Statistical and computational methodology for visualizing high-dimensional digital data

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Canadians have access to enormous numbers of digital data objects in the form of documents, web sites, images, sound, and video. A set of instances of any one of these forms may be represented as a set of data points in a high-dimensional space (possibly 10s or 100s or 1000s of dimensions); the structure of the relative positions of these points in the space matches conceptual relationships between the original objects. The proposed research is directed towards the development of statistical methodology and software which will allow one to meaningfully explore any such collection by presenting it visually as a sequence of low-dimensional views (e.g. 1d, 2d, 3d, or even 4d). These views are arranged as if in a roadmap where each stop is a low dimensional view and the road between a smooth transition from one view to another -- the corresponding mathematical object is a graph with nodes (stops) and edges (roads) connecting them. This research is directed toward identifying interesting stops, roads, road trips, and entire roadmaps which, when explored, will reveal unanticipated structure in the set of digital data objects. The research is directed toward determining how best to construct new, though fewer (50 or fewer say), dimensions with which to represent the data, then from these how to choose pairs (or triples) of dimensions that reveal the most interesting structure, and finally how best to display such structure and to provide an interactive roadmap and other software tools so that the user may actively explore the visual structure presented and meaningfully relate it to the original objects. The research will also be developed for application to more abstract digital data objects such as statistical models of varying complexity (or dimensions), and to particular machine learning tasks. The latter include combining algorithmic methods with interactive visualization methods to better enable discovery or confirmation of clusters in the data, to develop rules capable of classifying new objects to known classes, and to do both together when the classes are known for some objects but not for others (semi-supervised learning).