  An ultrametric topology formalizes the notion of hierarchical structure. An
ultrametric embedding, referred to here as ultrametricity, is implied by a
natural hierarchical embedding. Such hierarchical structure can be global in
the data set, or local. By quantifying extent or degree of ultrametricity in a
data set, we show that ultrametricity becomes pervasive as dimensionality
and/or spatial sparsity increases. This leads us to assert that very high
dimensional data are of simple structure. We exemplify this finding through a
range of simulated data cases. We discuss also application to very high
frequency time series segmentation and modeling.
