Delaunay–Rips Persistence Diagrams

• Delaunay–Rips persistence diagrams are topological tools that combine Delaunay complexes with Rips filtrations to analyze Euclidean point clouds.
• They offer efficient computation by leveraging sparse Delaunay triangulations, substantially reducing memory and runtime compared to full Rips or Čech filtrations.
• The framework extends to chromatic (labeled) point clouds, maintaining local stability and high accuracy for multi-category data analysis.

The Delaunay-Rips persistence diagram is a tool in topological data analysis (TDA) that leverages the combinatorial sparsity of the Delaunay complex combined with Rips-style filtration on Euclidean point cloud data. Introduced as the Delaunay–Rips (DR) filtration, it provides a computationally favorable alternative to full Rips or Čech filtrations, especially for large point clouds in low to moderate ambient dimensions. The DR filtration also generalizes to the setting of labelled (chromatic) point clouds, providing theoretically justified, locally stable, and empirically efficient alternatives for multi-category data analysis. The persistence diagrams produced by this filtration approximate those of the standard Rips filtration while substantially reducing memory and runtime requirements in practice, up to feasible dimension (Clémot et al., 19 Dec 2025, Mishra et al., 2023, Natarajan et al., 29 May 2024).

1. Formal Definition and Construction

Given a finite point set $X \subset \mathbb{R}^d$, let $\mathrm{Del}(X)$ denote its (abstract) Delaunay complex, the nerve of the Voronoi cells. For any simplex $\sigma\subset X$, its diameter is $\delta(\sigma) = \max_{x,y\in\sigma}\|x-y\|$. The Delaunay–Rips filtration is defined as the nested sequence of complexes

$$DR_r(X)=\bigl\{\sigma\in\mathrm{Del}(X)\mid \delta(\sigma)\le 2r\bigr\} = R_r(X)\cap \mathrm{Del}(X),$$

where $R_r(X)$ is the Vietoris–Rips complex at scale $r$. This construction yields a filtration $\{DR_r(X)\}_{r\ge0}$ suitable for the persistent homology pipeline, producing a persistence diagram $\mathrm{dgm}_k(DR_\bullet(X))$ for each homological degree $k$ (Clémot et al., 19 Dec 2025, Mishra et al., 2023).

For labelled or chromatic point clouds, a coloring $\ell:X\to\{0,1,\dots,s\}$ is used to define chromatic Delaunay–Rips complexes, constructed by lifting each point to a higher-dimensional space encoding color and then projecting the Delaunay triangulation back to the original space. The filtered complex is:

$\{\Del\VR_r(X,\ell)\}_{r\ge0}, \quad \text{where} \quad \Del\VR_r(X,\ell) = \VR_r(X) \cap \Del(X,\ell)$

(Natarajan et al., 29 May 2024).

2. Algorithmic Approaches and Complexity

The Delaunay–Rips persistence diagram can be computed without enumerating the full Rips complex:

• Step 1: Construct the Delaunay triangulation of $X$, typically using libraries such as CGAL or SciPy’s Qhull back-end.
• Step 2: Assign each simplex $\sigma$ the Rips-style weight $\delta(\sigma)$.
• Step 3: Sort simplices by weight (and dimension for tie-breaking).
• Step 4: Apply standard persistence algorithms (e.g., boundary matrix reduction using PHAT, Gudhi, or custom MSA-based schemes) to extract persistence pairs.

In generic dimension $d$, the expected size of the Delaunay complex is $O(n^{\lceil d/2 \rceil})$; the boundary matrix reduction step is orders of magnitude cheaper than for the Rips complex, as far fewer simplices are involved. The presented algorithm also introduces minimum-spanning acycle (MSA) techniques and Urquhart simplices to fuse reductions and avoid large matrix operations. For $k=0$, Kruskal’s algorithm on the Urquhart graph provides the persistence diagram directly (Clémot et al., 19 Dec 2025, Mishra et al., 2023).

For chromatic filtrations, the initial Delaunay triangulation is carried out in $\mathbb{R}^{d+s}$, then projected; subsequent steps mirror the non-chromatic case. Time complexity remains favorable, with median runtimes $\sim10^0$--$10^1$ seconds up to $n=10^4$ in low $d$ (Natarajan et al., 29 May 2024).

3. Theoretical Guarantees and Stability

The DR persistence diagram approximates that of the full Rips diagram up to a provable bound. Specifically, for $k\ge1$: $d_B(\log\dgm_k^R,\log\dgm_k^{DR})\le\log(k+1)$ where $d_B$ denotes bottleneck distance, and equivalently

$d_B(\dgm_k^R,\dgm_k^{DR})\le k \cdot \diam(X)$

on the original scale, with $\diam(X) = \max_{x,y\in X}\|x-y\|$ (Clémot et al., 19 Dec 2025).

