Mapper

The construction

Mapper turns a point cloud into a small graph in three steps:

1. Filter. Choose a function on the data — density, a principal component, a disease score, anything real-valued — and evaluate it at every point.
2. Cover. Split the filter’s range into overlapping intervals.
3. Cluster and connect. Cluster the points in each interval separately. Every cluster becomes a node; two nodes are joined when they share a point, which they can because the intervals overlap.

The result is a graph small enough to look at, in which shape is legible: a loop in the graph means a loop in the data, and a flare means a subpopulation that separates along the filter.

Its appeal is that it is a visualization that survives high dimension. The output is meant to be looked at, which is unusual in this corpus and accounts for much of Mapper’s uptake outside mathematics.

Its relationship to persistent homology

Mapper is the nerve of a pullback cover, so it is a topological summary in the same family as persistent-homology — but it behaves very differently in practice, and the difference is not subtle.

Persistent homology takes no parameters beyond the complex and returns a stable, comparable invariant. Mapper takes a filter, an interval count, an overlap fraction, and a clustering algorithm with its own parameters — and the output graph changes substantially with all of them. There is no stability theorem doing for Mapper what the stability theorem does for barcodes.

That is the honest headline: Mapper is exploratory. It is very good at suggesting structure and poor at establishing it. A Mapper graph is a hypothesis-generating picture, and treating one as a result is the characteristic misuse.

Using it responsibly

• The filter is the model. Everything Mapper shows is structure relative to the filter you chose. Choosing it after seeing which choice gives a satisfying picture is circular, and it is the most common failure.
• Vary the parameters and show it. A feature that survives a range of interval counts and overlaps is worth attention; one that appears at a single setting is not. Reporting one graph from one parameter set, without saying what else was tried, hides the only diagnostic available.
• Confirm findings by other means. The appropriate output of a Mapper analysis is a subpopulation or a relationship to go and test, not a conclusion.

Used this way it earns its place, particularly early in an analysis when you do not yet know what question to ask. Used as evidence, it will eventually embarrass you.

What implements it here

• kmapper-environment — KeplerMapper, the standard Python implementation, with interactive HTML output.
• giotto-tda-environment — Mapper alongside persistence, in a scikit-learn-shaped API, so the parameter sweep above can be run as an ordinary grid search.
• scikit-tda-environment — bundles KeplerMapper with ripser.py and persim.

Related

persistent-homology is the stable, non-visual counterpart, and the right tool once you know what you are measuring. spectral-geometry is the other route from high-dimensional data to a low-dimensional picture; it produces coordinates rather than a graph, and is metric rather than topological.