Structured Learning on Mapper Representations

George Babus; Farzana Nasrin

stat.MLAug 22, 2026

Structured Learning on Mapper Representations

Authors: George Babus, Farzana Nasrin

Organizations: Department of Mathematics, University of Tennessee, Knoxville, TN -37916, USA

Abstract

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	$d_{\mathrm{loc}}(R_{1},R_{2})$	Measures differences between local attributes of refined pullback regions, such as cluster summaries, statistical moments, geometric descriptors, or domain specific features.
Overlap	$d_{\mathrm{ov}}(R_{1},R_{2})$	Measures differences in overlap organization among refined regions, capturing the strength and structure of interactions between neighboring regions.
Relational	$d_{\mathrm{rel}}(R_{1},R_{2})$	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	$0.807\pm 0.055$
	No-edge	10	0.2	$0.796\pm 0.060$
	Edge-perturbed	16	0.6	$0.826\pm 0.066$
	Original Mapper	7	0.2	$0.832\pm 0.057$
HAR	Global	13	0.5	$0.804\pm 0.000$
	No-edge	7	0.6	$0.852\pm 0.037$

Table 3: Maximum observed test Macro-F1 over the evaluated Mapper parameter grid

Panel of Figure 6 — 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 ( $\delta_{A_{\theta}}$ ) for three edge perturbation mechanisms.

Dataset	Perturbation	$\widehat{\beta}_{1}$	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	$10^{-4}$	$5\times 10^{-3}$

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

Panel of Figure 10 — Figure 10: Parameter wise predictive performance across the complete Mapper parameter grid for all four datasets.

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A prevailing paradigm in modern representation learning is the map-first approach, in which a representation map is learned from reconstruction, embedding, or task objectives. At the optimum, when the learned map accurately recovers a global coordinate chart, it should exhibit three structural properties whose geometric meaning can be illustrated through a face-editing example: Commutativity requires that changing pose and then expression gives the same result as applying them in the reverse order; Time Coherence requires that the same variation along one coordinate induces the same expression change across faces; Common-Reference requires that all faces are organized relative to a common reference face. However, small approximation errors in the learned map need not translate into small errors in these structural properties, and can therefore disrupt the global organization of the representation. Based on this observation, we consider the converse of the map-first formulation and ask whether a global representation can instead emerge by directly learning these properties. We represent variations along individual coordinates through vector fields defined in the ambient space and introduce a non-contraction condition preventing one transformation from destroying directions associated with the others. We derive an unsupervised objective that learns these structural properties and establish theoretical results connecting its minimization to tangent-space recovery. Experiments on controlled manifolds validate the predicted tangent-space recovery and global structure, while an autoencoder baseline shows that small map-first errors can still produce substantial violations of the targeted properties.

Abstract

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