We introduce a novel interactive framework for visualizing and exploring high-dimensional datasets based on subspace analysis and dynamic projections. We assume the high-dimensional dataset can be represented by a mixture of low-dimensional linear subspaces with mixed dimensions, and provide a method to reliably estimate the intrinsic dimension and linear basis of each subspace extracted from the subspace clustering. Subsequently, we use these bases to define unique 2D linear projections as viewpoints from which to visualize the data. To understand the relationships among the different projections and to discover hidden patterns, we connect these projections through dynamic projections that create smooth animated transitions between pairs of projections. We introduce the view transition graph, which provides flexible navigation among these projections to facilitate an intuitive exploration. Finally, we provide detailed comparisons with related systems, and use real-world examples to demonstrate the novelty and usability of our proposed framework.
Dimension reduction techniques are essential for feature selection and feature extraction of complex highdimensional data. These techniques, which construct low-dimensional representations of data, are typically geometrically motivated, computationally efficient and approximately preserve certain structural properties of the data. However, they are often used as black box solutions in data exploration and their results can be difficult to interpret. To assess the quality of these results, quality measures, such as co-ranking [LV09], have been proposed to quantify structural distortions that occur between high-dimensional and low-dimensional data representations. Such measures could be evaluated and visualized point-wise to further highlight erroneous regions [MLGH13]. In this work, we provide an interactive visualization framework for exploring high-dimensional data via its twodimensional embeddings obtained from dimension reduction, using a rich set of user interactions. We ask the following question: what new insights do we obtain regarding the structure of the data, with interactive manipulations of its embeddings in the visual space? We augment the two-dimensional embeddings with structural abstractions obtained from hierarchical clusterings, to help users navigate and manipulate subsets of the data. We use point-wise distortion measures to highlight interesting regions in the domain, and further to guide our selection of the appropriate level of clusterings that are aligned with the regions of interest. Under the static setting, point-wise distortions indicate the level of structural uncertainty within the embeddings. Under the dynamic setting, on-thefly updates of point-wise distortions due to data movement and data deletion reflect structural relations among different parts of the data, which may lead to new and valuable insights.
Linear projections are one of the most common approaches to visualize high-dimensional data. Since the space of possible projections is large, existing systems usually select a small set of interesting projections by ranking a large set of candidate projections based on a chosen quality measure. However, while highly ranked projections can be informative, some lower ranked ones could offer important complementary information. Therefore, selection based on ranking may miss projections that are important to provide a global picture of the data. The proposed work fills this gap by presenting the Grassmannian Atlas, a framework that captures the global structures of quality measures in the space of all projections, which enables a systematic exploration of many complementary projections and provides new insights into the properties of existing quality measures.
We demonstrate that highly oriented pyrolytic graphite is a potentially useful material for plasmonic applications in the terahertz (THz) spectral range. Using THz time-domain spectroscopy, we studied the transmission properties of a $7.5 lm thick graphite film made via mechanical exfoliation and found that the complex dielectric constant follows the Drude model with a plasma frequency p $ 34 THz. To assess the graphite plasmonic properties, we fabricated a subwavelength periodic aperture array in the graphite film and in a free-standing stainless steel film. Despite the low conductivity of graphite compared to conventional metals, the resonant transmission properties are similar. V
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