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The small-scale structure of the fluctuating passive scalar field in a turbulent boundary layer
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Linear-Size Approximations to the Vietoris-Rips Filtration
The Vietoris-Rips filtration is a versatile tool in topological data
analysis. It is a sequence of simplicial complexes built on a metric space to
add topological structure to an otherwise disconnected set of points. It is
widely used because it encodes useful information about the topology of the
underlying metric space. This information is often extracted from its so-called
persistence diagram. Unfortunately, this filtration is often too large to
construct in full. We show how to construct an O(n)-size filtered simplicial
complex on an -point metric space such that its persistence diagram is a
good approximation to that of the Vietoris-Rips filtration. This new filtration
can be constructed in time. The constant factors in both the size
and the running time depend only on the doubling dimension of the metric space
and the desired tightness of the approximation. For the first time, this makes
it computationally tractable to approximate the persistence diagram of the
Vietoris-Rips filtration across all scales for large data sets.
We describe two different sparse filtrations. The first is a zigzag
filtration that removes points as the scale increases. The second is a
(non-zigzag) filtration that yields the same persistence diagram. Both methods
are based on a hierarchical net-tree and yield the same guarantees
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