Persistent Cross Entropy

Sijin Yeom; Jae-Hun Jung

cs.LGAug 25, 2026

Persistent Cross Entropy

Authors: Sijin Yeom, Jae-Hun Jung

Organizations: Artificial Intelligence Graduate School, Pohang University of Science and Technology, Pohang, Korea · Department of Mathematics, Pohang University of Science and Technology, Pohang, Korea

Abstract

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.

Figures & tables

Figure 1: Two-loop experiment with equal persistent entropy. The first row shows the point clouds $X,Y_{1},\ldots,Y_{4}$ , and the second row shows $D_{X},D_{Y_{1}},\ldots,D_{Y_{4}}$ ; the square and triangle in $D_{X}$ mark the large- and small-loop features, respectively, and circles denote the remaining noise features. Each diagram is annotated with its persistent entropy. The third row shows the actual augmented probability masses $p_{X}^{X}(u),p_{X}^{Y_{1}}(u),\ldots,p_{X}^{Y_{4}}(u)$ for $u\in D_{X}$ on a common color scale and reports the corresponding PCE. These masses are not renormalized within $D_{X}$ . The bar below each panel completes the probability by partitioning unit mass between $D_{X}$ and the unexplained event $\partial$ .

Figure 2: Spring–mass directional comparison. Each point represents one coupling pair: blue denotes the independent case, orange denotes $A\to B$ , green denotes $B\to A$ , and red denotes bidirectional coupling. The dashed line is $y=x$ ; under the common plotting convention, the lower-right half favors $A\to B$ , the upper-left half favors $B\to A$ , and points near the diagonal show little directional preference. The two PCE-based panels are computed only from $D_{A}$ and $D_{B}$ . They plot $(\Delta H_{B}^{A},\Delta H_{A}^{B})$ and $(p_{B}^{A}(\partial),p_{A}^{B}(\partial))$ , respectively. Each coordinate in the second panel is the fraction of persistence mass not explained in that direction. Every remaining panel uses the joint diagram and plots $(d(D_{AB},D_{A}),d(D_{AB},D_{B}))$ . In order, the panels use bottleneck distance, $2$ -Wasserstein distance, $L^{2}$ distances between Betti curves, persistence landscapes, and persistence silhouettes, the persistence scale-space kernel distance, and Euclidean distance between persistence images.

Method	Four regimes $R^{2}$	Two directions $R^{2}$
Entropy excess	$\mathbf{0.560}$	$0.740$
Unexplained mass	$0.449$	$\mathbf{0.993}$
Bottleneck	$0.180$	$0.733$
Wasserstein	$0.050$	$0.662$
Betti curve	$0.091$	$0.513$
Persistence landscape	$0.197$	$0.876$

Table 1: Separation of the coupling regimes in the two-dimensional representations shown in Figure 2 . Larger PERMANOVA

R^{2}

indicates stronger separation. The best result in each evaluation is shown in bold.

Method	Final	Last-10 mean
TopKD	$70.99$	$71.000$
PCE	$71.25$	$71.223$
EM-PCE	$\mathbf{71.49}$	$\mathbf{71.606}$

Table 2: Seed-

7

CIFAR-100 Top-1 accuracy (%). Final is the accuracy at epoch

240

, and Last-10 mean is the mean over epochs

231

–

240

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Data	Point-cloud size	Number of $H_{1}$ points	Persistent entropy
$X$	239	12	1.500261
$Y_{1}$	758	64	1.500305
$Y_{2}$	695	70	1.500447
$Y_{3}$	605	59	1.500307
$Y_{4}$	84	13	1.500474

Appendix

Table 3: Two-loop data and persistence-diagram summary.

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Persistence diagrams (PDs) provide stable and interpretable summaries of multiscale topological structure. While substantial progress has been made in the statistical analysis of PDs, existing literature often treats diagrams as static objects and provide limited frameworks for probabilistic modeling and stochastic evolution on PD space. We introduce a reinforcement learning framework for stochastic dynamics on PD space, where diagrams evolve through topology aware local edit operations. The dynamics define controlled Markov processes on spaces of finite PDs with variable cardinality. We establish conditions under which the induced Markov chains are irreducible, aperiodic, and geometrically ergodic, implying the existence of unique stationary probability laws on PD space. To guide the dynamics toward scientifically relevant topological targets, we formulate objectives that encompass distribution matching, task specific topological statistics, and structure-preserving compression. The resulting rewards balance task specific distributional targets, diagram fidelity, and complexity reduction, and yield a framework for adaptive topological simplification and probabilistic modeling. Experiments on synthetic and neuroimaging PDs demonstrate that the proposed framework can preserve dominant topological structure while reducing diagram complexity.

Abstract

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