# Multi-parameter persistence landscapes: From theory to application

**By Ulrike Tillmann**

Persistent homology has proved an effective tool when detecting shape in data, and barcodes provide a faithful and visual presentation. But for use in more automated and statistical analysis it is helpful to express the same information as a vector, so that for example averages can be taken. One such method are the persistent landscapes of Bubenik: the familiar bars are converted into a series of piecewise linear maps the graphs of which look like simplified mountain landscapes. These persistence landscapes can then be used for example as input to machine learning algorithms. [1] [2]

This all works well for persistent homology built on a single parameter filtration. In applications however often there are two or more parameters present that need to be considered, such as time and density, or distance and intensity. The theory of multi-parameter persistence is however vastly more complicated, and to find effective and computable invariants in this setting has been an important and intensely studied problem.

In a paper published in JMLR, Oliver Vipond, one of the Centre’s doctoral students, has generalised landscapes from the single to the multi-parameter setting, thus defining a new stable and computable invariant in the multi-parameter setting. Closely related to the known rank invariant it furthermore remains interpretable. [3]

In collaboration with researchers from the NIHR Oxford Biomedical Research Centre we are applying this new tool to study the infiltration of immune cells in tumours. To this end point clouds of different immune cells (CD8, CD68 and FoxP3) around necrotic regions of tumour sections are analysed. In each case, in addition to the usual radial parameter, varying oxygen levels as well as codensity provide a clinically significant second parameter thus giving rise to bi-filtered Cech-complexes. As the point clouds are too large for a single computation, multiple subsamples have to be taken. A statistical analysis based on the multi-parameter persistence landscapes suggests among other things that certain immune cells (CD8 and FoxP3) are excluded to a greater extend from hypoxic regions than others (CD68). [4]

[1] Bubenik, JMLR, https://arxiv.org/pdf/1207.6437.pdf

[2] Bubenik, Dlotko, JSC, https://arxiv.org/abs/1501.00179

[3] Vipond, JMLR, https://arxiv.org/pdf/1812.09935.pdf

[4] Vipond, Bull, Macklin, Tillmann, Pugh, Byrne, Harrington, forthcoming