Source-linked AI summary
LandscapeSHAP: Which Persistent Homology Class Gets the Credit?
Nikola Milićević
TL;DR
Persistence diagrams are not vector representations, and credit assigned to persistence-landscape coordinates does not reveal which diagram points matter. LandscapeSHAP assigns Shapley credit to those points through landscape-based cooperative games, with exact closed forms for linear models and axiomatically unique values for any model. For nonlinear models it offers sampling-based approximation, alongside stability results.
Problem
Persistence diagrams are not vectors or Hilbert-space elements, and attributing credit to landscape coordinates does not identify the responsible persistent homology classes.
Source: §1 · p. 2, Related works · p. 7Method
LandscapeSHAP treats persistence-diagram points as players in landscape-based cooperative games and allocates model-prediction credit using their Shapley values.
Source: §1 · p. 3, Our contributions · p. 4Results
LandscapeSHAP gives exact closed-form credit for linear models, uniquely axiomatically characterized credit for any model, stability results, and approximation guarantees for nonlinear models.
Source: §8 · pp. 37–38Takeaways & Limitations
The method offers a way to explain landscape-trained predictions in terms of persistence-diagram points, including diagrams with different numbers of points.
Source: §8 · p. 37For nonlinear models, exact credit requires evaluating 2^N−1 coalitions, which is infeasible for realistic persistence diagrams.
Source: Our contributions · p. 5
Abstract
from arXiv · showhide
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.
1 Introduction
Persistence diagrams are difficult to use directly for statistical learning and model explanations, while landscape-coordinate attributions do not identify the responsible persistent homology classes. LandscapeSHAP treats diagram points as players in landscape-based cooperative games and allocates model-prediction credit to them.
1 Introduction
- Persistence diagrams are not vectors or Hilbert-space elements, and landscape coordinates alone do not identify which persistent homology classes matter to a model.The introduction notes that a 20-layer landscape sampled at 100 grid points has 2,000 coordinates.Source: §1 · p. 2, Related works · p. 7
Our contributions
- LandscapeSHAP models diagram points as players in a collection of cooperative games defined by persistence landscapes, with each landscape coordinate treated as an order-statistic game.The value of a coalition for a coordinate is the corresponding ranked tent value, with zero padding when needed.Source: Our contributions · pp. 3–4
- The landscape games have an exact closed-form Shapley allocation, uniquely characterized by Efficiency, Symmetry, Linearity, and the Null player axiom.For k = 1 the formula recovers the Littlechild-Owen expression; the general-k closed form is presented as new.Source: Our contributions · p. 4
- The paper establishes 2-Lipschitz stability for landscape games and provides stability results for model-based credit allocations.The stated landscape-game bound concerns changes in persistence diagrams measured by bottleneck distance.Source: Our contributions · pp. 4, 6
- Linear models on landscapes admit closed-form point credit, while nonlinear models have a unique four-axiom allocation that can be approximated by Monte Carlo sampling.Exact nonlinear computation requires evaluating 2^N−1 coalitions, which is infeasible for realistic diagram sizes.Source: Our contributions · p. 5
- Experiments apply the method to curvature regression and dynamical-system classification, where the authors report explanations based on real observable topological features.The nonlinear model is reported to use a similar set of topological features as the linear model on the same data.Source: Our contributions · p. 6
Related works
- For biological TDA, the authors contrast LandscapeSHAP's domain-independent axiomatic attribution with existing approaches described as typically ad hoc and lacking uniqueness or stability guarantees.They also distinguish their target—diagram points and model predictions—from Shapley homology's fixed-scale Betti-number attribution.Source: Related works · pp. 7–8
- The paper positions LandscapeSHAP as complementary to persistence heatmaps, which allocate credit to raw point-cloud pixels and have been examined for efficiency but not the other fairness axioms.The authors suggest that diagram-point attributions could provide an importance signal for heatmap construction.Source: Related works · p. 7
2 Preliminaries
The preliminaries define fair player credit through coalitions and Shapley values, then explain how filtrations yield persistence diagrams and landscapes provide a function-based representation.
