-dimensional persistence diagram of the Rips filtration of the high-dimensional data, ensuring that the filtrations generate the same connected components when applied to the original as well as projected data. The presented case studies show that the topological guarantee provided by TopoMap not only brings confidence to the visual analytic process but also can be used to assist in the assessment of other projection methods.
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