Decomposition of exact pfd persistence bimodules

TL;DR

This paper characterizes when two-dimensional persistence modules can be decomposed into simple block-shaped components, introducing an exactness condition that generalizes known one-dimensional results to the more complex product order setting.

Contribution

It establishes a necessary and sufficient local condition called exactness for decomposing pfd 2D persistence modules into block summands, extending classical decomposition theorems.

Findings

Decomposition into block summands is equivalent to exactness for pfd modules.

The proof adapts one-dimensional techniques to the two-dimensional setting with new arguments.

Results facilitate stability analysis in zigzags and interlevel-sets persistence modules.

Abstract

We characterize the class of persistence modules indexed over $\mathbb{R}^2$ that are decomposable into summands whose support have the shape of a {\em block}---i.e. a horizontal band, a vertical band, an upper-right quadrant, or a lower-left quadrant. Assuming the modules are pointwise finite dimensional (pfd), we show that they are decomposable into block summands if and only if they satisfy a certain local property called {\em exactness}. Our proof follows the same scheme as the proof of decomposition for pfd persistence modules indexed over $\mathbb{R}$, yet it departs from it at key stages due to the product order on $\mathbb{R}^2$ not being a total order, which leaves some important gaps open. These gaps are filled in using more direct arguments. Our work is motivated primarily by the stability theory for zigzags and interlevel-sets persistence modules, in which block-decomposable bimodules play a key part. Our results allow us to drop some of the conditions under which that theory holds, in particular the Morse-type conditions.