2.1 Shapley values
- The Shapley value averages player i’s marginal contribution v(S ∪ {i}) − v(S) over all coalitions S not containing i and is the unique fair allocation satisfying four fairness axioms.The defining sum ranges over 2^(|N|−1) subsets, and is computationally intractable except for very small player sets.Source: §2.1 · p. 9
2.2 Persistent homology
- A filtration is a nested family of simplicial complexes; persistent homology yields birth-death intervals that form a persistence diagram.Each feature is born when it first appears in a homology group and dies when it becomes homologous to an older feature or a boundary.Source: §2.2 · pp. 9–10
- Vietoris–Rips uses pairwise distances without ambient coordinates, whereas planar alpha complexes have O(n) simplices versus up to O(2n) for Vietoris–Rips but require coordinates.Alpha-complex construction costs grow quickly with ambient dimension.Source: §2.2 · p. 10
- Persistence landscapes map diagrams to functions in a separable Banach space: each point contributes a tent, and λ_k(m) is the k-th largest tent value, or zero if fewer than k points are present.Landscapes thus provide a function-valued representation of the birth-death pairs.Source: §2.2 · p. 10
3 Persistence landscapes as cooperative games
LandscapeSHAP treats persistence diagram points as players in cooperative games defined by landscape coordinates. It computes fair point-level credits while preserving the rank changes caused by removing points from a coalition.
3.1 Coalitions and games from tent functions
- Each coalition’s landscape is recomputed from its points, allowing remaining points to shift rank instead of freezing coordinate ownership.The authors identify fixed ownership as problematic because it ignores reranking and can assign nearly identical tent values winner-take-all credit through arbitrary tie-breaking.Source: §3.1 · p. 11
3.2 Closed form solution
- A closed form gives exact Shapley credits for each point in every landscape-coordinate game, uniquely characterized by Efficiency, Symmetry, Linearity, and Null player.Credits depend on rank and consecutive tent-value gaps, so nearly equal values receive nearly equal attribution rather than arbitrary tie-break credit.Source: §3.2 · pp. 11, 13
- For K ≥ N, a point’s credits summed over all landscape layers recover its tent function, partitioning that function among points according to the landscape’s order-statistic structure.The pointwise components are nonnegative and piecewise linear in m.Source: §3.2 · p. 15
3.3 Monotonicity
- Every landscape-coordinate game is monotone, so its point-level Shapley values are nonnegative; negative point credits for a linear model therefore come from its coefficients.The game’s nonnegative marginal contributions yield nonnegative Shapley values.Source: §3.3 · pp. 15–16
3.4 Stability
- The paper establishes stability results for point attributions under diagram perturbations, comparing matched points across diagrams.For coordinate games, the rank-based theorem compares players by rank; a separate result addresses stability for a fixed matched player.Source: §3.4 · p. 17
3.5 Computation
- At each grid point, the computation sorts active tent values in O(N log N) time and then obtains each player’s value by an O(1) lookup.The algorithm computes values for each layer after sorting and preparing the required sums.Source: §3.5 · p. 18
4 Linear models on landscapes
For a fixed affine model on discretized persistence landscapes, LandscapeSHAP combines exact coordinate-level point credits into model-prediction credits. The closed form depends on the empty-diagram baseline, and linearity also permits credit allocation across averaged landscapes.
4.1 Exactness and uniqueness
- For a linear model, the exact point attribution is the weighted sum of the Shapley values for its landscape-coordinate games.The model coefficients are fixed across coalitions; its bias cancels from marginal contributions.Source: §4.1 · p. 19, §4 · p. 19
- Efficiency, Symmetry, Null player, and Linearity uniquely determine the point-credit rule for a linear model, while the chosen baseline affects the game.The empty persistence diagram is presented as a natural baseline for crediting persistent homology classes; a dataset-average baseline requires redefining the game and does not yield the same direct closed-form argument.Source: §4.1 · p. 19
4.2 Computation
- For a few hundred points, 40 layers, and 100 grid points, the computation is on the order of 10^5 sorting operations plus 10^6 lookups.The paper describes this workload as trivial for a modern machine.Source: §4.2 · p. 20
4.3 Credit allocation for an averaged landscape
- For an averaged landscape and a linear model, each diagram’s point credits are its isolated-diagram credits scaled by that diagram’s weight, with no cross-diagram interaction terms.The decomposition relies on linearity of the model.Source: §4.3 · pp. 20–21
- For nonlinear models, the linear decomposition across diagrams generally fails, so exact attribution must consider the total player set and has exponential cost in its size.The method’s closed-form result requires the model transformation to be a fixed affine map across coalitions.Source: §4 · p. 19, §4.3 · p. 21
5 Nonlinear models on persistence landscapes
For nonlinear models, LandscapeSHAP retains a uniquely characterized Shapley attribution but lacks the linear model’s closed form, motivating efficient permutation sampling. The paper gives sample guarantees and stability results under stated conditions.
