Parallel coordinate plots (PCPs) are among the most useful techniques for the visualization and exploration of high-dimensional data spaces. They are especially useful for the representation of correlations among the dimensions, which identify relationships and interdependencies between variables. However, within these high-dimensional spaces, PCPs face difficulties in displaying the correlation between combinations of dimensions and generally require additional display space as the number of dimensions increases. In this paper, we present a new technique for high-dimensional data visualization in which a set of low-dimensional PCPs are interactively constructed by sampling user-selected subsets of the high-dimensional data space. In our technique, we first construct a graph visualization of sets of well-correlated dimensions. Users observe this graph and are able to interactively select the dimensions by sampling from its cliques, thereby dynamically specifying the most relevant lower dimensional data to be used for the construction of focused PCPs. Our interactive sampling overcomes the shortcomings of the PCPs by enabling the visualization of the most meaningful dimensions (i.e., the most relevant information) from high-dimensional spaces. We demonstrate the effectiveness of our technique through two case studies, where we show that the proposed interactive low-dimensional space constructions were pivotal for visualizing the high-dimensional data and discovering new patterns.
Several recent studies have applied SPs for the representation of in which each dot in the SP represents a single dimension in the space. @cite_21 @cite_41 presented a dual SP model to visualize both the items and dimensions spaces. Similarly, @cite_29 presented a interactive mechanism to select low-dimensional subspaces on the SP display in which each dot corresponds to a different dimension. We also represent the relationships among the dimensions in a 2D space; however, our technique applies a graph rather than a SP.
@cite_41 @cite_29 @cite_21
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We prove strong lower bounds for the space complexity of ( spl epsi , spl delta )-approximating the number of distinct elements F sub 0 in a data stream. Let m be the size of the universe from which the stream elements are drawn. We show that any one-pass streaming algorithm for ( spl epsi , spl delta )-approximating F sub 0 must use spl Omega (1 spl epsi sup 2 ) space when spl epsi = spl Omega (m sup -1 (9 + k) ), for any k > 0, improving upon the known lower bound of spl Omega (1 spl epsi ) for this range of spl epsi . This lower bound is tight up to a factor of log log m for small spl epsi and log 1 spl epsi for large spl epsi . Our lower bound is derived from a reduction from the one-way communication complexity of approximating a Boolean function in Euclidean space. The reduction makes use of a low-distortion embedding from an l sub 2 to l sub 1 norm. High-dimensional data visualization is receiving increasing interest because of the growing abundance of highdimensional datasets. To understand such datasets, visualization of the structures present in the data, such as clusters, can be an invaluable tool. Structures may be present in the full high-dimensional space, as well as in its subspaces. Two widely used methods to visualize high-dimensional data are the scatter plot matrix (SPM) and the parallel coordinate plot (PCP). SPM allows a quick overview of the structures present in pairwise combinations of dimensions. On the other hand, PCP has the potential to visualize not only bi-dimensional structures but also higher dimensional ones. A problem with SPM is that it suffers from crowding and clutter which makes interpretation hard. Approaches to reduce clutter are available in the literature, based on changing the order of the dimensions. However, usually this reordering has a high computational complexity. For effective visualization of high-dimensional structures, also PCP requires a proper ordering of the dimensions. In this paper, we propose methods for reordering dimensions in PCP in such a way that high-dimensional structures (if present) become easier to perceive. We also present a method for dimension reordering in SPM which yields results that are comparable to those of existing approaches, but at a much lower computational cost. Our approach is based on finding relevant subspaces for clustering using a quality criterion and cluster information. The quality computation and cluster detection are done in image space, using connected morphological operators. We demonstrate the potential of our approach for synthetic and astronomical datasets, and show that our method compares favorably with a number of existing approaches. In this paper, we propose to model an action based on both the shape and the motion of the performing object. When the object performs an action in 3D, the points on the outer boundary of the object are projected as 2D (x, y) contour in the image plane. A sequence of such 2D contours with respect to time generates a spatiotemporal volume (STV) in (x, y, t), which can be treated as 3D object in the (x, y, t) space. We analyze STV by using the differential geometric surface properties to identify action descriptors capturing both spatial and temporal properties. A set of action descriptors is called an action sketch. The first step in our approach is to generate STV by solving the point correspondence problem between consecutive frames. The correspondences are determined using a two-step graph theoretical approach. After the STV is generated, actions descriptors are computed by analyzing the differential geometric properties of STV. Finally, using these descriptors, we perform action recognition, which is also formulated as graph theoretical problem. Several experimental results are presented to demonstrate our approach. Several state-of-the-art image representations consist in averaging local statistics computed from patch-level descriptors. It has been shown by that such average statistics suffer from two sources of variance. The first one comes from the fact that a finite set of local statistics are averaged. The second one is due to the variation in the proportion of object-dependent information between different images of the same class. For the problem of object classification, these sources of variance affect negatively the accuracy since they increase the overlap between class-conditional probabilities. Our goal is to include information about the spatial layout of images in image signatures based on average statistics. We show that the traditional approach to including the spatial layout - the spatial pyramid (SP) - increases the first source of variance while only weakly reducing the second one. We therefore propose two complementary approaches to account for the spatial layout which are compatible with our goal of variance reduction. The first one models the spatial layout in an image-independent manner (as is the case of the SP) while the second one adapts to the image content. A significant benefit of these approaches with respect to the SP is that they do not incur an increase of the image signature dimensionality. We show on PASCAL VOC 2007, 2008 and 2009 the benefits of our approach.