The sublevel Flood bifiltration: towards scalable 2-parameter persistent homology

Mattéo Clémot; Julie Digne; Julien Tierny

math.ATOct 4, 2026

The sublevel Flood bifiltration: towards scalable 2-parameter persistent homology

Authors: Mattéo Clémot, Julie Digne, Julien Tierny

Organizations: Université Lyon 1 · CNRS, Université Lyon 1 · CNRS, Sorbonne Université

Abstract

Multiparameter persistent homology is a rapidly developing branch of topological data analysis that improves the robustness of single-parameter persistent homology to outliers, while still capturing the metric characteristics of the data. However, a notable limitation is its lack of scalability. In this paper, we introduce a novel approach for efficiently computing 2-parameter persistent homology on large point sets. Our work extends the Flood filtration, originally developed for single-parameter persistence. Our construction, called the sublevel Flood bifiltration, offers a scalable approximation of the sublevel offset bifiltration. We show that it benefits from theoretical stability properties and describe how to compute it efficiently. We demonstrate the performance of our approach in classification tasks on low-dimensional synthetic datasets, where density awareness is critical, as well as on real-world time series datasets.

Figures & tables

Panel of Figure 1 — Figure 1 : (a) A point set $X\subset\mathbb{R}^{2}$ colored by its kernel density estimate (KDE). (b) The persistence diagram of its Delaunay–(Čech) filtration captures many small cycles (blue dots) with similar persistence (between $0.01$ and $0.02$ ). (c) Multiparameter module approximation (MMA) of the sublevel Delaunay bifiltration. (d) MMA of our sublevel Flood bifiltration (100 landmarks). The yellow (c) and green (d) large shapes both distinctively capture the prominent ring (a) with high density and large diameter (up to $0.8$ ).

Figure 2 : $\flood_{r,s}(X,L,\gamma)$ is represented with black vertices, black edges and dark gray triangles for $L\subset X\subset\mathbb{R}^{2}$ ( $|X|=400$ , $|L|=30$ ), and $\gamma:X\to\mathbb{R}$ a codensity function obtained by a Gaussian kernel density estimator. The sublevel offset bifiltration $\mathcal{O}_{r,s}(X,\gamma)=\mathcal{O}_{r}(X_{s})$ is shown in light gray.

Figure 3 : Running time of the bifiltration computation and its multiparameter module approximation. The homology is computed for degrees

k\in\{0,1\}

\mathbb{R}^{2}

(left) and degrees

k\in\{0,1,2\}

\mathbb{R}^{3}

(right).

Figure 4 : Matching distance between the modules of sublevel Flood and sublevel Delaunay bifiltrations, when the number of landmarks

|L|

increases.

X

has 10,000 points distributed over the noisy ring in

\mathbb{R}^{2}

(left), the noisy sphere in

\mathbb{R}^{3}

(center) and the noisy torus in

\mathbb{R}^{3}

(right).

Figure 5 : Accuracy scores (averaged over 10 splits) of single-parameter PH (diameterPD, densityPD and bothPD) and multiparameter PH (Flood and Delaunay). Left: Synthetic noisy swisscheese toy datasets in

\mathbb{R}^{2}

parameterized by difficulty

x

. The dataset contains here 800 sets equally split into four classes characterized by the diameter of their low-density regions. Each set contains 5,000 points. Right: Synthetic porous material datasets on a

64^{3}

grid. The problem difficulty increases with the parameter

p

that quantifies the level of ambient noise. We also show the result obtained with the neural network-based method PVT Zhang et al. (2022) for reference.

