On Embedding of Finite Metric Spaces into Hilbert Space

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 main criteria for the quality of an embedding is its average distortion over all pairs. A celebrated theorem of Bourgain 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, is this the case for the average distortion ? Our main result is a strengthening of Bourgain’s theorem providing an embedding with constant average distortion for arbitrary metric spaces. 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.