ADTDA, lecture 3

\(\def\Real{\mathbb{R}}
\def\ph{\mathtt{PH}}
\def\dim{\mathrm{dim}}
\)

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.

Persistence jitter

While originally a tool to describe topological spaces, persistent diagrams is now used also as a tool to characterize functions. This is especially pronounced in material sciences. With this change of focus, short bars become a signal, a descriptor of the function, not nuisance to get rid of.

Natural question is to ask what to expect from a “generic” function, for example: how the short bars accumulate towards the diagonal.

A definition of PH dimension is one measure of this accumulation.

Namely, let \(\mu_k(f)\) be the (counting) measure associated with the \(k\)-th persistence diagram of the function.
We take the persistence-dimension of the function \(f\) to be
\[
\ph_k\dim(f):=inf\{p:\int (d-b)^p_+d\mu_k(f)<\infty\}.
\]

Example: For the univariate Lipschitz function \(x^{q+1}\cos(x^{-q}\), one has \(\ph_k\dim(f)=q/(q+1)\)

We prove that the general estimates (in high generality) hold for all Lipschitz or Hölder functions on a compact finite-dimensional polyhedron \(X\): \(\ph_*\dim(f)<\dim X/\alpha\), where \(\alpha\) is the Hölder parameter (\(=1\) for Lipschitz functions).

Moreover, it turns out that for generic Lipschitz or Hölder functions (outside of a meager set), this estimate is precise: a generic function has the highest possible persistence dimension.

Persistence for univariate functions

Persistence diagrams are rather incomplete descriptor for a univariate function. If one restrict attention to descriptors that are reparametrization invariant, the merge tree associated to the function is: Harris walk is the inverse transformation.

Also, persistence diagrams for univariate functions can be constructed fast, essentially using an online algorithm with a pair of stacks recording local minima and maxima.

The structure of jitter for one class of random functions is especially easy to investigate: Brownian trajectories.

Let \(f(t)\) be the Brownian motion with constant drift. Then \(\mu_0(f)\) is a random point process, and of \(\ph_0\dim\) at most 2.

We give a quite detailed description of this process, basing on this preprint.

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