Abstract In this article, we describe techniques for visual data mining based on differential topology. Data scientists have been working long on the analysis of data obtained from a wide variety of sources. The data is often represented as discrete sample points of a function R n → R m $oldsymbol {R}^{n} ightarrow oldsymbol {R}^{m}$ , while the dimensions of the data domain and range have rapidly increased due to recent advancement in computational power and measurement technology. Mathematical formulations of differential topology effectively help us to analyze such data in a hierarchical fashion and to visually extract significant features from it. We present new algorithms and application examples as well as existing ones, including the authors’ recent results, so that we can fully elucidate the potential power of this approach especially in data visualization.
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