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.
Figures & tables
Figure 1: Overview of the proposed framework. A dataset is transformed into a structured multiscale decomposition through the Mapper construction. The resulting representation serves as the foundation for three interconnected components of the framework.
Figure 2: Overview of the Mapper construction used to obtain topological representations of datasets. Starting from a high-dimensional dataset X (A), a lens function maps the data to a lower-dimensional space where an overlapping cover is constructed (B). Each pullback set is clustered, and clusters become vertices of the Mapper graph, with edges indicating overlapping clusters (C). The resulting graph summarizes the multiscale structural organization of the dataset (D).
Component
Notation
Interpretation
Local attribute
dloc(R1,R2)
Measures differences between local attributes of refined pullback regions, such as cluster summaries, statistical moments, geometric descriptors, or domain specific features.
Overlap
dov(R1,R2)
Measures differences in overlap organization among refined regions, capturing the strength and structure of interactions between neighboring regions.
Relational
drel(R1,R2)
Measures differences in the nerve structure, including connectivity, branching patterns, and cycle organization.
Table 1: Component wise structural dissimilarities between Mapper induced representations.
Figure 3: Representative samples from the four benchmark datasets used in the experimental evaluation.
Dataset
Lens
Distance
Clustering method
HAR
RMS of acc. and gyr. coordinates (2D)
Euclidean
HDBSCAN
WISDM
Acceleration RMS
Euclidean
HDBSCAN
MUTAG
Spectral
induced-graph connectivity
Connected component
PROTEINS
diffusion-map lenses (2D)
induced-graph connectivity
Connected component
Table 2: Fixed Mapper construction and learning settings used in the experiments.
Figure 4: Richness of Mapper induced representations considered in the experimental study.
Figure 5: Average predictive performance of the four representation variants across the evaluated Mapper parameter configurations. Points indicate mean test Macro-F1 across the 30 configurations, and error bars indicate one standard deviation across configurations. Thus, the error bars characterize sensitivity to Mapper construction parameters rather than variability across repeated training runs.
Dataset
Representation
Intervals
Overlap
Test Macro-F1
WISDM
Global
7
0.6
0.807±0.055
No-edge
10
0.2
0.796±0.060
Edge-perturbed
16
0.6
0.826±0.066
Original Mapper
7
0.2
0.832±0.057
HAR
Global
13
0.5
0.804±0.000
No-edge
7
0.6
0.852±0.037
Table 3: Maximum observed test Macro-F1 over the evaluated Mapper parameter grid
Figure 6: Structural organization of Mapper representations across the parameter grid. The heatmap displays average pairwise composite distances, while the dendrogram summarizes the hierarchy induced by those distances.
Figure 7: Two-dimensional classical MDS visualization of the Mapper representation distance matrix. Marker shape identifies cover overlap and labels indicate the number of cover intervals.
Figure 8: Relationship between topological complexity and test Macro-F1 across the four benchmark datasets.
Figure 9: Estimated robustness slopes relating test Macro-F1 to perturbation level ( δAθ ) for three edge perturbation mechanisms.
Dataset
Perturbation
β1
95% CI
WISDM
Deletion
−0.004
[−0.016,0.008]
Addition
0.001
[−0.0002,0.002]
Rewiring
−0.009
[−0.022,0.005]
HAR
Deletion
−0.002
[−0.006,0.002]
Addition
−0.009
[−0.019,0.001]
Rewiring
−0.014
[−0.031,0.003]
Table 4: Estimated robustness slopes for test Macro-F1 as a function of relative spectral perturbation. Confidence intervals are reported at the 95% level.
Appendix figures & tables4 assets
Supplementary material from the paper’s appendix.
Appendix
Feature category
WISDM
HAR
MUTAG
PROTEINS
Mapper node size
✓
✓
✓
✓
Lens statistics
✓
✓
Acceleration statistics
✓
✓
Gyroscope statistics
✓
Temporal change statistics
✓
✓
Node attribute statistics
✓
✓
Appendix
Table 5: Categories of node attributes associated with the refined pullback regions for each benchmark dataset. All datasets additionally include Mapper-node size and Mapper structural features.
Setting
WISDM / HAR
MUTAG / PROTEINS
Architecture
2 GCNConv + linear
2 GCNConv + linear
Hidden dimension
64, 64
32, 32
Dropout
0.25
0.50
Optimizer
Adam
Adam
Learning rate
0.001
0.003
Weight decay
10−4
5×10−3
Appendix
Table 6: Learning configurations used for the attributed Mapper representations.
Dataset
Original feature representation
Classifier
Macro-F1
WISDM
109 statistical, temporal, correlation, and spectral features extracted from each original 200-sample, three-axis accelerometer window.
SVM
0.885
HAR
Official 561-dimensional UCI HAR feature vectors computed from the original accelerometer and gyroscope signals.
SVM
0.872
MUTAG
36-dimensional native-graph summary containing pooled statistics of the original node attributes and basic graph-topology descriptors.
Random forest
0.857
PROTEINS
24-dimensional native-graph summary containing pooled statistics of the original node attributes and basic graph-topology descriptors.
SVM
0.756
Appendix
Table 7: Reference predictive performance obtained directly from the original feature representation without constructing a Mapper object. Values for cross-validated experiments are mean Macro-F1
Figure 10: Parameter wise predictive performance across the complete Mapper parameter grid for all four datasets.
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