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Structured Learning on Mapper Representations
George Babus, Farzana Nasrin
TL;DR
The paper addresses the limited treatment of Mapper as a complete learning representation rather than an intermediate graph or fixed-dimensional summary. It develops mathematical and empirical tools for structured Mapper representations, finding that preserving original connectivity generally improves predictive accuracy while sensitivity remains dataset- and perturbation-dependent.
Problem
Mapper is typically reduced to graph or vector summaries, leaving the complete construction's structural components underused for learning and comparison.
Method
The framework treats the full Mapper construction as a representation and studies equivalence, structural complexity, representation geometry, and learning-oriented stability.
Results
Across four benchmark datasets, representations preserving original connectivity generally achieve stronger predictive accuracy than simplified variants, with sensitivity depending on dataset and perturbation.
Takeaways & Limitations
Mapper representations can be systematically compared and analyzed through representation geometry, structural complexity, and predictive stability rather than treated as fixed preprocessing outputs.
Takeaways & Limitations
The stability analysis concerns graph-based predictors on attributed Mapper graphs, not the stability of the Mapper construction itself.
Abstract
from arXiv · showhide
Modern machine learning (ML) methods are highly effective for prediction tasks, but many commonly used representations reduce complex data to fixed dimensional embeddings that may suppress multiscale structural organization. The Mapper algorithm from topological data analysis (TDA) provides a different perspective by decomposing data into overlapping local regions connected through a nerve construction, producing a structured representation that captures geometric organization, local statistical behavior, and relational connectivity simultaneously. In this work, we develop a framework for learning over Mapper induced structured representations. Rather than treating Mapper as a preprocessing step that produces a graph for downstream learning, we treat the full Mapper construction as part of the representation itself. We study mathematical properties of these representations, including invariance under relabeling, a distance functional on the space of Mapper representations, structural complexity of multiscale decompositions, and learning oriented stability under representation perturbations. Experiments on time series and graph classification datasets validate the proposed framework through controlled studies of representation ablation, Mapper parameter sensitivity, and the geometry of the induced representation space. Together, these results demonstrate how the proposed mathematical framework enables systematic comparison, interpretation, and analysis of Mapper representations, providing practical tools for studying representation geometry, structural complexity, and learning stability in learning tasks.
1 Introduction
The paper reframes the complete Mapper construction as a structured representation for learning, addressing the common reduction of Mapper outputs to graphs or fixed-dimensional vectors. It develops geometry, complexity, stability, and empirical tools for analyzing how representation structure affects prediction.
- Motivation: Machine-learning representations determine what information is retained, how datasets are compared, and what can be learned.
- Motivation: Mapper produces an interpretable multiscale decomposition through filtering, overlapping covers, clustering, and nerve construction.
- Learning over Mapper representations: The paper treats the full Mapper construction—not only its nerve graph—as the representation, retaining covers, pullback decompositions, clusters, overlaps, connectivity, and attributes.
- Mathematical framework: The framework introduces permutation-invariant equivalence relations, structural distances, kernels, complexity notions, and stability results for Mapper representations.
- Related work: The related-work gap is a unified framework for representing, comparing, measuring complexity of, and statistically analyzing Mapper-derived representations.
- Experiments: Experiments on timeseries and graph classification datasets examine representation components, Mapper parameters, and representation organization under the proposed distance.
2 Preliminaries
The preliminaries define a broad supervised learning setting and describe Mapper as a sequence from lens mapping through overlapping covers, pullback clustering, and nerve construction. The resulting graph captures local organization and global connectivity.
- Data domains and notation: The framework applies to datasets including point clouds, images, signals, time series, and graph vertex sets with associated features.
- Data domains and notation: In supervised learning, each dataset is paired with a target, and the goal is to learn a predictor that generalizes to unseen datasets.
- The Mapper construction: Mapper combines a low-dimensional data representation, an overlapping cover, and clustering within overlapping regions.
- The Mapper construction: A lens maps each data point to a coordinate vector, after which overlapping regions partition the lens image and induce pullback sets in the original dataset.
- The Mapper construction: Clustering refines each pullback set into clusters, which become vertices of the Mapper graph; edges encode overlaps through the nerve construction.
- The Mapper construction: The Mapper graph is the 1-skeleton of the nerve of the refined pullback cover and captures local organization together with global connectivity patterns.
3 Predictive Learning on Mapper Induced Decompositions
The paper formulates prediction on the full Mapper-induced decomposition rather than on the graph alone, allowing local, structural, and relational information to be studied separately. Learning and ablation experiments test how these components influence predictive behavior.
- Representation components: The entire Mapper-induced decomposition is treated as the learning representation, including local summaries, overlap structure, and relational connectivity.
- Predictive formulation: A representation map assigns each dataset to a structured object, after which a learning map predicts from that object.
- Predictive formulation: Mapper construction parameters control the decomposition, while predictor parameters control how the resulting structured representation is used for prediction.
- Representation components: Local-only, graph-based, and richer models isolate the predictive roles of local attributes, nerve connectivity, overlap strengths, cover locations, and multiscale information.
