23 Summary and further reading
“We shall not cease from exploration”
— T. S. Eliot, in Little Gidding
23.1 What we can now calculate
We began with a cloud of observations and the question of its shape. The square showed how to turn distances into simplices, simplices into boundary matrices, and cycles modulo boundaries into homology classes. A filtration then let us follow these classes as the scale changed.
The computations added choices that the abstract definitions alone do not make: units, preprocessing, metric, coefficients, filtration, sample size, and cutoff. We compared Rips, alpha, and cubical inputs; learned to read finite and essential intervals; and used shared-grid vectorizations and diagram distances.
ToMATo follows density peaks through merges. Mapper builds overlapping groups from a filter and a local clustering rule. Ball Mapper builds a cover around landmarks. Their plots tell us about those constructions. Edges in a displayed graph do not automatically certify holes in the underlying space, and a graph layout is not a faithful geometric map.
The applications tested those distinctions. The Target benchmark exposed a clustering failure. The digit experiment separated training, validation, and test transformations and included a pixel baseline. Delay embeddings required attention to sampling and independence. Shape comparisons required common vectorization coordinates and a decision about geometric scale.
Finally, the uncertainty chapter separated a stability theorem, empirical sampling variation, and a test under a stated null model. A long interval is a lifetime; its scientific interpretation remains a question about the data.
23.2 A workflow worth carrying to your next dataset
- State the question and sampling units. What phenomenon would a component, loop, or cavity represent? Which observations are independent?
- Choose and record the representation. Inspect missing values, duplicates, units, and outliers. Justify the metric and filtration rather than accepting their defaults silently.
- Predict a small result. Use a synthetic control or a hand-calculable example. Check conventions, including edge length versus ball radius and foreground versus background.
- Separate fitting from evaluation. Fit preprocessing and diagram vectorizers on training data, select parameters with validation data, and reuse the fitted transformations on test data.
- Check sensitivity and alternatives. Vary sample size and scientifically plausible parameters. Compare with classical methods and, when inference is intended, a justified null model or confidence procedure.
- Report what the result cannot distinguish. A diagram is a compressed summary. Different objects can have the same diagram, and different constructions on one dataset can answer different questions.
The six JuliaTDA packages used here are frozen with the book’s environment. The setup notes explain how to reproduce this edition and how to switch to development sources.
23.3 Further reading
23.3.1 Books and surveys
For algorithms and the underlying mathematics, read Computational Topology by Edelsbrunner and Harer (Edelsbrunner and Harer 2010) or Topology for Computing by Zomorodian (Zomorodian 2005). Topological Data Analysis with Applications is by Carlsson and Vejdemo-Johansson (Carlsson and Vejdemo-Johansson 2021).
Hatcher’s Algebraic Topology develops the homology theory beyond our finite-field examples (Hatcher 2002). Ghrist’s Elementary Applied Topology offers a visual route into applied topology (Ghrist 2014).
Carlsson’s Topology and Data (Carlsson 2009) and Ghrist’s Barcodes: The Persistent Topology of Data (Ghrist 2008) are survey papers, useful companions to a first pass through this book.
23.3.2 Follow the question into the original papers
| If you want to understand… | Start with… |
|---|---|
| Stability under function perturbations | Cohen-Steiner, Edelsbrunner, and Harer (Cohen-Steiner et al. 2007) |
| Stability for metric-space filtrations | Chazal, de Silva, and Oudot (Chazal et al. 2014) |
| Confidence sets and statistical guarantees | Fasy and colleagues (Fasy et al. 2014) |
| Density peaks and persistence-based clustering | Chazal and colleagues, the 2013 ToMATo paper (Chazal et al. 2013) |
| Covers, filters, and Mapper | Singh, Mémoli, and Carlsson (Singh et al. 2007) |
| Landmark covers and Ball Mapper | Dłotko (Dłotko 2019) |
| Vector features on a common diagram domain | Adams and colleagues (Adams et al. 2017); Bubenik (Bubenik 2015) |
| Sliding windows and periodicity | Perea and Harer (Perea and Harer 2015) |
| Quasiperiodic delay embeddings | Gakhar and Perea (Gakhar and Perea 2024) |
23.3.3 Topics beyond this book
One-parameter persistence is not the end of the story. Multiparameter persistence can track, for example, distance and density together; its invariants are more complicated than a single barcode. Zigzag persistence permits objects to be added and removed, useful for changing observations over time. Cohomology and circular coordinates can turn a persistent loop into a useful coordinate rather than only a lifetime.
For geometric shape analysis, the persistent homology transform compares many directional filtrations; this is a broader construction than our small family of image directions. For other computational budgets, witness and sparse complexes and approximation guarantees deserve study. Distance-to-measure filtrations address robustness using a different geometric construction, rather than expecting ordinary Rips persistence to ignore outliers.
Sheaves and cosheaves extend local-to-global reasoning, while differentiable topology brings topological summaries into optimization. These are directions for further study, not prerequisites for the examples we have worked through.
Go forth and find some holes in your data! Then ask why they are there.