Delaunay-Rips Filtration in TDA

• Delaunay–Rips filtration is a technique that builds filtered simplicial complexes by intersecting Delaunay triangulation with the Vietoris–Rips complex from Euclidean point clouds.
• It offers significant computational advantages and approximation guarantees through multiplicative interleaving, reducing the number of simplices compared to full Rips filtrations.
• The method extends to chromatic settings and is applied in machine learning pipelines, demonstrating faster runtimes and lower memory usage in practical TDA applications.

The Delaunay–Rips filtration is a family of filtered simplicial complexes constructed from Euclidean point clouds by intersecting the Delaunay triangulation with the Vietoris–Rips complex at each scale parameter. By definition, the Delaunay–Rips complex at scale $r$ is given by restricting the Rips complex to those simplices that are faces in the Delaunay triangulation, or equivalently, those with an empty circumsphere and diameter at most $2r$. This filtration has emerged as an efficient and accurate alternative to the full Rips filtration in persistent homology, with significant computational advantages for low- to moderate-dimensional data and with practical justification both in standard and labelled (chromatic) settings (Mishra et al., 2023, Clémot et al., 19 Dec 2025, Natarajan et al., 29 May 2024).

1. Formal Definition and Construction

Let $X \subset \mathbb{R}^d$ be a finite point cloud. The Delaunay triangulation $\mathrm{Del}(X)$ consists of simplices whose vertices’ Voronoi cells have nonempty intersection, equivalently, those with empty circumspheres. For a simplex $\sigma \subset X$, define its diameter as $\delta(\sigma) = \max_{x,y\in\sigma} \|x-y\|_2$. The Delaunay–Rips filtration $\{DR_r(X)\}_{r \ge 0}$ is then

$$DR_r(X) = \{ \sigma \in \mathrm{Del}(X) : \delta(\sigma) \le 2r \}$$

forming an increasing sequence of simplicial complexes as $r$ increases. Persistent homology is computed on this filtration in the standard way, yielding persistence diagrams $\mathrm{dgm}_k^{DR}(X)$ in each homology dimension $k$ (Clémot et al., 19 Dec 2025, Mishra et al., 2023).

For a labelled point cloud $(X, \mu)$ with $\mu: X\to\{0,1,\dots,s\}$, the chromatic Delaunay–Rips filtration is defined by lifting each $x\in X_i$ to $(x, e_i)\in\mathbb{R}^{d+s}$, forming the Delaunay triangulation in this higher-dimensional space, and intersecting the resulting complex with the Rips complex at each scale. The filtration is denoted $\{DR_r(X, \mu)\}_{r \ge 0}$ (Natarajan et al., 29 May 2024).

2. Theoretical Properties and Approximation Guarantees

The Delaunay–Rips filtration provides a sparsified version of the Rips filtration with strong approximation properties. For any $k < d$, there are simplicial inclusions

$$DR_r^{(k)}(X) \subset R_r^{(k)}(X) \subset DR_{(k+1)r}^{(k)}(X)$$

(using $(\cdot)^{(k)}$ to denote all simplices of dimension at most $k$). This induces multiplicative $(k+1)$–interleaving of the persistence modules of $DR$ and $R$, with the following bound: $$W_\infty\left(\log \mathrm{dgm}_k(R), \log \mathrm{dgm}_k(DR)\right) \le \log(k+1)$$ or, equivalently in the original scale,

$$W_\infty\left(\mathrm{dgm}_k(R), \mathrm{dgm}_k(DR)\right) \le k \delta(X)$$

where $\delta(X)$ denotes the diameter of $X$ (Clémot et al., 19 Dec 2025). For many random or noisy point sets, the empirical bottleneck distance between log–diagrams is much smaller than the worst-case bound.

In the chromatic setting, Bauer–Edelsbrunner–type results are extended: the Čech, chromatic Delaunay–Čech, and chromatic alpha filtrations are related by sequences of elementary collapses (strong deformation retracts), and Delaunay–Rips-based filtrations preserve the underlying homotopy types up to these collapses (Natarajan et al., 29 May 2024).

3. Stability and Instability

Classical Rips filtrations satisfy Lipschitz stability of persistence diagrams under the Hausdorff (or Gromov–Hausdorff) distance on point clouds: $$W_\infty(\mathrm{dgm}(R; X), \mathrm{dgm}(R; Y)) \le 2d_{GH}(X, Y)$$ By contrast, the Delaunay–Rips filtration exhibits stability only when the Delaunay triangulation remains fixed under perturbation. If $X$ and $X'$ are $\varepsilon$-perturbations with $\mathrm{Del}(X) = \mathrm{Del}(X')$, then

$$W_\infty(\mathrm{dgm}_k^{DR}(X), \mathrm{dgm}_k^{DR}(X')) \le 2 d_H(X, X')$$

(Mishra et al., 2023). However, generic point configurations may induce combinatorial changes in the Delaunay complex (e.g., edge flips in 2D), causing true discontinuities in DR persistence diagrams. For instance, in $\mathbb{R}^2$, four near-cocircular points can produce a nontrivial jump in $H_1$-persistence due to an infinitesimal perturbation, violating Lipschitz (or even continuity) of the mapping from point cloud to persistence diagram (Clémot et al., 19 Dec 2025).

