Summarization and Search Over Geometric Spaces

The last decade has seen an explosion in the amount of data being generated, in part due to the prevalence of image and video sensors. As a result, searching through these data for relevant information or even getting a gist of the data is increasingly becoming difficult. The task is further complicated when the data have a non-Euclidean geometric interpretation. In this chapter, we address these limitations by discussing techniques to (a) summarize the data and (b) search through the data to find the nearest neighbor, in the general case of data lying on non-Euclidean manifolds. First, we consider the “precis” problem of sampling K representative yet diverse data points from a large data set. We formulate a general theory which encompasses not only traditional techniques devised for vector spaces but also non-Euclidean manifolds, thereby enabling these techniques for shapes, human activities, textures, and many other image and video-based data sets. We discuss the intrinsic manifold measures for measuring the quality of a selection of points with respect to their representative power, and their diversity. We also extend our formulation to the infinite-dimensional manifolds. We then address the problem of nearest-neighbor search in curved spaces. Towards this end, we discuss geodesic hashing which employs intrinsic geodesic-based functions to hash the data for realizing approximate but fast nearest-neighbor retrieval. The proposed family of hashing functions, although intrinsic, is optimally selected to empirically satisfy the locality sensitive hashing property.