Subsampling Methods for TDA

• Subsampling in TDA is a process designed to handle large datasets by selecting representative subsets that maintain key topological features.
• Techniques like landmark-based reduction and average persistence landscapes improve scalability while providing statistical reliability.
• These methods reduce runtime and memory usage, offering theoretical guarantees and empirical robustness in complex data analysis.

Topological Data Analysis (TDA) leverages algebraic-topological tools to extract robust topological and geometric information from complex datasets represented as point clouds or finite metric spaces. The combinatorial complexity of classical constructions, such as Vietoris–Rips or Čech complexes, necessitates subsampling approaches for scalability and statistical reliability. Subsampling methods provide tractable approximations, reduce memory and runtime costs, and, under precise conditions, maintain provable fidelity to the topological features of the underlying spaces.

1. Foundational Concepts and Motivation

Subsampling in TDA refers to strategies for working with manageable subsets or representations of large datasets, either by selecting representative points, constructing summarizing structures, or averaging over multiple subsamples. The primary motivations are computational—the number of simplices in classical filtrations grows exponentially with the dataset size—and statistical, as subsampling can smooth out noise and allows confidence assessments via repeated draws or aggregation (Chazal et al., 2014, Stier et al., 3 Sep 2025, Minian, 26 Nov 2025).

Key notions across the literature include:

• Statistical risk/approximation: The tradeoff between computational tractability and topological accuracy is formalized through risk bounds and stability theorems for persistent diagrams, landscapes, or Betti curves under subsampling, bottleneck/Wasserstein distances, and landscape-norm deviations (Chazal et al., 2014, Park et al., 6 Dec 2025).
• Landmark-based reduction: Selection of a subset (“landmarks”) to serve as the basis for nerve, witness, or other complexes, aiming to preserve geometric coverage or neighborhood structure (Brunson et al., 2022).
• Graph-theoretic and topological simplification: Strong collapse and core reductions systematically prune redundant points while preserving homological features at given scales (Minian, 26 Nov 2025).

2. Statistical Subsampling and Average-Based Approaches

A central paradigm is to compute persistent homology on many small random or structured subsamples and aggregate the resulting invariants:

• Empirical Average Landscape / Subsample Averaging: Given a large point cloud $X_N$, repeatedly sample small subsets $S_i^m$, compute their persistence diagrams or landscapes, and form averages such as $$\overline{\lambda_n^m}(t) = \tfrac{1}{n} \sum_{i=1}^n \lambda_{S_i^m}(t)$$. The expectation under i.i.d. draws converges under mild conditions to the “true” topological signature of the underlying measure or support (Chazal et al., 2014, Stier et al., 3 Sep 2025).
• ALBATROSS Protocol: Implements a stochastic-subsampling and averaging scheme for Betti curves, drawing many small subsets, computing VR/Čech filtrations and topological summaries, and averaging to obtain robust, memory-efficient topology estimates. Theoretical justification rests on stability and central limit properties, and empirical results show negligible degradation for stable features once the subsample size exceeds the dimension of interest (Stier et al., 3 Sep 2025).
• Confidence Bounds via Subsampling: In persistent homology for time-delay embeddings, the subsampling approach is further harnessed to construct non-asymptotic confidence sets for persistent diagrams, using quantiles of subsample-to-full sample bottleneck/Hausdorff distances and leveraging stability and regularity assumptions on the underlying manifold (Park et al., 6 Dec 2025).

3. Geometric and Combinatorial Subsampling Frameworks

Beyond stochastic subsampling, several geometric methods target coverage and combinatorial complexity:

• δ-core Subsampling via Strong Collapse: The δ-core approach eliminates “dominated” points whose δ-neighborhoods are subsumed by others, yielding a minimal vertex set for the Vietoris–Rips complex at scale δ. This operation leverages the strong collapse framework to guarantee that global and local homological structure at δ is preserved exactly at the core, with uniqueness up to δ-equivalence. The δ-core is computationally efficient (near-linear in common regimes), with empirical bottleneck and Wasserstein distances consistently outperforming landmark or witness-based reductions (Minian, 26 Nov 2025).
• Landmark Selection: Maxmin and Lastfirst: The classical maxmin algorithm selects landmarks to maximize coverage with minimal radius balls, suitable in Euclidean or metric settings. The lastfirst method generalizes to arbitrary (possibly non-symmetric) dissimilarities and data with variable density, constructing covers of uniform neighborhood cardinality using rank-based out-neighborhoods. Both methods induce combinatorial coverings with established minimality and separation properties, and are particularly effective for scalable nerve/witness complexes or interpretable cohort construction (Brunson et al., 2022).

