Filtrations & persistence

Persistence diagrams are precise but hard to read cold. The intuition behind them is a sweep: pick a scale, connect everything closer than that scale, and watch the shape of the data change as the scale grows.

See it

Each point grows an “influence ball” of radius scale / 2. When two balls touch, their points connect — and separate pieces merge (a component, H₀, dies). When a ring of points is enclosed but not yet filled, there is a loop (H₁) alive; once the interior fills with triangles, the loop dies. The counts below are the Betti numbers H₀ (components) and H₁ (loops) at the current scale, from the same reducer as src/spatial/persistence.ts.

Switch the view between the flat 2-D picture, a cones scale-space — each point becomes a cone of its distance function, and a slice plane at the current scale cuts out the 2-D disks, so sweeping the plane up the cones is the filtration — or both side by side. Drag the cones to rotate.

view2DconesbothH₀ –  H₁ –

Vietoris–Rips complex

colour = connected component (H₀)

persistence diagram

filled = alive at this scale · H₀ / H₁

⏸scale0.0

Illustrative, not yet primitive-driven

The Betti counts come from the real src/spatial/persistence.ts reducer, but the drawing is a hand-built three.js approximation — it isn’t yet rendered from the toolbox’s own GPU primitives. So a few visual details can drift from the homology: for instance the outline may briefly show a loop the count has already closed, where the union of fat disks disagrees with the combinatorial complex.

How to read it

• H₀ (components) starts high — every point is its own island — and falls as balls merge islands together. The “two clusters” shape parks at H₀ = 2 for a while before merging to 1: that plateau is what makes two clusters a real, persistent feature rather than noise.
• H₁ (loops) appears when a cycle of connections encloses empty space, and dies when that space fills in. The “ring” holds one loop across a wide band of scales; the “figure-eight” holds two. A feature that survives a wide band of scales is the robust topology — exactly what a persistence diagram records as a point far from the diagonal.

This is the filtration a Vietoris–Rips persistence reducer walks through. The toolbox’s job is to build this filtration efficiently — including the density- adaptive CkNN variant — and hand it to the reducer, which reads off how long each component and loop survives.

Hard balls vs fuzzy fields

The sweep above grows hard balls: a point is in the set or it isn’t, and the boundary of a union of balls has corners where two balls meet. That hard threshold is the same boxcar we replaced with a smooth window everywhere else in this toolbox (see Fuzzy TDA), and it has the same costs: the boundary is non-smooth, and a single stray outlier spawns a full-strength ball at radius 0 — a long, important-looking bar that is really just noise.

The fuzzy filtration replaces each hard ball with a smooth Gaussian kernel and thresholds the resulting density field — the superlevel-set filtration of a KDE. Now the boundary is C-infinity (no corners), and how strongly a region counts depends on how much density is piled there, so sparse noise is demoted toward the diagonal. The robust-statistics version, distance-to-measure (DTM), makes this a theorem: its persistence is provably stable to outliers in the bottleneck/Wasserstein sense, which Čech/Vietoris–Rips is not.

Sweep the same parameter through both. Turn outliers on and watch the hard diagram (left) sprout points far from the diagonal while the fuzzy diagram (right) keeps the noise near it and the true feature prominent. Switch the fuzzy field between KDE and DTM.

The second row of controls exposes the parameters that decide what counts as structure. Falloff is the kernel bandwidth σ (or, for DTM, the neighbour count k): widen it and nearby features merge; narrow it and a smooth ring shatters into a necklace of separate density beads (H₀ climbs). Resolution is the grid the field is sampled on. Noise band is the persistence threshold below which a feature is treated as noise — the shaded strip along the diagonal — i.e. the line between signal and chatter that a hard filtration forces you to draw by hand.

add outliersfuzzy fieldKDEDTMCPU

bandwidth σ1.9noise band0.08

hard — union of balls distance, sublevel

hover a component or a hole to inspect it

fuzzy — KDE superlevel set density, superlevel

hover a component or a hole to inspect it

H₀ –  H₁ – · spurious bars cluster off the diagonal

H₀ –  H₁ – · noise stays near the diagonal

⏸sweep0%

This one is primitive-driven

Unlike the sweep above, this demo imports the toolbox’s own library directly — the fields come from src/spatial/scalarField.ts (gaussianKdeField — the CPU twin of the KDE splat primitive — distanceField, dtmField) and the diagrams from src/spatial/sublevelsetPersistence.ts, a cubical sublevel-set reducer (the field counterpart of the point-cloud Vietoris–Rips reducer). Both the hard ball-union and the fuzzy field are reduced by that one primitive: the only thing that changes is which field you sublevel-set.

When your browser exposes WebGPU (badge top-right), the fields are computed by a compute shader — the same additive-Gaussian splat / brute-force gather the KDE splat primitive uses — and the badge turns green; without it, the CPU library runs instead (toggle it to compare — the diagram is identical). Any backend that returns a bad field is validated and silently falls back to the CPU, and the persistence reduction itself is inherently sequential and always stays on the CPU.

Sources & further reading

• This toolbox — the Betti counts come from src/spatial/persistence.ts (standard boundary-matrix reduction over ℤ/2); the density-adaptive scale variant is CkNN. The diagram conventions follow the persistence diagram primitive.
• R. Ghrist, Barcodes: the persistent topology of data, Bull. Amer. Math. Soc. 45 (2008) — the most readable entry point.
• G. Carlsson, Topology and data, Bull. Amer. Math. Soc. 46 (2009).
• H. Edelsbrunner & J. Harer, Computational Topology: An Introduction (AMS, 2010) — the standard textbook for the reduction algorithm and Vietoris–Rips/Čech complexes.
• N. Otter et al., A roadmap for the computation of persistent homology, EPJ Data Science 6 (2017) — practical, computation-focused (open access).

On the fuzzy / robust filtrations in the second demo:

• F. Chazal, D. Cohen-Steiner & Q. Mérigot, Geometric inference for probability measures, Found. Comput. Math. 11 (2011) — the distance-to-measure function.
• F. Chazal, B. Fasy, F. Lecci, B. Michel, A. Rinaldo & L. Wasserman, Robust Topological Inference: Distance To a Measure and Kernel Distance, JMLR 18 (2017) — stability of DTM- and KDE/kernel-distance filtrations to noise and outliers.
• H. Anai, F. Chazal, M. Glisse, Y. Ike, H. Inakoshi, R. Tinarrage & Y. Umeda, DTM-based filtrations, SoCG 2019 — the weighted-Rips construction this demo’s DTM field is the field analogue of.