Searching for the embedded manifolds in high- dimensional data, problems and unsolved questions

Starting from a recall of several classical - and less classical - remarks about high dimensional data spaces, this paper gives a bird's eye view over various techniques of data reduction, from linear multidimensional scaling to non-linear and non-parametric methods. Two kinds of approaches will be presented, the first one operating in the feature space, the second one operating in the dissimilarity space. A special attention will be devoted to the CCA algorithm, in a version which aims at capturing the mean manifold spanned by the data vectors. Some examples from artificial and real data are given.