Chromatic Delaunay–Rips filtrations nevertheless demonstrate local stability in general position: for $(X, \mu), (Y, \nu)$ with colour-preserving correspondences and small chromatic Hausdorff distance, the filtration diagrams are $d_C(X, Y)$–interleaved for sufficiently small perturbations (Natarajan et al., 29 May 2024).

4. Algorithms and Complexity

Algorithmic construction of the Delaunay–Rips filtration in the classical setting proceeds as follows (Mishra et al., 2023, Clémot et al., 19 Dec 2025):

1. Compute the Delaunay triangulation $\mathrm{Del}(X)$ (e.g., using CGAL or qhull); in low dimensions this is $O(n \log n)$ expected time.
2. For each simplex $\sigma\in\mathrm{Del}(X)$ up to dimension $k$, compute its diameter and assign it as the filtering value.
3. Sort simplices by diameter, breaking ties by dimension.
4. Compute persistent homology via boundary matrix reduction (PHAT, Ripser, Gudhi), but only on Delaunay simplices.

A scalable, dimension-generic union-find–based algorithm is presented in (Clémot et al., 19 Dec 2025) with key steps including:

• Construction of minimum spanning $k$–acycles (generalizing MST),
• Extraction of Urquhart subcomplexes,
• Sparse matrix reduction confined to a restricted set of “cells” built on Delaunay $(k+1)$-simplices.

Complexity is governed by the number of Delaunay simplices, which is $O(n^{\lceil d/2\rceil})$ in worst case but $O(n)$ expected for random point sets in low $d$. This yields substantial computational savings compared to the Rips complex, which involves $O(n^{k+1})$ simplices.

Chromatic variants require chromatic lifting to a higher-dimensional space but retain similar algorithmic approaches; the practical bottleneck remains the Delaunay triangulation step (Natarajan et al., 29 May 2024).

5. Empirical Performance and Machine Learning Applications

Empirical evaluation demonstrates that DR filtration is markedly faster and more memory-efficient than classical Rips and Alpha filtrations for low-dimensional point clouds. For example, in $\mathbb{R}^3$ with $10^5$ points, the method of (Clémot et al., 19 Dec 2025) is approximately $3\times$ faster than PHAT and $5\times$ faster than Gudhi, with $\sim 30\%$ of the memory usage; Ripser becomes infeasible beyond several thousand points. Parallel implementations with 20 threads show speedups up to $4\times$ (Clémot et al., 19 Dec 2025).

Experiments on machine learning pipelines for shape classification and physiological (sleep/wake) state discrimination find that Delaunay–Rips-based persistence performs comparably to Rips and Alpha filtrations in terms of classification accuracy, area under ROC, and feature importance heatmaps—differences in statistical metrics are not significant (p-values $>0.1$). On synthetic and real datasets, accuracy and robustness of DR are closely matched to established filtrations (Mishra et al., 2023).

In chromatic applications, DR and its chromatic generalizations enable orders-of-magnitude speedups on multi-class data while maintaining detection of relational topological features, such as interlocking loops of different classes, which are captured more efficiently than by chromatic Alpha or Čech filtrations (Natarajan et al., 29 May 2024).

6. Practical Guidelines, Limitations, and Further Directions

Delaunay–Rips filtration is preferred when preservation of most Rips-type topology is desired at reduced computational cost, especially in moderate ambient dimensions ($d \le 5$) where Delaunay complexes remain tractable. Instabilities due to Delaunay triangulation combinatorics are mitigated by ensuring general position, e.g., via preprocessing or small random jitter. In high dimension ($d \gg 10$), computation of $\mathrm{Del}(X)$ quickly dominates and alternative sparsification methods (e.g., witness complexes) are advised (Mishra et al., 2023, Clémot et al., 19 Dec 2025).

Key decision criteria and caveats are summarized in Table 1.

Criterion	When to Use DR	When to Avoid DR
Point cloud dimension	$d \le 5$ (moderate)	$d > 10$ (high)
Required stability	Can tolerate small instabilities	Need full Lipschitz continuity
Computation budget	Limited	Large, but Rips needed
Application type	Low-dim. TDA, ML pipelines	High-dim. or streaming data

Table 1: Usage recommendations for Delaunay–Rips filtration (criteria direct from (Clémot et al., 19 Dec 2025, Mishra et al., 2023)).

Extensions include efficient cell-reduction in generalized flag filtrations, GPU acceleration, streaming/online updates, and use as differentiable layers inside geometric neural networks (Clémot et al., 19 Dec 2025). In chromatic contexts, the framework scales to multi-class relational TDA with robust local stability and computational advantages (Natarajan et al., 29 May 2024).

7. Context within Topological Data Analysis

The Delaunay–Rips filtration structurally interpolates between Rips (combinatorially universal but exponentially large) and Čech/Alpha (homotopically faithful but computationally expensive due to minimum-enclosing-ball computations). DR achieves major reductions in simplex counts and runtime (especially in $d=2,3$), with theoretical guarantees on persistence diagram approximation and homotopy equivalence (via collapses, in the chromatic case) to more classical filtrations (Clémot et al., 19 Dec 2025, Natarajan et al., 29 May 2024). Its practical effectiveness and public C++/Python implementation (GeoPH) position it as a method of choice for persistent homology in low- to moderate-dimensional Euclidean data analysis.