The Density of Cross-Persistence Diagrams and Its Applications

Alexander Mironenko; Evgeny. Burnaev; Serguei Barannikov

cs.AIMar 12, 2026

The Density of Cross-Persistence Diagrams and Its Applications

Authors: Alexander Mironenko, Evgeny. Burnaev, Serguei Barannikov

Organizations: Skolkovo Institute of Science and Technology, Moscow 121205, Russia · AIRI, Moscow 123317, Russia · CNRS, IMJ, Paris University, Paris 75205, France

Abstract

Topological Data Analysis (TDA) provides powerful tools to explore the shape and structure of data through topological features such as clusters, loops, and voids. Persistence diagrams are a cornerstone of TDA, capturing the evolution of these features across scales. While effective for analyzing individual manifolds, persistence diagrams do not account for interactions between pairs of them. Cross-persistence diagrams (cross-barcodes), introduced recently, address this limitation by characterizing relationships between topological features of two point clouds. In this work, we present the first systematic study of the density of cross-persistence diagrams. We prove its existence, establish theoretical foundations for its statistical use, and design the first machine learning framework for predicting cross-persistence density directly from point cloud coordinates and distance matrices. Our statistical approach enables the distinction of point clouds sampled from different manifolds by leveraging the linear characteristics of cross-persistence diagrams. Interestingly, we find that introducing noise can enhance our ability to distinguish point clouds, uncovering its novel utility in TDA applications. We demonstrate the effectiveness of our methods through experiments on diverse datasets, where our approach consistently outperforms existing techniques in density prediction and achieves superior results in point cloud distinction tasks. Our findings contribute to a broader understanding of cross-persistence diagrams and open new avenues for their application in data analysis, including potential insights into time-series domain tasks and the geometry of AI-generated texts. Our code is publicly available at https://github.com/Verdangeta/TDA_experiments

Figures & tables

Fig. 1: Standard pipeline for Cross-barcode vectorization

Panel of Figure 2 — Fig. 2: In these pictures, the density of $MTD(Q_{1},Q_{1})$ is represented by dashed lines, and all other densities are represented by continuous lines. For each dataset, there is only one picture presented with one core cloud.

Panel of Figure 3 — Fig. 3: Densities of $MTD(Q_{1},Q_{1})$ (dashed) and $MTD(Q_{1},Q_{s})$ (solid). As can be seen from these images, the probability that our approach suggests that the value of $MTD(Q_{1},Q_{k})$ was sampled from the same distribution is greater than the classical significance threshold (0.05).

Panel of Figure 4 — Fig. 4: Densities of $MTD(Q_{1},Q_{1})$ (dashed) and $MTD(Q_{1},Q_{s})$ (solid), where Gaussian noise is applied only to the right argument (i.e., $MTD(\text{Pure}_{i},\text{Noised}_{i})$ ). From left to right, the relative noise norms $||\xi||/||x||$ are $[0\%,25\%,50\%,75\%]$ . As the noise intensity increases, the density of $MTD(Q_{1},Q_{1})$ gradually shifts to the right and eventually merges with the others once the objects become indistinguishable.

Fig. 5: The process of adding Gaussian noise to COIL20 images. From left to right, the relative noise levels $||\xi||/||x||$ are $[0\%,25\%,50\%,75\%]$

Panel of Figure 6 — Fig. 6: Densities of $MTD(Q_{1},Q_{1})$ (dashed) and $MTD(Q_{1},Q_{s})$ (solid). The relative noise norm is $||\xi||/||x||=50\%$ . These plots illustrate three regimes of point cloud comparison: without noise, with noise applied only to the right point cloud, and with noise applied to both.

Panel of Figure 7 — Fig. 7: Densities of $MTD(Q_{1},Q_{1})$ (dashed) and $MTD(Q_{1},Q_{s})$ (solid), where Gaussian noise is applied only to the right argument (i.e., $MTD(\text{Pure}_{i},\text{Noised}_{i})$ ). From left to right, the relative noise norms $||\xi||/||x||$ are $[0\%,25\%,50\%,75\%]$ . In this high-complexity regime, direct overlap comparison becomes less visually informative due to substantial distributional heterogeneity. However, the overlap values remain far from saturation, indicating persistent geometric discrepancy. The relative concentration (dispersion) of $MTD(Q_{1},Q_{1})$ compared to $MTD(Q_{1},Q_{s})$ provides additional descriptive insight into geometric variability. See Appendix C for discussion of this regime.

