Properties of Algorithmic Information Distance

The domain-independent universal Normalized Information Distance based on Kolmogorov complexity has been (in approximate form) successfully applied to a variety of difficult clustering problems. In this paper we investigate theoretical properties of the un-normalized algorithmic information distance $d_{K}$ . The main question we are asking in this work is what properties this curious distance has, besides being a metric. We show that many (in)finite-dimensional spaces can(not) be isometrically scale-embedded into the space of finite strings with metric $d_{K}$ . We also show that $d_{K}$ is not an Euclidean distance, but any finite set of points in Euclidean space can be scale-embedded into $(\{0,1\}^{*},d_{K})$ . A major contribution is the development of the necessary framework and tools for finding more (interesting) properties of $d_{K}$ in future, and to state several open problems.