dorsal/arxiv
View SchemaHilbert Space Becomes Ultrametric in the High Dimensional Limit: Application to Very High Frequency Data Analysis
| Authors | Fionn Murtagh |
|---|---|
| Categories | |
| ArXiv ID | physics/0702064 |
| URL | https://arxiv.org/abs/physics/0702064 |
Abstract
An ultrametric topology formalizes the notion of hierarchical structure. An ultrametric embedding, referred to here as ultrametricity, is implied by a natural 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.
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"abstract": "An ultrametric topology formalizes the notion of hierarchical structure. An\nultrametric embedding, referred to here as ultrametricity, is implied by a\nnatural hierarchical embedding. Such hierarchical structure can be global in\nthe data set, or local. By quantifying extent or degree of ultrametricity in a\ndata set, we show that ultrametricity becomes pervasive as dimensionality\nand/or spatial sparsity increases. This leads us to assert that very high\ndimensional data are of simple structure. We exemplify this finding through a\nrange of simulated data cases. We discuss also application to very high\nfrequency time series segmentation and modeling.",
"arxiv_id": "physics/0702064",
"authors": [
"Fionn Murtagh"
],
"categories": [
"physics.data-an"
],
"title": "Hilbert Space Becomes Ultrametric in the High Dimensional Limit: Application to Very High Frequency Data Analysis",
"url": "https://arxiv.org/abs/physics/0702064"
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