On Dimension Reduction Mappings for Approximate Retrieval of Multi-dimensional Data

Abstract

Approximate retrieval of multi-dimensional data, such as documents, digital images, and audio clips, is a method to get objects within some dissimilarity from a given object. We assume a metric space containing objects, where distance is used to measure dissimilarity. In Euclidean metric spaces, approximate retrieval is easily and efficiently realized by a spatial indexing/access method R-tree. First, we consider objects in discrete L ₁ (or Manhattan distance) metric space, and present embedding method into Euclidean space for them. Then, we propose a projection mapping H-Map to reduce dimensionality of multi-dimensional data, which can be applied to any metric space such as L ₁ or L∞ metric space, as well as Euclidean space. H-Map does not require coordinates of data unlike K-L transformation. H-Map has an advantage in using spatial indexing such as R-tree because it is a continuous mapping from a metric space to an L∞ metric space, where a hyper-sphere is a hyper-cube in the usual sense. Finally we show that the distance function itself, which is simpler than H-Map, can be used as a dimension reduction mapping for any metric space.