  Understanding the response of an output variable to multi-dimensional inputs
lies at the heart of many data exploration endeavours. Topology-based methods,
in particular Morse theory and persistent homology, provide a useful framework
for studying this relationship, as phenomena of interest often appear naturally
as fundamental features. The Morse-Smale complex captures a wide range of
features by partitioning the domain of a scalar function into piecewise
monotonic regions, while persistent homology provides a means to study these
features at different scales of simplification. Previous works demonstrated how
to compute such a representation and its usefulness to gain insight into
multi-dimensional data. However, exploration of the multi-scale nature of the
data was limited to selecting a single simplification threshold from a plot of
region count. In this paper, we present a novel tree visualization that
provides a concise overview of the entire hierarchy of topological features.
The structure of the tree provides initial insights in terms of the
distribution, size, and stability of all partitions. We use regression analysis
to fit linear models in each partition, and develop local and relative measures
to further assess uniqueness and the importance of each partition, especially
with respect parents/children in the feature hierarchy. The expressiveness of
the tree visualization becomes apparent when we encode such measures using
colors, and the layout allows an unprecedented level of control over feature
selection during exploration. For instance, selecting features from multiple
scales of the hierarchy enables a more nuanced exploration. Finally, we
demonstrate our approach using examples from several scientific domains.
