278 research outputs found
Parametrized Homology via Zigzag Persistence
This paper develops the idea of homology for 1-parameter families of
topological spaces. We express parametrized homology as a collection of real
intervals with each corresponding to a homological feature supported over that
interval or, equivalently, as a persistence diagram. By defining persistence in
terms of finite rectangle measures, we classify barcode intervals into four
classes. Each of these conveys how the homological features perish at both ends
of the interval over which they are defined
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
Homology and Robustness of Level and Interlevel Sets
Given a function f: \Xspace \to \Rspace on a topological space, we consider
the preimages of intervals and their homology groups and show how to read the
ranks of these groups from the extended persistence diagram of . In
addition, we quantify the robustness of the homology classes under
perturbations of using well groups, and we show how to read the ranks of
these groups from the same extended persistence diagram. The special case
\Xspace = \Rspace^3 has ramifications in the fields of medical imaging and
scientific visualization
Quantifying Transversality by Measuring the Robustness of Intersections
By definition, transverse intersections are stable under infinitesimal
perturbations. Using persistent homology, we extend this notion to a measure.
Given a space of perturbations, we assign to each homology class of the
intersection its robustness, the magnitude of a perturbations in this space
necessary to kill it, and prove that robustness is stable. Among the
applications of this result is a stable notion of robustness for fixed points
of continuous mappings and a statement of stability for contours of smooth
mappings
The Importance of Forgetting: Limiting Memory Improves Recovery of Topological Characteristics from Neural Data
We develop of a line of work initiated by Curto and Itskov towards
understanding the amount of information contained in the spike trains of
hippocampal place cells via topology considerations. Previously, it was
established that simply knowing which groups of place cells fire together in an
animal's hippocampus is sufficient to extract the global topology of the
animal's physical environment. We model a system where collections of place
cells group and ungroup according to short-term plasticity rules. In
particular, we obtain the surprising result that in experiments with spurious
firing, the accuracy of the extracted topological information decreases with
the persistence (beyond a certain regime) of the cell groups. This suggests
that synaptic transience, or forgetting, is a mechanism by which the brain
counteracts the effects of spurious place cell activity
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