Stability holds locally: under $\varepsilon$–perturbations preserving the Delaunay triangulation, the bottleneck distance between diagrams is at most $2\varepsilon$ (Mishra et al., 2023). However, the DR diagram is not uniformly stable under arbitrarily small perturbations when the Delaunay triangulation flips, and discontinuous jumps may occur. Explicit cocircular (planar) examples show that the persistence of certain features can change by an amount proportional to $\diam(X)$ under small changes in point configuration (Clémot et al., 19 Dec 2025, Mishra et al., 2023). In the chromatic case, local stability is quantified by the color-preserving distortion metric, ensuring the interleaving distance between chromatic Delaunay–Rips filtrations does not exceed the pointwise displacement (Natarajan et al., 29 May 2024).

4. Empirical Performance and Applications

Empirical benchmarks, across various persistent homology computation frameworks (Ripser, PHAT, Gudhi, Alpha complex-based pipelines), demonstrate that Delaunay–Rips is substantially more efficient than full Rips or alpha filtrations for large $n$ in low $d$:

• Memory: DR uses approximately 30% the memory of PHAT on 3D data of size $n\sim 10^5$.
• Speed: DR is 3–5× faster than PHAT and can be orders of magnitude faster than Ripser as $n$ increases.
• Parallelism: Parallel implementations achieve an additional 3–4× speedup on multicore systems (Clémot et al., 19 Dec 2025).

In machine learning pipelines, DR persistence diagrams enable competitive classification accuracy (within a few percentage points of Alpha and Rips), both in synthetic shape recognition and biomedical time series (e.g., EKG-based sleep/wake classification), with negligible practical loss of information (Mishra et al., 2023).

Chromatic Delaunay–Rips filtrations are particularly efficient for multi-species or multi-class datasets, offering speedups of one to two orders of magnitude over chromatic alpha complexes in large planar datasets (Natarajan et al., 29 May 2024).

5. Comparative Perspective and Topological Relations

The Delaunay–Rips filtration interpolates between Čech/alpha filtrations (costly due to full Voronoi computations) and Rips filtrations (combinatorially explosive). In low-to-moderate $d$, DR offers a rigorous, sparse, and computationally tractable alternative, with persistence diagrams provably approximating those of Rips up to interleaving constants.

Via generalized discrete Morse theory, chromatic DR filtrations are shown to be related by simplicial collapses to Čech and chromatic alpha filtrations. Thus, their persistent homology is equivalent up to simple homotopy, enabling efficient computation with theoretical guarantees of topological fidelity (Natarajan et al., 29 May 2024).

6. Practical Guidelines and Implementation Considerations

The DR filtration is most advantageous for:

• Large point clouds in $d\le 4$ (with practical limits at $d\le8$ due to Delaunay triangulation cost).
• Applications where Delaunay triangulation is readily available or can be efficiently computed.
• Scenarios requiring rapid, memory-efficient computation of persistence diagrams without significant loss in topological information.

Practitioners should avoid near-degenerate configurations if strict stability is required, since Delaunay flips can induce bottleneck jumps in the persistence diagram. For high-dimensional or degenerate-data scenarios, the exponential cost of Delaunay construction may negate DR’s advantages. Chromatic DR filtrations are recommended for multi-class point cloud datasets where capturing inter-class relations is relevant (Clémot et al., 19 Dec 2025, Mishra et al., 2023, Natarajan et al., 29 May 2024).

7. Summary Table: Key Properties of Delaunay–Rips Persistence Diagrams

Property	Delaunay–Rips Filtration	Full Rips/Alpha Complexes
Simplicial size	$O(n^{\lceil d/2\rceil})$	$O(n^{k+1})$ (Rips), $O(n^{\lceil d/2\rceil})$ (Alpha)
Approximation bound	Provable: $d_B(\log\dgm_k^R,\log\dgm_k^{DR})\le\log(k+1)$	—
Stability	Local (if Delaunay unchanged)	Uniform (Rips/Alpha)
Computational cost	Low ($d\leq8$), parallelizable	High for large $n$ or $k$
Applicability	Best for $d\leq8$, large $n$	Small $n$ or exact topology

The Delaunay–Rips persistence diagram framework provides a computationally and theoretically robust approach for large-scale topological data analysis in Euclidean settings, and generalizes efficiently to multi-category data with chromatic filtrations. Its use is now supported by rigorous approximation, stability analysis, and documented algorithmic and empirical advances (Clémot et al., 19 Dec 2025, Mishra et al., 2023, Natarajan et al., 29 May 2024).