Dataset	Flood100	Flood300	Flood500	Flood1000	Delaunay
NSC-2D-5k	38.3 / 17.4	85.3 / 32.3	142.7 / 49.1	312.6 / 57.8	1484.6 / 91.8
NSC-2D-20k	91.3 / 11.5	150.2 / 12.9	219.3 / 19.4	416.4 / 38.3	OOM
NSC-3D-5k	183.9 / 24.8	800.7 / 43.6	2309.3 / 53.0	13990.6 / 130.3	OOM
rocks64	602.7 / 11.3	1086.7 / 16.1	1815.3 / 41.8	5925.6 / 70.3	OOM

Table 1 : Timing in seconds for the noisy swisscheese datasets with 5,000 and 20,000 points ( NSC-*D-*k ) and the porous rocks dataset ( rocks64 ), averaged over all their difficulties. The left number is the computation time of MMAs of the whole dataset. The right number is the classification time from the MMAs (averaged over the 10 splits). OOM means that memory was exceeded. A point set in rocks64 has on average 69,000 points for

p=0

and 130,000 points for

p=0.9

Figure 8

Dataset	ED	DTWc	Čech	$\mathcal{D}$ /MPL	$\flood$ /MPL	$\mathcal{D}$ /SCDR	$\flood$ /SCDR
DPOAG	0.626	0.770	0.698	0.741	0.669	0.698	0.719
DPOC	0.717	0.717	0.699	0.641	0.649	0.659	0.732
DPTW	0.633	0.590	0.647	0.576	0.647	0.597	0.640
IPD	0.955	0.950	0.710	0.722	0.724	0.747	0.724
PC	0.933	0.878	0.889	0.744	0.728	0.883	0.911
LN2	0.754	0.869	0.721	0.525	0.541	0.607	0.705

Table 2 : Accuracy scores on UCR time series datasets.

\mathcal{D}

and

\flood

denote the Delaunay and Flood bifiltrations, respectively.

Appendix figures & tables17 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 6 — Figure 6 : Two bifiltrations over a filled triangle indexed over $\llbracket 0,3\rrbracket^{2}$ . The left one is 1-critical while the right one is 3-critical. The blue (resp. red) dots represent the minimal set of bigrades $\bigrades(\sigma)$ of the blue (resp. red) edge. Their respective supports $\supp(\sigma)$ are highlighted with the same color.

Panel of Figure 7 — Figure 7 : Illustration of the tensor operations that compute the set of bigrades of a triangle $\sigma\in\mathrm{Del}(L)$ . Top left: the sample $P_{\sigma}$ (black dots) has size 10 and the set of points $X^{\sigma}=\{x_{0}^{\sigma},\ldots,x_{11}^{\sigma}\}$ has size 12 and is sorted and colored by $\gamma$ value. Bottom left: the support of $\sigma$ (in gray, with the minimal set of bigrades $\bigrades(\sigma)$ shown with black dots) is computed from the distance matrix $D_{\sigma}=\mathrm{d}(P_{\sigma},X^{\sigma})$ of which we take the cumulative minimum operation along each row, and then the maximum along each column.

Figure 8 : Running time of the bifiltration computation (plus its minimal presentation for Rips and Delaunay) for

X\subset\mathbb{R}^{2}

(left) and

X\subset\mathbb{R}^{3}

(right). Contrary to Fig. 3 , this does not include multiparameter module approximation.

Figure 9 : Top line: size of the bifiltrations, i.e., total number of stored bigrades. For 1-critical bifiltrations (all except sublevel Flood), this corresponds to the number of simplices. For the minimal presentations, the total number of generators is shown instead (all homology dimensions). Middle line: total number of summands in the multiparameter module approximation (all homology dimensions). Bottom line: average criticality

|\bigrades(\sigma)|

in the sublevel Flood bifiltrations.

Panel of Figure 10 — Figure 10 : Stability of the sublevel Flood bifiltration with respect to $X$ , as measured by the matching distance between $\floodmodule(X,L,\gamma_{X})$ and $\floodmodule(X^{\prime},L^{\prime},\gamma_{X^{\prime}})$ with $X^{\prime}=X+E$ and with $\gamma_{X}$ and $\gamma_{X^{\prime}}$ obtained as KDEs. In the top line, $L=L^{\prime}$ is obtained as a FPS on $X$ (fixed landmarks, as in Theorems 1 and 2 ). In the bottom line, $L^{\prime}$ is computed as a FPS from $X^{\prime}$ . $L$ has 1,000 points and $X$ has 10,000 points distributed over the noisy ring in $\mathbb{R}^{2}$ (left), the noisy sphere in $\mathbb{R}^{3}$ (center) and the noisy torus in $\mathbb{R}^{3}$ (right).