Fig. 8: Illustration of the proposed architectures. Let $N$ and $M$ denote the sizes of the input point clouds, and let $W=N+M$ . The parameter $K$ represents the output dimensionality of the dimension reduction method applied to the distance matrix, which contains zeros for all pairs of points within the right cloud. (a) RipsNet includes only the combined cloud block. (b) Cross-RipsNet includes all blocks except the distance features. (c) Cross-RipsNet with distance matrix incorporates all input blocks, including both point coordinates and distance-based features. See Appendix H for the ablation study of the inputs.

Symmetrical KL distance averaged among test data
Model	Synthetic	3D shapes	Textual data
RipsNet	$2.60\pm 0.10$	$0.69\pm 0.01$	$0.90\pm 0.03$
Cross-RipsNet	$2.34\pm 0.05$	$0.52\pm 0.01$	$0.62\pm 0.03$
Cross-RipsNet/PCA	$2.40\pm 0.04$	$0.57\pm 0.01$	$0.66\pm 0.02$
Cross-RipsNet/MAX	$2.38\pm 0.04$	$0.54\pm 0.02$	$0.66\pm 0.05$
Cross-RipsNet/Quantiles	2.05 $\pm$ 0.08	0.48 $\pm$ 0.03	0.56 $\pm$ 0.01

TABLE I: Comparison of methods for predicting the density of cross-persistence diagrams across three domains. The

RipsNet

and

Cross-RipsNet

models do not use the distance matrix as input, while the other methods reduce its dimensionality to

K=60

using different approaches.

Panel of Figure 10 — Fig. 9: Examples of densities and corresponding pairs of point clouds for the cross-persistence diagrams in the synthetic dataset. All plots share the same spatial boundaries.

Panel of Figure 11 — Fig. 10: Results for the 3D shapes dataset. For each part of data, the left column shows the real persistence image, and the right column shows the predicted persistence image. Cross-RipsNet accurately predicts the center and shape of the density.

Panel of Figure 12 — Fig. 11: Results for text data. For each part of data, the left column shows the real persistence image, and the right column shows the predicted persistence image. The Cross-RipsNet model captures the size and shape of the density accurately.

Panel of Figure 13 — Fig. 12: Results for text data for zero-dimensional homology case. For each part of data, the left column shows the real persistence image, and the right column shows the predicted persistence image. Cross-RipsNet effectively learns the specificity of the data and achieves good results.

Domain	Cross-Barcodes	Training + Predicting	Time Gain
3D	2.60	0.40	6.5 $\times$
Text	3.00	0.74	4.0 $\times$
Synthetic	0.60	0.55	1.15 $\times$

TABLE II: Time comparison of cross-barcode density computation for each domain using classical methods versus Cross-RipsNet (trained on

80\%

of data, predicting

20\%

). Times are reported in hours.

Panel of Figure 15 — Fig. 13: Example of gravitational wave noise process. (a) Gravitational wave with noise. (b) Time-delay embedded gravitational wave visualized in 3D.

Model	Accuracy (train)	ROC (train)	Accuracy (valid)	ROC (valid)
Persistence Entropy (Giotto)	$0.658\pm 0.004$	$0.703\pm 0.005$	$0.650\pm 0.018$	$0.700\pm 0.059$
Cross-Entropies	$0.666\pm 0.001$	$0.720\pm 0.004$	$0.700\pm 0.030$	$0.747\pm 0.040$
MTDs	$0.686\pm 0.007$	$0.746\pm 0.006$	$0.704\pm 0.054$	$0.772\pm 0.052$
MTDs+Cross-Entropies	0.692 $\pm$ 0.004	0.776 $\pm$ 0.005	0.711 $\pm$ 0.040	0.783 $\pm$ 0.040

TABLE III: Results of experiments with gravitational waves. Best scores are shown in bold