- Experimental design: Graph neural networks, ablated models, and perturbed representations study prediction changes as different levels of the Mapper decomposition become available.
- Experimental design: The No-edge representation preserves node attributes while removing edge connectivity, isolating the predictive contribution of relational connectivity.
- Experimental design: Parameter-variation and representation-modification experiments analyze structural contributions, graph-connectivity robustness, and Mapper effects on learning behavior.
4 Structural Geometry of Mapper Induced Representations
The paper defines Mapper representations intrinsically, removing arbitrary labels and equipping their quotient space with structural distances, kernels, and multiscale organization for comparison and learning.
- Intrinsic geometry: Mapper representations are compared through intrinsic equivalence, structural dissimilarity, similarity, and multiscale organization rather than implementation-dependent labels.The framework targets similarity search, clustering, retrieval, and kernel-based learning directly on representation space.
- Equivalence: Representation equivalence preserves region intersections, attached attributes, overlap relations, and nerve connectivity under bijective relabeling.This identifies representations that differ only in indexing of refined pullback regions.
- Equivalence: The quotient space removes arbitrary cluster-label dependence, so distances, kernels, and complexity functionals depend on intrinsic decomposition structure.Invariant functionals on labeled representations induce well-defined functionals on equivalence classes.
- Structural distances: Structural distances decompose dissimilarity into interpretable components whose weighted combination emphasizes different sources of structural variation.Local statistical attributes, relational structure, and multiscale organization may contribute differently across applications.
- Structural kernels: Structural kernels complement distances by measuring shared structural characteristics through quotient-invariant feature maps, without requiring explicit vectorization of the full decomposition.Features can encode cluster-size distributions, nerve degree statistics, branching, and cycle summaries.
- Summary: Together, equivalence, distances, kernels, and refinement provide a mathematical geometry for comparing and organizing complete Mapper representations independently of a particular predictive model.The framework is intended to support systematic structural comparison and multiscale analysis.
5 Complexity and Stability of Mapper Representations
The paper characterizes Mapper complexity through local resolution, topology, and overlap, and analyzes how attributed-graph predictor outputs respond to adjacency and feature perturbations under stated assumptions.
- Motivation: Mapper complexity reflects a trade-off between structural richness and predictive robustness, with coarse decompositions potentially missing local structure and fine ones becoming noise-sensitive.The section treats complexity and robustness as complementary considerations for decomposition-aware learning.
- Topological complexity: The proposed complexity functional combines refined-region count, nerve-graph cycle structure, and overlap interaction as complementary mechanisms of representation complexity.These correspond to local resolution, global topology, and interaction structure.
- Overlap interaction: Normalized overlap measures shared membership relative to refined-region sizes, with larger values indicating stronger interaction among neighboring regions.Alternative measures include raw overlap counts and Jaccard overlap.
- Predictive stability: The stability analysis models Mapper graphs with weighted adjacency and node-feature matrices, then propagates perturbations through shared-weight message-passing layers, pooling, and output maps.Using the same trained weights isolates sensitivity to input-representation changes rather than retraining variation.
- Predictive stability: The explicit perturbation bound quantifies how node-feature and adjacency changes accumulate across successive layers and influence final predictions.The result follows from recursive layerwise bounds under Lipschitz and boundedness assumptions.
- Scope: This stability analysis applies to predictors acting on attributed Mapper graphs, not to perturbations of the complete Mapper construction itself.Extending it to models operating directly on covers, pullback organization, and overlap structure remains future work.
6 Experiments
Experiments evaluate Mapper representations across time-series and graph-classification benchmarks, testing richness, geometry, complexity, parameter sensitivity, and robustness. Results show that structural components contribute differently across datasets and Mapper constructions, while representation distances organize parameterized families coherently.
- Experimental Design: The experiments use four benchmark datasets spanning multivariate time-series and graph classification, with fixed Mapper and learning settings for controlled comparisons.WISDM and HAR are human-activity datasets, while MUTAG and PROTEINS are graph-classification benchmarks.
- Representation Richness: Representation richness distributes predictive information across multiple Mapper components, with local attributes contributing substantially and relational structure adding dataset-dependent information.The comparisons progressively retain structural components while holding the underlying Mapper object and learning conditions fixed.
- Representation Richness: Richer representations do not guarantee monotone performance gains because component utility depends on the dataset and the scale of the Mapper construction.Graph-based variants enable controlled connectivity comparisons under the same GCN, whereas the Global summary uses an SVM and is a broader baseline.
- Representation Geometry: The three structural descriptors capture complementary rather than redundant aspects of variation in the representation space.The setting-level distance averages distances between matched representations, allowing parameterized Mapper families to be compared independently of the downstream predictor.
- Representation Geometry: The representation-distance hierarchy has cophenetic correlation 0.915, with nearby parameter settings generally closer and seven-interval constructions separated from higher-resolution representations.Overlap interacts with interval count, so the organization reflects combined Mapper parameters rather than cover resolution alone.
