ADTDA, lecture 2

\(\def\Real{\mathbb{R}}
\)

Topological Inference, continued

While NSW provides theoretical pipe to recovering an embedded manifold topology with large enough sample, its reliance on Čech complexes is (one of) the computational obstacles.

Vietoris-Rips complexes are much easier to compute, and they make sense in arbitrary metric spaces.

The caveats exist: unlike Čech complexes, they do not necessarily reflect faithfully the topology of the space covered by the metric balls: spurious homologies can emerge.

However, if the sample is dense enough, approximation guarantees with Vietoris-Rips complexes do exist: Hausmann, Latschev.

The biggest problem with the “sample from the manifold” model is the noise: the data points that are deviating from the underlying space, and present spurious topological patterns at all scales.

Persistent homology theory

An approach to address the spurious topological features without assuming a particular scale was proposed, under the name of “persistent homology”.

(Basic facts rely on Edelsbrunner-Harer)

Definitions: filtration (an exhaustive family of subsets) of a topological space allows one to arrange the topological features at all scales, ant to track those that are stable, – persistent.

As filtration grows, so does the linear spaces of cycles and boundaries. So one can ask when a particular homology class emerged, and when it dies, – i.e. becomes bounded by a chain. This fits naturally into a neat linear algebra formalism, known as “quiver representation theory”, and leads to a notational device for the persistent classes, known as “persistence diagrams”.

In simplicial complexes, there are distinguished bases in the spaces of chains, and one can track them using an efficient algorithm (cubic in the number of simplices).

Stability theorem establishes that the persistent diagrams vary little (in the so-called bottleneck metric) when the functions is perturbed little in \(L_\infty\) norm.

Projects

:

      • Find the homological dimension of Weierstraß function \(\sum a^{-k}\sin(b^k x)\), and generalize to higher dimensions (consider
        \[
        \sum a^{-k}\sin(b^k x)+\sum c^{-l}\sin(d^l y).
        \]
        etc…).
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