the persistence stability theorem
A method that turns noisy data into a topological summary is only trustworthy if a small wiggle in the data causes only a small wiggle in the summary. The persistence stability theorem is the guarantee that makes topological data analysis a science rather than a curiosity: perturb your data a little and the persistence diagram moves only a little, by an amount you can bound by the size of the perturbation. Without this, a single misplaced point could rearrange the whole barcode and the method would be useless.
Precisely, in its cleanest form (Cohen-Steiner, Edelsbrunner, Harer): for two real-valued functions f and g on the same space, with sublevel-set filtrations, the bottleneck distance between their persistence diagrams is at most the sup-norm distance between the functions, d_B(Dgm(f), Dgm(g)) <= ||f - g||_infinity. There is a stronger algebraic version: the interleaving distance between two persistence modules upper-bounds the bottleneck distance between their diagrams, and in fact the two are equal (the isometry theorem). For point-cloud TDA, a companion result bounds the diagram change by the Gromov-Hausdorff distance between the point sets, so resampling or jittering the data perturbs the barcode by a controlled amount. The bottleneck distance is the right metric here precisely because the theorem is stated in it: it matches diagram points within epsilon and lets near-diagonal (low-persistence) points be created or destroyed freely.
This theorem is why long bars can be read as signal and short bars dismissed as noise with a clear conscience: noise of magnitude epsilon can only create or destroy bars of length at most 2*epsilon, so any bar much longer than the noise level is genuine. It underpins statistical inference on diagrams, confidence sets, and machine-learning pipelines. The honest caveats: stability is in the bottleneck (or Wasserstein) metric, and a stable diagram does not imply the data are 'close' in any stronger sense — very different shapes can have nearby diagrams; the function-stability bound assumes a fixed underlying space, while point-cloud stability uses a coarser Gromov-Hausdorff control; and 'small perturbation' must be measured in the norm the theorem actually uses, not in some intuitive notion of closeness.
Add jitter of magnitude epsilon = 0.05 to every point of the noisy-circle sample. The stability theorem guarantees each point of the persistence diagram moves by at most 0.05 in the bottleneck distance, so the one long H_1 bar (lifespan, say, 0.8) barely shifts and is never created or destroyed by the jitter, while only the tiny near-diagonal bars can wink in and out — exactly the bars we already discard.
Small data perturbations move diagram points by at most the perturbation size; long bars are safe.
Stability is a statement in the bottleneck (or Wasserstein) metric and does not run backward: nearby diagrams do NOT imply similar data, since very different shapes can have nearly identical persistence. Measure 'small perturbation' in the norm the theorem uses (sup-norm of functions, or Gromov-Hausdorff for point clouds), not in an intuitive sense of closeness.