Figure 11 : Accuracy scores (averaged over 10 splits) of single-parameter PH (diameterPD, densityPD and bothPD) and multiparameter PH (Flood) on the synthetic noisy swisscheese toy datasets parameterized by difficulty

x

. The dataset contains here 800 sets equally split into four classes characterized by the diameter of their low-density regions. The black lines show the score of a random classifier. Left: Each point set contains 20,000 points in

\mathbb{R}^{2}

. Right: Each point set contains 5,000 points in

\mathbb{R}^{3}

Dataset	ED	DTWc	Čech	$\mathcal{D}$ /MPL	$\flood$ /MPL	$\mathcal{D}$ /SCDR	$\flood$ /SCDR
PPOAG	0.785	0.805	0.785	0.829	0.785	0.815	0.815
PPOC	0.808	0.783	0.832	0.663	0.674	0.814	0.821
PPTW	0.707	0.756	0.737	0.776	0.756	0.741	0.771
ECG200	0.880	0.770	0.710	0.740	0.790	0.830	0.830
MI	0.684	0.737	0.584	0.533	0.547	0.628	0.514
Plane	0.962	1.000	0.876	0.629	0.790	0.933	0.610

Appendix

Table 3 : Accuracy scores on additional UCR time series datasets.

\mathcal{D}

and

\flood

denote the Delaunay and Flood bifiltrations, respectively.

Figure 12 : Three sampling approaches to get 500 landmarks (red dots) from an input $X\subset\mathbb{R}^{2}$ of size 10,000 (black dots).

	Noisy ring ( $\mathbb{R}^{2}$ )		Noisy sphere ( $\mathbb{R}^{3}$ )
	$\matchd$ ( $H_{0}$ )	$\matchd$ ( $H_{1}$ )	$\matchd$ ( $H_{0}$ )	$\matchd$ ( $H_{1}$ )	$\matchd$ ( $H_{2}$ )
FPS	$0.022\pm 0.005$	$0.020\pm 0.005$	$0.059\pm 0.011$	$0.052\pm 0.005$	$0.067\pm 0.009$
random	$0.012\pm 0.004$	$0.012\pm 0.004$	$0.044\pm 0.007$	$0.032\pm 0.011$	$0.045\pm 0.017$
DA-FPS	$0.012\pm 0.003$	$0.009\pm 0.003$	$0.049\pm 0.008$	$0.035\pm 0.004$	$0.038\pm 0.007$

Appendix

Table 4 : Matching distance between the sublevel Delaunay and Flood bifiltrations for several landmark sampling techniques: classical and density-aware furthest point sampling, and random sampling. The input point sets have 10,000 points and 500 landmarks are selected. The matching distances are averaged over 100 runs.

Figure 13 : Accuracy score for two classification problems using the sublevel Flood bifiltration, as a function of the number of landmarks.

Panel of Figure 14 — Figure 14 : Accuracy score for the classification problem involving counting the number of voids, using the sublevel Flood bifiltration, as a function of the number of landmarks. On the right, we show the Delaunay triangulation of the set of landmarks when the classification is unsuccessful ( $|L|=100$ ) or successful ( $|L|=316$ ).

Figure 15 : Multiparameter module approximation when the number of landmarks increases, for a point set

X

of size 2000 sampled as a noisy ring in

\mathbb{R}^{2}

Figure 16 : Comparison of runtimes for computing the sublevel Flood bifiltration (no MMA) on a GPU and a CPU (we used a Core i7-13850HX CPU for this experiment).

