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We provide a rigorous mathematical treatment to the crowding issue in datavisualization when high dimensional data sets are projected down to lowdimensions for visualization. By properly adjusting the capacity of highdimensional balls, our method makes right enough room to prepare for theembedding. A key component of the proposed method is an estimation of thecorrelation dimension at various scales which reflects the data densityvariation. The proposed adjustment to the capacity applies to any distance(Euclidean, geodesic, diffusion) and can potentially be used in many existingmethods to mitigate the crowding during the dimension reduction. We demonstratethe effectiveness of the new method using synthetic and real datasets.

Capacity Preserving Mapping for High-dimensional Data Visualization