Model	Worms	WormsTwoClass	Computers	Earthquakes	RefrigerationDevices
catch22	$0.766\pm 0.011$	$0.784\pm 0.019$	$0.764\pm 0.032$	$0.804\pm 0.012$	$0.639\pm 0.019$
TopGen	$0.701\pm 0.010$	$0.728\pm 0.017$	$0.657\pm 0.017$	$0.816\pm 0.019$	$0.494\pm 0.016$
FreshPrince	$0.779\pm 0.027$	$0.777\pm 0.036$	0.833 $\pm$ 0.024	0.821 $\pm$ 0.020	0.692 $\pm$ 0.020
catch22+TopGen	0.779 $\pm$ 0.024	0.784 $\pm$ 0.028	$0.758\pm 0.030$	0.814 $\pm$ 0.020	0.661 $\pm$ 0.020
Fresh+TopGen	0.783 $\pm$ 0.026	0.784 $\pm$ 0.022	$0.825\pm 0.027$	$0.820\pm 0.021$	$0.690\pm 0.020$

TABLE IV: Experimental results on UCR datasets. Best scores are shown in bold ; improved scores using TopGen features are underlined .

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

Fig. 14: Illustration of the overlap metric $\mathcal{O}(p,q)$ on four representative pairs of densities $p$ and $q$ .

Panel of Figure 19 — Fig. 15: Comparison of cross-persistence diagram densities for different input combinations. A higher level of diversity is evident in the densities corresponding to human texts on the left side.

Panel of Figure 20 — Fig. 16: Projections of cross-persistence densities onto a 2D space using different dimensionality reduction techniques. Human vs Any text densities (blue) and AI vs Any text densities (orange) exhibit partial separation, confirming their distinct characteristics.

Panel of Figure 21 — Fig. 17: Projections of concatenated persistence images onto a 2D space using different dimensionality reduction techniques. It can be seen, that with usage of common persistence images, detection wether left cloud originates from AI or Human became much more harder, as support of Human points in this case are not localized.

Model	Accuracy (train)	ROC (train)	Accuracy (valid)	ROC (valid)
Wiki dataset (Wikipedia prompts)
Persistence Entropy (baseline)	$0.839\pm 0.003$	$0.942\pm 0.001$	$0.841\pm 0.027$	$0.945\pm 0.007$
Cross-Entropies	$0.948\pm 0.004$	$0.970\pm 0.001$	$0.950\pm 0.009$	$0.977\pm 0.007$
MTDs	$0.955\pm 0.001$	$0.975\pm 0.001$	$0.960\pm 0.007$	$0.980\pm 0.006$
MTDs + Cross-Entropies	0.958 $\pm$ 0.002	0.976 $\pm$ 0.001	0.966 $\pm$ 0.007	0.981 $\pm$ 0.006
Reddit dataset (question-based prompts)

Appendix

TABLE V: Results of AI vs. human text classification on Wiki and Reddit datasets. We report mean

\pm

standard deviation over repeated runs. Wiki texts are generated using Wikipedia prompts, while Reddit texts are generated from question-based prompts following the experimental protocol of [ 8 ] . Best validation scores within each dataset block are shown in bold .

Panel of Figure 23 — Fig. 18: Cross-RipsNet’s performance in predicting MTD density, for the synthetic dataset and two randomly chosen pairs of clouds from the test set.

Fig. 19: COIL20 sensitivity analysis: mean intersection between cross-persistence density estimates across all classes vs. noise level. Blue curve: pure vs. noise (only the right point cloud is noised). Orange curve: noise vs. noise (both point clouds are noised).

Panel of Figure 25 — Fig. 20: Boxplot showing the distribution of KL divergence values for cross-persistence density prediction across multiple random seeds on the test set. The plot compares the original Cross-RipsNet model (using all inputs) with a modified version in which the input corresponding to the second point cloud has been removed.

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Persistent entropy is the Shannon entropy of a persistence-based probability measure defined on a persistence diagram. However, its cross-entropy version is not naturally defined because two persistence diagrams generally have different event spaces. To bridge these event spaces, we combine a similarity function with persistence weighting to define an induced probability. The induced probability reflects information from one diagram on the event space of the other diagram and assigns unexplained probability mass to the unexplained event. Using the induced probability, we extend cross entropy to persistence diagrams, called persistent cross entropy (PCE). We establish the main properties of both the induced probability and PCE and prove stability theorems for both. Through three numerical studies, we show that PCE distinguishes diagrams with the same persistent entropy, separates causal directions in dynamical systems without constructing a joint persistent diagram, and can be used as a directional topology loss for knowledge distillation.

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