$\|X\|$	$\|L\|$	Runtime (s)		$\matchd$ ( $H_{0}$ )		$\matchd$ ( $H_{1}$ )		$\matchd$ ( $H_{2}$ )
$\|X\|$	$\|L\|$	M	E	M	E	M	E	M	E
$1000$	$100$	$0.14$	$0.39$	$0.17$	$0.12$	$0.04$	$0.05$	$0.05$	$0.02$
	$300$	$0.46$	$1.38$	$0.09$	$0.09$	$0.03$	$0.03$	$0.06$	$0.01$
	$500$	$0.76$	$2.46$	$0.07$	$0.07$	$0.03$	$0.03$	$0.06$	$0.01$
	$1000$	$1.70$	$5.07$	$0.01$	$0.01$	$0.03$	$0.03$	$0.02$	$0.01$
$20000$	$100$	$0.55$	$5.03$	$0.05$	$0.05$	$0.08$	$0.10$	$0.10$	$0.11$

Appendix

Table 5 : Comparison between the exact (E) and masked (M) approaches (see Section 3.3 ) in terms of runtime and approximation of the sublevel Delaunay bifiltration.

$h$	Flood (300 l.)	Delaunay
$0.004$	$0.793\pm 0.027$	$0.817\pm 0.019$
$0.01$	$0.891\pm 0.016$	$0.920\pm 0.017$
$0.04$	$0.916\pm 0.015$	$0.923\pm 0.017$
$0.1$	$0.774\pm 0.029$	$0.794\pm 0.025$
$0.4$	$0.429\pm 0.035$	$0.454\pm 0.031$

Appendix

Table 6 : Average classification accuracy on the 2-dimensional swisscheese dataset with the sublevel Flood (300 landmarks) and Delaunay bifiltrations, using Gaussian KDE with several bandwidths

h

Figure 17 : Two examples of each class in the noisy_swiss_cheese dataset in $\mathbb{R}^{2}$ , with 10,000 points in each point set and difficulty parameter $x=0.14$ .

Dataset	UCR dataset full name	Train	Test	Length	Classes	Category
DPOAG	DistalPhalanxOutlineAgeGroup	400	139	80	3	image
DPOC	DistalPhalanxOutlineCorrect	600	276	80	2	image
DPTW	DistalPhalanxTW	400	139	80	6	image
PPOAG	ProximalPhalanxOutlineAgeGroup	400	205	80	3	image
PPOC	ProximalPhalanxOutlineCorrect	600	291	80	2	image
PPTW	ProximalPhalanxTW	400	205	80	6	image

Appendix

Table 7 : The list of the UCR datasets used in this paper, with their acronym and their full name, the size of train and test sets, the length of the time series and the number of classes.

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Abderrahim Bendahi, Alexandre Duplessis, Arnaud Fickinger

ENS Ulm, PSL Paris, France · UC Berkeley Berkeley, CA, USA

Topology Optimization Persistent Homology Shape Optimization Voronoi-Type Losses Graph Laplacians

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Persistence-based topological optimization deforms a point cloud

X \subset \mathbb{R}^d

by minimizing objectives of the form

L(X) = \ell(\mathrm{Dgm}(X))

, where

\mathrm{Dgm}(X)

is a persistence diagram. In practice, optimization is limited by two coupled issues: persistent homology is typically computed on subsamples, and the resulting topological gradients are highly sparse, with only a few anchor points receiving nonzero updates. Motivated by diffeomorphic interpolation, which extends sparse gradients to smooth ambient vector fields via Reproducing Kernel Hilbert Space (RKHS) interpolation, we propose a more scalable pipeline that improves both subsampling and gradient extension. We introduce subsampling via random slicing, a lightweight scheme that promotes iteration-wise geometric coverage and mitigates density bias. We further replace the costly kernel solve with a fast Nadaraya-Watson (NW) Gaussian convolution, producing a globally defined smooth update field at a fraction of the computational cost, while being more suited for topological optimization tasks. We provide theoretical guarantees for NW smoothing, including anchor approximation bounds and global Lipschitz estimates. Experiments in

2

D and

3

D show that combining random slicing with NW smoothing yields consistent speedups and improved objective values over other baselines on common persistence losses.

Abstract

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