A Novel Approach to Embedding of Metric Spaces

An embedding of one metric space (X, d) into another (Y, ρ) is an injective map f : X → Y . The central genre of problems in the area of metric embedding is finding such maps in which the distances between points do not change “too much”. Metric Embedding plays an important role in a vast range of application areas such as computer vision, computational biology, machine learning, networking, statistics, and mathematical psychology, to name a few. The mathematical theory of metric embedding is well studied in both pure and applied analysis and has more recently been a source of interest for computer scientists as well. Most of this work is focused on the development of bi-Lipschitz mappings between metric spaces. In this work we present new concepts in metric embeddings as well as new embedding methods for metric spaces. We focus on finite metric spaces, however some of the concepts and methods may be applicable in other settings as well. One of the main cornerstones in finite metric embedding theory is a celebrated theorem of Bourgain which states that every finite metric space on n points embeds in Euclidean space with O(log n) distortion. Bourgain’s result is best possible when considering the worst case distortion over all pairs of points in the metric space. Yet, it is natural to ask: can an embedding do much better in terms of the average distortion? Indeed, in most practical applications of metric embedding the main criteria for the quality of an embedding is its average distortion over all pairs. In this work we provide an embedding with constant average distortion for arbitrary metric spaces, while maintaining the same worst case bound provided by Bourgain’s theorem. In fact, our embedding possesses a much stronger property. We define the `q-distortion of a uniformly distributed pair of points. Our embedding achieves the best possible `q-distortion for all 1 ≤ q ≤ ∞ simultaneously. The results are based on novel embedding methods which do well in another aspect: the dimension of the host space into which we embed (usually Lp spaces). The dimension of an embedding is of very high importance in particular in applications and much effort has been invested in analyzing it. Our embedding methods yield a tight O(log n) dimension. In fact, they shed new light on another fundamental question in metric embedding, which is: whether the metric dimension of a metric space is related to its intrinsic dimension ? I.e., whether the dimension in which it can be embedded in some real normed space is related to the intrinsic dimension, which is captured by the inherent geometry of the metric space, measured by its doubling dimension. The existence of such an embedding, where the distortion depends only on the intrinsic dimension as well, was conjectured by Assouad and was later posed as an open problem by others. Our embeddings give the first positive result of this type showing that every finite metric space attains a low distortion (and constant average distortion) embedding in Euclidean space of dimension proportional to its doubling dimension. We also consider the basic problem of how well a tree can approximate the distances induced by a graph, of particular interest is the case where the tree is a spanning tree of the graph. Unfortunately, such approximation can suffer linear distortion in the worst case, even for very simple graphs. We show an embedding of any metric into a tree metric (in fact an ultrametric), and embed any weighted graph into a spanning tree both with constant average distortion.