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In this paper we present a theoretical foundation for a representation of adata set as a measure in a very large hierarchically parametrized family ofpositive measures, whose parameters can be computed explicitly (rather thanestimated by optimization), and illustrate its applicability to a wide range ofdata types. The pre-processing step then consists of representing data sets assimple measures. The theoretical foundation consists of a dyadic productformula representation lemma, a visualization theorem. We also define anadditive multiscale noise model which can be used to sample from dyadicmeasures and a more general multiplicative multiscale noise model which can beused to perturb continuous functions, Borel measures, and dyadic measures. Thefirst two results are based on theorems. The representation uses the verysimple concept of a dyadic tree, and hence is widely applicable, easilyunderstood, and easily computed. Since the data sample is represented as ameasure, subsequent analysis can exploit statistical and measure theoreticconcepts and theories. Because the representation uses the very simple conceptof a dyadic tree defined on the universe of a data set and the parameters aresimply and explicitly computable and easily interpretable and visualizable, wehope that this approach will be broadly useful to mathematicians,statisticians, and computer scientists who are intrigued by or involved in datascience including its mathematical foundations.

Product Formalisms for Measures on Spaces with Binary Tree Structures: Representation, Visualization, and Multiscale Noise