5.1 Uniqueness
- For any model, Efficiency, Symmetry, Linearity, and Null player uniquely characterize the credit allocation, but uniqueness is relative to the chosen baseline.Unlike the linear case, a general nonlinear model has no exact closed-form attribution formula in this treatment.Source: §5.1 · p. 21
5.2 Approximation
- Exact nonlinear attribution requires averaging marginal contributions over 2^N coalitions, while the efficient sampler reduces each permutation’s landscape-update cost to O(N|M|K log K).The sampler maintains each grid point’s top K tent values under insertion; evaluating the model adds a separate model-dependent cost.Source: §5.2 · p. 22
5.3 Rate of convergence
- A point’s insertion changes the landscape by at most an amount bounded by its own tent function, independent of coalition size, and the sampling bound’s R_i term does not grow with N.The authors state that the sample-count dependence on accuracy is ε^-2, the ordinary rate for a Monte Carlo mean estimator.Source: §5.3 · pp. 22, 24
5.4 Tree ensembles
- Tree ensembles permit exact feature Shapley values in polynomial time, but combining those values into persistence-diagram point attributions remains open.The cited dynamic-programming method applies to feature attribution for tree ensembles.Source: §5.4 · p. 25
5.5 Stability
- For an L_f-Lipschitz model on the flattened landscape vector, the paper proves stability of point Shapley values across matched diagrams.The stability result is stated under the 1-norm condition on the K × |M| landscape vector.Source: §5.5 · pp. 25–26
6 Experiments
The experiments illustrate LandscapeSHAP on regression and classification tasks.
6 Experiments
- The paper demonstrates LandscapeSHAP on both a regression task and a classification task.Hyperparameter choices are detailed in Section 6.3.Source: §6 · p. 27
6.1 Curvature regression
The curvature study uses average persistence landscapes to predict Gaussian curvature, then applies LandscapeSHAP to explain a linear SVR’s point-level credits. The attributions track curvature direction and align with the expected role of persistence classes.
6.1.1 Average landscapes detect curvature
- For each curvature value κ, 100 runs produced persistence diagrams whose average landscapes formed 101 vectors; the first principal component was almost perfectly correlated with κ (r = −0.9986).Source: §6.1.1 · p. 28
6.1.2 Explaining a linear SVR
- Negative-curvature classes receive negative Shapley values and positive-curvature classes receive positive values, while the κ = 0 contributions cancel toward a prediction near 0.The sum of Shapley values equals the prediction minus the empty-diagram baseline, scaled by 1/100.Source: §6.1.2 · pp. 28–29
- Classes with larger death values receive negative credits and those with smaller death values positive credits, consistent with average hyperbolic distances exceeding spherical distances.Source: §6.1.2 · p. 29
- Some of the highest-absolute-credit classes are shorter bars, consistent with the mathematical theory in.The authors conclude that the SVR’s predictions are reasonable in light of these attribution patterns.Source: §6.1.2 · p. 29
6.2 Dynamical system classifier
The dynamical-system study applies LandscapeSHAP to linear and nonlinear SVM classifiers trained on persistence landscapes. Attributions reveal which loops drive predictions, how uncertainty appears, and how credit allocation varies across models.
6.2.1 Linear model
- At r = 4, the linear SVM misclassified the example as r = 3.5, and its weaker top attributions indicate uncertainty.The authors note the model was not tuned to be the best predictor and suggest model choice, landscape layers, or adding dimension-0 features as possible avenues to investigate.Source: §6.2.1 · p. 31
- For r = 4.1 and r = 4.3, one or two large loops dominate the linear SVM’s attributions and push predictions toward higher r.For r = 2 and 3.5, the largest five absolute attributions are an order of magnitude smaller and mixed in sign; the most persistent point at r = 2 has near-zero attribution.Source: §6.2.1 · pp. 30–31, Figure 8 · p. 32
6.2.2 Linear model approximation
- Monte Carlo attribution error decays at O(1/√n) across five diagrams, while agreement in ranking and direction saturates faster than attribution magnitude.The diagrams each had more than 250 points; 3,000 permutations were reported as sufficient for the linear-model approximation study.Source: Figure 10 · p. 33, §6.2.2 · p. 33
6.2.3 Nonlinear model approximation
- At r = 2, no point receives an attribution larger than ±0.03 in the nonlinear model, suggesting a diffuse decision signal; at r = 3.5, five loops receive consistently positive, moderate attributions.For r = 3.5, this is a more distributed positive signal than the linear model’s small, sign-mixed top five; overall, nonlinear credit varies diagram by diagram.Source: §6.2.3 · p. 34
- For r = 4.1 and r = 4.3, the nonlinear SVM also assigns almost the entire prediction to one dominant loop or two comparably large loops.The remaining top-five absolute attributions are near zero, so these examples are qualitatively similar to the linear model’s explanations.Source: §6.2.3 · p. 34, Figure 11 · p. 35
6.3 Hyperparameters
The experiments fix landscape construction and model settings rather than tuning them. Curvature regression and dynamical-system classification use different landscape layer counts and feature dimensions.
