LandscapeSHAP: Which Persistent Homology Class Gets the Credit?

Nikola Milićević

math.ATSep 25, 2026

LandscapeSHAP: Which Persistent Homology Class Gets the Credit?

Authors: Nikola Milićević

Abstract

Shapley values, a solution concept from cooperative game theory, have recently become a standard tool for feature credit allocation in machine learning. They provide an axiomatically justified method to fairly distribute a model's prediction among the data features. Shapley values have not yet been applied to explain machine learning models trained on features from topological data analysis. We develop what we believe is the first such approach, focusing on the persistence landscape featurization of persistence diagrams. Because each landscape coordinate is a rank statistic, crediting a model's prediction back to individual persistent homology classes (persistence diagram points) is nontrivial. We introduce LandscapeSHAP, a method for fair credit allocation to persistence diagram points based on a model's prediction. For linear models on persistence landscapes, LandscapeSHAP has a closed form expression that gives the exact Shapley value of every persistence diagram point. In particular, there is no coalition sampling required. We further prove that the four Shapley "fairness" axioms uniquely characterize this credit allocation for any model, not only linear ones. For a general nonlinear model, this unique value can only be calculated exactly from its defining coalition averaging formula, which requires considering all $2^N$ many coalitions, where $N$ is the number of points in the persistence diagram. This is computationally intractable for persistence diagrams of realistic size. We complement the exact linear model result with an efficient Monte Carlo sampling of persistence diagram coalitions. We give convergence rates in terms of number of samples needed to approximate to a desired degree of accuracy. We also prove stability results for the LandscapeSHAP credit allocation, for any model.

Figures & tables

Panel of Figure 1 — Figure 1: The TDA to machine learning pipeline. Explainable AI methods based on Shapley values are well established for attributing model predictions back to featurization coordinates (blue brace), but pulling this credit back further to the individual points of the persistence diagram (red dashed arrow) has not been addressed.

Panel of Figure 2 — Figure 2: Persistence diagram and landscape example. (panel a) and their associated tent functions $\ell_{A}$ , $\ell_{B}$ and $\ell_{C}$ together with the three persistence landscape layers (panel b).

S_{N+1}=0,\quad S_{j}=S_{j+1}+\dfrac{t_{(j,m)}-t_{(j+1,m)}}{j},\quad j=N,\dots,1

Algorithm 1 LandscapeSHAP(m)

\varphi_{i}\leftarrow\varphi_{i}+w_{(k,m)}\cdot S_{\max(r_{i},k)}

Algorithm 2 LandscapeSHAP for linear models

Figure 3: Samples from discs of radius $1$ of constant curvatures. Left, $1000$ points sampled from disc of curvature $-2$ , middle $1000$ points sampled from curvature $0$ , right $1000$ points sampled from curvature $2$ .

Figure 4: The first principal component ( $y$ -axis) plotted against the true underlying curvature values associated with the average persistence landscape vectors.

Figure 5: LandscapeSHAP explaining (linear) curvature regression. Top, a persistence diagram coming from a point cloud sampled from curvature $\kappa=-2$ , colored by using the Shapley values of the points. The top five Shapley values (in absolute value) are selected and the corresponding representative cycles are shown. Middle, a persistence diagram coming from a point cloud sampled from curvature $\kappa=0$ , colored by using the Shapley values of the points. The top five Shapley values (in absolute value) are selected and the corresponding representative cycles are shown. Bottom, a persistence diagram coming from a point cloud sampled from curvature $\kappa=2$ , colored by using the Shapley values of the points. The top five Shapley values (in absolute value) are selected and the corresponding representative cycles are shown.

Figure 6: Point clouds with $1000$ points each of the dynamical system for $r\in\{2,3.5,4,4.1,4.3\}$ going from left to right.

Figure 7: LandscapeSHAP explaining (linear) dynamical system classification. First row, a persistence diagram coming from a point cloud sampled from dynamics with $r=2$ , colored by using the Shapley values of the points. Second row, a persistence diagram coming from a point cloud sampled from dynamics with $r=3.5$ , colored by using the Shapley values of the points. Third row, a persistence diagram coming from a point cloud sampled from dynamics with $r=4$ , colored by using the Shapley values of the points. Fourth row, a persistence diagram coming from a point cloud sampled from dynamics with $r=4.1$ , colored by using the Shapley values of the points. Fifth row, a persistence diagram coming from a point cloud sampled from dynamics with $r=4.3$ , colored by using the Shapley values of the points. The top five Shapley values (in absolute value) are selected and the corresponding representative cycles are shown, for each row.

Figure 8: Estimated Shapley values vs exact Shapley values of the linear SVM model for the point clouds in Figure 7 .

Figure 9: Convergence analysis of estimated Shapley values for a linear SVM. Left, the plot of mean error normalized by mean absolute Shapley values for five example diagrams against the number of permutations used on the $\log$ scale. Right, the Pearson correlation between estimates and exact Shapley values (across 8 different random seed estimations) plotted against the number of permutations used on the $\log$ scale.

Figure 10: LandscapeSHAP explaining (nonlinear) dynamical system classification. First row, a persistence diagram coming from a point cloud sampled from dynamics with $r=2$ , colored by using the Shapley values of the points. Second row, a persistence diagram coming from a point cloud sampled from dynamics with $r=3.5$ , colored by using the Shapley values of the points. Third row, a persistence diagram coming from a point cloud sampled from dynamics with $r=4$ , colored by using the Shapley values of the points. Fourth row, a persistence diagram coming from a point cloud sampled from dynamics with $r=4.1$ , colored by using the Shapley values of the points. Fifth row, a persistence diagram coming from a point cloud sampled from dynamics with $r=4.3$ , colored by using the Shapley values of the points. The top five Shapley values (in absolute value) are selected and the corresponding representative cycles are shown, for each row.

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