Representation Mapper: A TDA Framework

Updated 5 January 2026

Representation Mapper is a mathematical and algorithmic framework that summarizes high-dimensional data by constructing interpretable simplicial graphs using a filter-cover-cluster-nerve pipeline.
It reveals structural features such as clusters, decision boundaries, and subpopulation patterns by mapping complex embeddings to a discretized topological space related to Reeb spaces.
Its applications span model diagnostics, cross-modal alignment, graph pooling, and explainability in deep learning, providing actionable insights for complex datasets.

A Representation Mapper is a mathematical and algorithmic framework that leverages the Mapper construction from topological data analysis (TDA) to summarize and analyze the global structure of high-dimensional data spaces through the construction of a simplicial graph, or network, that reflects the topological and geometric organization of the original data. In the context of machine learning and representation learning, a Representation Mapper transforms complex data embeddings (e.g., neural network activations, contextualized LLM outputs, latent vectors) into an interpretable network whose nodes and edges systematically capture local and global features such as clusters, decision boundaries, and subpopulation structure. The framework is distinguished by its filter-cover-cluster-nerve pipeline, strong theoretical connections to Reeb spaces, and diverse applications including model diagnostics, cross-modal alignment, graph pooling, and explainability in deep learning.

1. Mathematical Foundations and Algorithmic Pipeline

A Representation Mapper defines a four-stage process on a data set $X \subset \mathbb{R}^d$ , known as the Mapper construction. The stages are:

Filter (Lens) Function: A continuous (or measurable) map $f : X \to \mathbb{R}^k$ (with $k \ll d$ ) projects high-dimensional data onto a lower-dimensional "lens" emphasizing a feature or semantic axis (e.g., principal component, prediction confidence, L2 norm, PageRank score). The choice and parametric structure of $f$ are central, as evidenced by work on filter optimization (Oulhaj et al., 2024).
Cover of the Filter Range: The image $f(X)$ is partitioned into overlapping bins or open sets $\{U_i\}_{i=1}^r$ , controlled by resolution ( $r$ or bin number) and overlap parameters ( $\varepsilon$ or gain $\alpha \in (0,1)$ ). In one dimension:

$U_i = [a_i, b_i],\quad a_i = (i-1)/r - \varepsilon \Delta, \quad b_i = i/r + \varepsilon \Delta, \quad \Delta = 1/r$

Overlap is essential for capturing the continuity and topology of the data.

Clustering in Pullback Sets: For each cover element $U_i$ , form the pullback $X_i = f^{-1}(U_i)$ and cluster $X_i$ using single-linkage, DBSCAN, HDBSCAN, k-means, or task-specific alternatives. Each resulting cluster $C_{i,j}$ becomes a candidate node.
Nerve Graph Construction: Construct a graph (1-skeleton of the simplicial nerve of the cover) with nodes representing clusters and edges representing shared data points (i.e., clusters $C_{i,j}$ and $C_{k,\ell}$ share at least one $x$ if and only if they are connected). The union across all bins and clusters forms the final Mapper graph $G = (V, E)$ .

The generic pseudocode structure for Mapper is:

Y = [f(x) for x in X]
U = [intervals_with_overlap(Y, r, epsilon)]
V, E = set(), set()
for i, Ui in enumerate(U):
    Xi = [x for x in X if f(x) in Ui]
    for cluster in Cluster(Xi):
        V.add(cluster)
for c1, c2 in combinations(V, 2):
    if c1.data_points & c2.data_points:
        E.add((c1, c2))
return Graph(V, E)

This framework abstracts across vector, graph, and sequence embedding spaces (Madukpe et al., 12 Apr 2025, Munch et al., 2015, Yan et al., 24 Jul 2025).

2. Theoretical Properties and Connections to Reeb Spaces

The Mapper graph serves as a discretization of a more general topological invariant called the Reeb space. For a continuous map $f : X \to \mathbb{R}^d$ , the Reeb space $R(f)$ quotients points in $X$ by both their image under $f$ and their path-connectedness in their level sets. Mapper is a categorical approximation of the Reeb space: with sufficient cover resolution and overlap, its interleaving distance to $R(f)$ can be made arbitrarily small (Munch et al., 2015):

If $\mathrm{res}(U) = \max_i \diam(U_i)$, then the interleaving distance $d_I$ between the categorical Mapper and categorical Reeb space satisfies $d_I \leq \mathrm{res}(U)$ .
As the discretization refines (i.e., $\mathrm{res}(U) \to 0$ ), Mapper converges to the Reeb space functorially.

This establishes Mapper's approximation guarantees for topological and homological features, providing theoretical rigor for empirical findings in representation analysis.

3. Parametric Variants and Optimization

Multiple developments address the sensitivity of Mapper outputs to filter choice, cover specification, and clustering algorithm:

Differentiable (Soft) Mapper (Oulhaj et al., 2024): Introduces smooth, stochastic cover assignments, parameterization of filters (linear or deep neural networks), and topological loss functions (e.g., persistence of nerve graphs), enabling gradient-based optimization of Mapper representations. As $\delta \to 0$ in the bump functions $q_j(x)$ , the Soft Mapper converges in law to the classical Mapper.
Ball Mapper, Fuzzy Mapper, V-Mapper, G-Mapper, D-Mapper (Madukpe et al., 12 Apr 2025): Address cover construction and stability via balls, fuzzy partitioning, adaptive interval splitting, and density modeling.
Ensemble-based Methods: Sample over parameter grids $(f, r, \varepsilon)$ , aggregate Mapper results via co-occurrence clustering, and select stable subgraphs.
Hierarchical Deep Graph Mapper (Bodnar et al., 2020): Integrates Mapper with GNNs for multi-level pooling and representation, exploiting equivalence with soft-assignment pooling (DiffPool, minCUT).