6.3 Hyperparameters
- Landscapes use 100 grid points over [0, G], with 25 layers for curvature regression and 15 for dynamical-system classification.The resulting feature dimensions are 2,500 and 1,500, respectively; points below the specified persistence threshold are filtered.Source: §6.3 · p. 35
- The curvature examples are averages of 100 diagrams, while classification uses 500 point clouds split 75%/25% into training and test sets.Landscape features are standardized using the training split; model coefficients are mapped back to original coordinates for the linear-model results.Source: §6.3 · p. 36
- The linear models use C = 1, the RBF SVM uses C = 10 and gamma="scale", and no hyperparameters were tuned.Nonlinear Shapley values use 3,000 permutations per diagram; the convergence study uses eight independent seeds.Source: §6.3 · p. 36
6.4 Representative cycles
The representative cycles visualize the top five points by absolute Shapley value. Their construction differs between the dynamical-system and curvature examples.
6.4 Representative cycles
- For each selected point, the authors recompute its birth edge and death triangle, then find an earlier-edge path to form a representative cycle.Source: §6.4 · p. 36
- Dynamical-system cycles are drawn over the alpha complex at their birth time, whereas curvature cycles are drawn over the point clouds without the Vietoris–Rips filtration.Source: §6.4 · p. 36
6.5 Data availability
The paper provides a GitHub repository for reproducing all experiments.
6.5 Data availability
- The code for reproducing all experiments is available at github.com/milicevic-nikola/LandscapeSHAP.The passage identifies the repository location.Source: §6.5 · p. 36
7 Additive featurizations
The section explains how additive persistence-diagram featurizations, illustrated by persistence images, support straightforward Shapley credit allocation. Normalizing persistence images changes the game by introducing interactions between points.
7 Additive featurizations
- In persistence images, each diagram point contributes to pixels through its own coordinates, independently of which other points are present.The point is mapped to birth-persistence coordinates before its contribution to each pixel is defined.Source: §7 · pp. 36–37
- Because unnormalized persistence-image contributions are additive, each point’s Shapley value is its contribution, and a linear model preserves this additivity.The passage extends the same argument to other additive featurizations, including Betti curves, total persistence, and Euler characteristic curves.Source: §7 · p. 37
- Normalizing persistence images makes pixel values ratios of additive games, so removing one point changes other points’ denominators and introduces interactions.The normalized game may be non-additive and non-monotone; exact computation would need a fresh derivation, otherwise the manuscript uses coalition subsampling.Source: §7 · p. 37
8 Discussion
The discussion summarizes LandscapeSHAP’s closed-form landscape credit allocation, fairness, stability, and approximation results, then identifies open directions for nonlinear models, point-cloud regions, and explainable featurization design.
8 Discussion
- The paper gives a closed-form Shapley credit allocation for persistence-diagram points in landscape games and explains linear-model predictions using it.The method also applies to diagrams with different numbers of points.Source: §8 · p. 37
- LandscapeSHAP credit is stable, and its values satisfy the four fairness axioms for any model; the paper also gives permutation-sample estimates for nonlinear models.The discussion reports experimental verification of these results.Source: §8 · p. 38
- For nonlinear tree ensembles, exact landscape-coordinate Shapley values are possible, but pulling interacting coordinate credits back to diagram points remains unclear.The discussion poses this as an open question about designing nonlinear models with better-understood coordinate interactions.Source: §8 · p. 38
- The paper leaves open how to pull credit from persistence-diagram points back onto regions of the original point clouds, as in [10].It asks for an allocation that is stable, fair, and computable or has known approximation bounds.Source: §8 · p. 38
- The discussion recommends designing persistence-diagram featurizations as collections of cooperative games to support axiomatically justified explainability.It presents this as a consideration for future featurization design in TDA.Source: §8 · p. 38