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GPU Spatial Primitives

Fuzzy TDA

Topological data analysis asks what shape a point set has — connected components, loops, voids — and how robustly those features persist as you connect points at growing scales. The standard tool, the Vietoris–Rips complex, connects two points iff they are within radius r. That is a boxcar in distance, and it inherits every weakness of a hard threshold: brittle to the radius, sensitive to outliers, and blind to how strongly two points are connected.

Soften the connectivity

Section titled “Soften the connectivity”

“Fuzzier” TDA replaces the binary edge with a smooth membership in [0,1]:

μ_ij = exp(-d_ij² / 2σ²)     instead of   [ d_ij ≤ r ]

An edge now fades in with proximity rather than snapping on at one threshold. Run persistence on the resulting weighted complex and the features that survive across a wide band of membership are the robust ones.

This is not exotic — it is the construction underneath UMAP (fuzzy simplicial sets), and it has relatives in weighted Vietoris–Rips persistence and distance-to-measure (density-aware, outlier-stable) filtrations.

Division of labour

Section titled “Division of labour”

The homology reduction itself is inherently sequential — leave it to a CPU library (Ripser / GUDHI). Everything upstream — building the fuzzy filtration — is embarrassingly parallel and is where this toolbox helps:

  • ✅ Fuzzy adjacency — the kernel-weighted graph; the weighted 1-skeleton.
  • ✅ Adaptive per-point bandwidth (UMAP-style) — each point gets its own σ_i from its k-th neighbour distance, symmetrised with the probabilistic t-conorm μ_ij = a + b − a·b (src/gpu/spatial/fuzzyAdjacencyAdaptive.ts).
  • ✅ Membership-sweep filtration: the GPU builds the (fuzzy) edge weights, a membership → distance map turns them into a filtration (d = 1 − μ), and the CPU Vietoris–Rips reducer sweeps it. All four steps are nodes in the operation-graph runtime — wire points → kthNeighborDistance → fuzzyAdjacencyAdaptive → membershipToDistance → vietorisRipsPersistence and pull the diagram.

So the rule mirrors crisp VR: GPU builds the (now fuzzy) filtration; CPU reduces it. Fuzzy adjacency is the first brick, and it falls straight out of the same windowing kernel used everywhere else in the toolbox.