Non-Euclidean Internet Coordinates Embedding For many applications it is desirable to be able to estimate latency in a decentralised network when it is not practical to explicitly measure it. It has previously been shown that latency can be approximated by assigning hosts coordinates in some geometric space such that the Euclidean distance between two hosts in this space is equivalent to latency, a method known as a Network Coordinate (NC) system. This is commonly achieved by a large scale distributed optimisation which seeks to minimise the error between latency and Euclidean distance. In this work we challenge the assumption of Euclidean space as a satisfactory model for embedding Internet-like networks, due to the curved nature of network distances. We present a novel distributed optimisation methodology: Non-Euclidean Internet Coordinates Embedding (NICE). NICE uses a polynomial regression model to explicitly learn the most effective distance function for latency estimation within a geometric space, in addition to a distributed non linear dimensionality reduction method. Dimensionality reduction is achieved via a variant of Landmark Multi Dimensional Scaling (LMDS) and a distributed optimisation algorithm. This allows the distributed system to create a set of coordinates for each of the participating hosts that can be used to accurately estimate latency. The system is implemented within the Java based PeerSim networksimulator using both real and artificially generated input topologies and then compared to two of the most widely implemented NC systems: GNP and Vivaldi. By experimental simulation we show that NICE is significantly more accurate than either method while still remaining robust in the face of real network conditions.