The choice of filter significantly impacts Mapper stability and feature visibility, motivating data-adaptive learning of filters (Oulhaj et al., 2024).

4. Metrics and Topological Interpretation

Representation Mapper enables quantitative assessment of structural properties in the embedding space using topological metrics:

Component Purity: For component $C$ , label type $\ell \in \{\mathrm{true}, \mathrm{pred}\}$ ,

$\operatorname{CP}_\ell(C) = \frac{1}{|X_C|} \sum_{x \in X_C} [\ell(x) = \operatorname{mode}_\ell(C)]$

Edge Agreement:

$\operatorname{EA}(G) = \frac{1}{|E|} \sum_{(n_i, n_j) \in E} [\operatorname{mode}_{\text{true}}(n_i) = \operatorname{mode}_{\text{true}}(n_j)]$

Majority Match:

$\operatorname{MM}(G) = \frac{1}{|\mathcal{C}|} \sum_{C \in \mathcal{C}} [\operatorname{mode}_{\text{pred}}(C) = \operatorname{mode}_{\text{true}}(C)]$

These metrics, when visualized as node or edge coloration, facilitate diagnosis of overconfident clustering, label ambiguity, and decision boundary collapse within model representations (Rair et al., 20 Oct 2025, Yan et al., 24 Jul 2025).

5. Applications across Learning Domains

Model Diagnostics and Explainability:

Mapper provides a diagnostic tool to reveal modular, non-convex regions in transformer-based models (e.g., RoBERTa-Large on MD-Offense), uncover overconfident clusters, and distinguish between robust and ambiguous subregions (Rair et al., 20 Oct 2025).
Mapper graphs support explainability for LLM embeddings, offering a measurable topology in which agents annotate nodes/edges and perturbations are used to assess the semantic or syntactic consistency of clusters (Yan et al., 24 Jul 2025).

Cross-Modal and Cross-Lingual Mapping:

In cross-lingual retrieval, mappers align transformer-leveraged representations across language domains via linear or neural mapping functions. Empirically, linear maps (Least Squares) deliver near-perfect mate retrieval for document-level aligned pairs, indicating that embedding spaces can be post-aligned with minimal training (Tashu et al., 2024).

Graph Representation Learning and Pooling:

Mapper serves as a pooling operator in GNN architectures, with mathematically proven equivalence to soft-assignment algorithms like DiffPool and minCUT, and demonstrated empirical performance on graph classification benchmarks (Bodnar et al., 2020).

Visual Scene Mapping and Robotics:

In spatial representation, the Trans4Map architecture transforms egocentric sensory streams into allocentric semantic maps using transformer encoders and BAM modules, with the representational mapping yielding state-of-the-art efficiency and accuracy for scene understanding (Chen et al., 2022).

Latent Space Control in Generative Models:

TD-GEM learns a residual mapping in GAN latent space, guided by text prompts and CLIP losses, allowing targeted, disentangled manipulations for fashion image editing (Dadfar et al., 2023).

Bioinformatics, Medicine, Neuroscience, Finance, and Environmental Science:

Mapper is applied to single-cell RNA-seq (trajectory inference and pseudotime), EHR clustering, fMRI state dynamics, financial fraud detection, and air quality monitoring, translating unsupervised high-dimensional structure into human-interpretable graphs (Madukpe et al., 12 Apr 2025).

6. Limitations, Open Problems, and Future Directions

Representation Mapper is sensitive to selection of filter, cover, and clustering parameters; small changes can dramatically affect output topology. Stability analysis via graph metrics and persistent homology is an area of active investigation. Theoretical guarantees regarding topology recovery from noisy or finite data remain limited except under strong assumptions. Computational complexity is nontrivial—particularly for large $r$ or high dimension—prompting research into efficient Mapper variants and ensemble averaging.

Open problems and research directions include:

Adaptive and differentiable filter and cover selection (Oulhaj et al., 2024)
End-to-end differentiable Mapper pipelines for integration with deep learning
Multi-lens and multiscale Mapper constructions for enhanced robustness
Unsupervised cross-lingual mapping without parallel data (Tashu et al., 2024)
Deeper theoretical links to persistent homology and full Reeb space recovery (Munch et al., 2015)
Toolkits for interactive exploration and explainability at scale (Yan et al., 24 Jul 2025)

7. Comparative Summary of Major Mapper Variants

Variant	Major Feature(s)	Application Domain
Classical Mapper	Filter-cover-cluster-nerve	TDA, bioinformatics, representation learning
Soft/Differentiable Mapper	Filter parameter optimization	Topology-aware ML, dataset-structure optimization
Ball Mapper	Ball-based cover, reduced params	Massive-scale data visualization
Fuzzy Mapper, G-Mapper	Adaptive/fuzzy covering	High-heterogeneity and noisy data
Deep Graph Mapper	GNN integration, soft pooling	Graph classification, hierarchical pooling
Explainable Mapper	LLM-based annotation/verification	LLM interpretability, embedding space analysis

Each of these variants balances computational tractability, interpretability, and fidelity to the underlying data topology. Representation Mapper, as a flexible, unifying framework, continues to influence both theory and practice at the intersection of TDA and representation learning across scientific domains (Madukpe et al., 12 Apr 2025, Oulhaj et al., 2024, Bodnar et al., 2020, Munch et al., 2015, Yan et al., 24 Jul 2025, Rair et al., 20 Oct 2025).