| One of the challenges for computer scientists and information scientists is how to display large-scale, multidimensional datasets on a computer screen in a manner that demonstrates the underlying data relationships. The broad area of study concerned with data placement is layout, where the objective is to create a mapping of h samples from a sample set S to representative points in a representation set R, so that similar samples in S are mapped close together in R. Most layout methods move points around in R in order to attempt to minimize an error function that describes how accurately proximity values in S are related to distances in R (for all pairs). Unfortunately, these layouts methods, generally, have run a time complexity of O(h2), where h is the number of samples. If we define faithfulness as preserving distances precisely, then the traditional methods also, generally, do not map samples faithfully in the degenerate case of a one and two-dimensional data set in S. We developed a noniterative layout method where samples in S are mapped into R based on the centroid of their geometric relationships to a set of user selected and placed anchors. The time complexity of this Projective Mapping method is O (kh) (where k is the number of user placed samples in R that are used for the mapping and h is the number of samples). The advantage of this method is that it preserves faithfulness in the degenerate cases. More precisely, our Projective Mapping method maps samples faithfully when the references are mapped faithfully for x-dimensional subspace of an n-dimensional space mapped into an x-dimensional reference space, when x ≤ m. |
