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Time, temperature, and data cloud geometry.

We demonstrate that the geometry of a data cloud is computable on multiple scales without prior knowledge about its structure. We show that the concepts of "time" and "temperature" are beneficial for constructing a hierarchical geometry based on local information provided by a similarity measure. We design two devices for construction of this hierarchy. Along the time axis, a regulated random walk incorporated with recurrence-time dynamics detects information about the number of clusters and the corresponding cluster membership of individual data nodes. Along the temperature axis we build the geometric hierarchy of a data cloud, which consists of only a few phase transitions. The base level of the hierarchy especially exhibits the intrinsic data structure. At each chosen temperature, we form an ensemble matrix that summarizes information extracted from many regulated random walks. This device constitutes the basis for constructing one corresponding level of the hierarchy by means of spectral clustering. We illustrate the construction of such geometric hierarchies using simulated and real data.