On the Computation of the Geodesic Distance with an Application to Dimensionality Reduction in a Neuro-Oncology Problem

Manifold learning models attempt to parsimoniously describe multivariate data through a low-dimensional manifold embedded in data space. Similarities between points along this manifold are often expressed as Euclidean distances. Previous research has shown that these similarities are better expressed as geodesic distances. Some problems concerning the computation of geodesic distances along the manifold have to do with time and storage restrictions related to the graph representation of the manifold. This paper provides different approaches to the computation of the geodesic distance and the implementation of Dijkstra’s shortest path algorithm, comparing their performances. The optimized procedures are bundled into a software module that is embedded in a dimensionality reduction method, which is applied to MRS data from human brain tumours. The experimental results show that the proposed implementation explains a high proportion of the data variance with a very small number of extracted features, which should ease the medical interpretation of subsequent results obtained from the reduced datasets.