- Complexity and Robustness: Topological complexity has no consistent relationship with test Macro-F1, with correlations of r = 0.33, −0.05, −0.33, and 0.40 for WISDM, HAR, MUTAG, and PROTEINS.The complexity functional summarizes structural aspects rather than directly predicting classification performance; task-aware alternatives remain a future direction.
- Complexity and Robustness: Robustness is representation- and dataset-dependent: some perturbations yield slopes near zero, whereas selected deletion, addition, or rewiring settings produce more negative slopes.Regression summarizes average directional sensitivity because random edge modifications need not cause monotone degradation in individual trials.
7 Discussion
Across benchmark datasets, retaining Mapper’s original connectivity generally improves predictive accuracy, while representation geometry and stability analyses show that construction parameters and perturbations have dataset-dependent effects.
- Predictive performance: Connectivity-preserving Mapper representations generally achieve stronger predictive accuracy than simplified variants across four benchmark datasets.The improvement magnitude varies across datasets and parameter settings, and relational information can provide signal beyond node-level descriptors.
- Representation geometry: A composite structural distance combining local, overlap, and relational descriptors organizes Mapper representations by construction parameters and identifies similarities between configurations.This geometry supports quantitative comparison of different Mapper constructions beyond predictive accuracy.
- Parameter dependence: Mapper construction parameters alter the resulting representation before downstream learning, making representation design part of the learning pipeline.The relevant choices include cover resolution, overlap, and retained structural information.
- Stability: Robustness sensitivity is dataset- and perturbation-dependent, with several configurations showing near-zero slopes and selected deletion or rewiring settings showing clearer degradation.Table 4 reports estimated robustness slopes for test Macro-F1 under relative spectral perturbation with 95% confidence intervals.
- Implications and future work: No single graph-complexity notion or parameter choice universally optimizes predictive performance, because behavior depends on interactions among resolution, overlap, and retained structure.The paper proposes extending the framework to richer topological objects, stronger guarantees beyond graph neural networks, and additional scientific domains.
A Additional Experimental Results
The appendix adds experimental references, dataset-specific node-attribute categories, and parameter-wise performance visualizations for the complete Mapper grid.
- Additional experimental results: The appendix reports reference performance without Mapper representations for comparison with the main experiments.It also summarizes node-attribute categories and presents parameter-wise performance heat maps.
- Additional experimental results: Parameter-wise performance heat maps cover the complete Mapper parameter grid.These visualizations support examination of performance across Mapper settings.
- Additional experimental results: Node-attribute categories are summarized for each dataset in the supplementary experimental details.The appendix does not list every individual statistic.
A.1 Node attribute categories
Refined pullback regions receive node attributes combining domain-specific descriptors with structural properties of their corresponding Mapper graphs.
- Node attribute categories: Node attributes combine domain-specific descriptors with structural properties of the corresponding Mapper graph.The attributes are associated with refined pullback regions.
- Node attribute categories: All datasets additionally include Mapper-node size and Mapper structural features among their attribute categories.Table 5 summarizes the dataset-specific feature categories rather than listing every statistic.
A.2 Learning and evaluation details
Attributed Mapper representations use standardized evaluation protocols, with GCNs for connectivity-aware variants, an SVM for the Global variant, and ten-fold cross-validation tailored to dataset type.
- Learning architecture: Attributed Mapper representations are evaluated with a two-layer GCN, global mean pooling, and a linear classifier.ReLU activation follows each graph convolution, and Table 6 summarizes dataset-specific learning configurations.
- Evaluation controls: The same hyperparameters are used across representation variants within each dataset group so performance differences reflect representation changes rather than model-specific tuning.This standardization applies to the learning configurations summarized in Table 6.
- Cross-validation: WISDM and HAR use subject-level ten-fold cross-validation, preventing samples from the same subject from appearing in training and test folds.Subject-level partitioning reduces leakage from repeated samples belonging to one subject.
- Cross-validation: MUTAG and PROTEINS use stratified ten-fold cross-validation, with one model trained per fold and ten fits per experimental condition.The complete cross-validation procedure is performed once.
- Representation-specific evaluation: The Global representation is evaluated separately with an SVM because it does not contain graph connectivity.No-edge, Perturbed, and Original representations use the described GCN architecture.
A.3 Reference performance on the original data
Table 7 provides reference predictive performance from the original input or feature representation without constructing a Mapper object. These results contextualize benchmark difficulty and support comparison with Mapper-based representations.
- Table 7 reports predictive performance obtained directly from the original input data without constructing a Mapper representation.
- The reference results provide context for the difficulty of each benchmark dataset.
- Cross-validated values are reported as mean Macro-F1.
A.4 Parameter wise performance across the Mapper parameter grid
The framework is evaluated across a common Mapper parameter grid to examine sensitivity to construction choices. Heat maps show test Macro-F1 across cover resolution and overlap percentage combinations.
- Every representation was evaluated over the common Mapper parameter grid described in Section 6.1.
- Each heat map reports test Macro-F1 for every combination of cover resolution and overlap percentage.
- The heat maps complement averaged main-text results by illustrating predictive-performance variation across Mapper constructions.