4. Data-Driven and Learning-Focused Subsampling Schemes

Recent work explores subsampling not solely for computational reduction, but also as part of learning pipelines or embedding discovery:

• Sparse-TDA via QR Pivoting: Constructs data matrices of vectorized persistent features (e.g., persistence images) and applies low-rank SVD plus pivoted-QR for column or pixel selection, yielding a sparse but informative feature subset. This reduces downstream classifier training costs dramatically and retains or outperforms state-of-the-art full-feature or kernel-based TDA in multi-way classification benchmarks (Guo et al., 2017).
• Robust Embedded Coordinates through Subsample Aggregation: Embedding algorithms (e.g., Isomap, Laplacian Eigenmaps) are applied over many subsamples and algorithmic hyperparameter settings. The resulting set of candidate embeddings is topologically clustered using bottleneck or Wasserstein distances on their persistence diagrams; robust representatives are then averaged via Generalized Procrustes Analysis (alternating least-squares) to yield consensus embeddings. This confers resilience to noise/outliers and yields “robustified” coordinate representations, with empirical validation on synthetic manifolds and high-dimensional genomics (Blumberg et al., 2 Aug 2024).

5. Theoretical Guarantees and Empirical Assessment

A broad spectrum of theoretical results establishes the conditions under which subsampling maintains statistical consistency and topological fidelity:

• Risk and Bias Bounds: Bias diminishes as $(\log m / m)^{1/b}$ with subsample size, while the variance falls as $1/\sqrt{n}$ in the average landscape approach, balancing statistical accuracy and compute (Chazal et al., 2014).
• Stability of Persistence under Subsampling: Wasserstein and Hausdorff-based inequalities establish continuity of landscapes and diagrams under measure/distributional perturbations, with explicit dependence on subset size and metric properties (Chazal et al., 2014, Minian, 26 Nov 2025, Park et al., 6 Dec 2025).
• CLT and Inferential Anchoring: Averaged Betti curves and associated scalar functionals obey central limit behavior under repeated subsampling, enabling formal hypothesis testing and p-value computation via omnibus statistics in high-throughput settings (Stier et al., 3 Sep 2025).
• Preservation under Collapse and Cores: δ-core and strong collapse methods yield exact recovery of persistent features at fixed scales and up to interleaving errors across filtrations, with empirical bottleneck discrepancies significantly below those from alternative reductions (Minian, 26 Nov 2025).

Extensive experimental validations span triangulated meshes, temporally indexed sensor data, high-dimensional biomedical signals, and functional genomics, demonstrating orders-of-magnitude acceleration, improved robustness to noise/outliers, and superior or comparable classification and feature detection rates.

6. Algorithmic Pipelines and Practical Guidance

Implementations of subsampling strategies are available in Python (e.g., tdasampling (Dufresne et al., 2018)), R (landmark (Brunson et al., 2022)), and other environments. Practical guidance includes:

• Choosing subsample size as large as acceptable for available TDA software, e.g., $m \approx 200$--$300$ for persistent homology;
• Number of iterations or resamples tailored to desired statistical noise level ($n \approx 50$--$200$) (Chazal et al., 2014, Stier et al., 3 Sep 2025);
• For δ-core, selecting δ at the 10–20th percentile of pairwise distances balances reduction and fidelity; tuning for data heterogeneity is suggested (Minian, 26 Nov 2025);
• For landmarkers, maxmin is preferred in homogeneous, metric spaces, while lastfirst is superior in density-heterogeneous or non-Euclidean settings (Brunson et al., 2022);
• In Procrustes or ensemble-embedding methods, use TDA-derived distances for clustering and outlier removal (no clean contractible cluster: flag instability) (Blumberg et al., 2 Aug 2024).

These pipelines yield monotonic computational savings—exponential in dimension for full complexes, polynomial for witness/landmark paradigms, near-linear for strong collapse cores—enabling routine analysis of datasets with hundreds of thousands of points or features.

7. Limitations, Trade-Offs, and Emerging Extensions

Notable caveats and research directions include:

• The global-choice of parameters (e.g., δ in δ-core, cover-radius in maxmin/lastfirst) can lead to under- or over-sampling in heterogeneous spaces; local or adaptive variants are an open area (Minian, 26 Nov 2025);
• Statistical bounds are tightest in settings with regular data distributions, positive reach, and sufficient noise separation; behavior near singularities or in highly non-uniform data remains a challenge (Chazal et al., 2014, Park et al., 6 Dec 2025);
• Outlier-robustness is generally good in average-based and lastfirst-like approaches but may be weak in closest-sample or purely geometric core reductions (Chazal et al., 2014, Brunson et al., 2022, Minian, 26 Nov 2025);
• Integration into multi-scale or density-based filtrations is being explored (e.g., iterative re-application of δ-core, ensemble/average landscapes across parameter grids) (Minian, 26 Nov 2025, Blumberg et al., 2 Aug 2024).
• Kernel, sparse, and matrix-decomposition-based approaches (e.g., pivoted-QR, low-rank SVD) occupy the intersection of statistical learning and TDA, actively expanding subsampling's role in classification, regression, and representation learning (Guo et al., 2017, Blumberg et al., 2 Aug 2024).

In sum, subsampling methods in TDA span probabilistic, combinatorial, and geometric domains, each with rigorous performance guarantees, scalable computation, and demonstrated empirical effectiveness across modalities and application fields. Their continued development underpins the tractable, principled extraction of topological information from modern, high-dimensional data sources.