  An ultrametric topology formalizes the notion of hierarchical structure. An
ultrametric embedding, referred to here as ultrametricity, is implied by a
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.
