TDA Feature Engineering Pipeline

• The TDA feature engineering pipeline is a modular workflow that converts raw data into geometric representations and persistent homology-based summaries.
• It integrates domain-specific filtration methods, such as Vietoris–Rips, cubical, and alpha complexes, to capture multiscale features for analysis.
• Vectorization techniques including persistence landscapes, Betti curves, and statistical summaries produce fixed-length features for machine learning tasks.

Searching arXiv for recent and foundational papers on TDA feature engineering pipelines, including time series, images, CT, and software/tooling. arxiv_search query: "topological data analysis feature engineering pipeline time series persistence homology images giotto-tda" A TDA feature engineering pipeline is a workflow in which raw data are converted into a geometric or functional representation, a filtration is constructed, persistent homology is computed, and the resulting topological summaries are transformed into numerical features for downstream learning. Across time series, images, volumetric CT, and customer-behavior data, the literature instantiates this pattern with delay embeddings, cubical complexes, Vietoris–Rips or alpha complexes, barcodes or persistence diagrams, and vectorizations such as persistence landscapes, Betti curves, silhouettes, persistence images, persistence entropy, and statistical summaries (Kim et al., 2018, Rivera-Castro et al., 2019, Garin et al., 2019, Tauzin et al., 2020, Hajij et al., 2021, Lin et al., 31 Mar 2025, Ali et al., 13 Dec 2025).

1. Conceptual architecture

The basic pipeline has a stable architecture even though its concrete realization is domain-specific. Raw observations are first recast as a topological object: a point cloud in a metric space, a cubical complex on a grid, or a filtration induced by a scalar function. A filtration is then formed as a nested sequence of spaces,

$$\mathcal{K}_{\epsilon_0} \subset \mathcal{K}_{\epsilon_1} \subset \dots \subset \mathcal{K}_{\epsilon_n},$$

or, in sublevel-set form for a function $f$,

$M^{(-\infty,b)} = \{x \in M : f(x) \le b\}.$

Persistent homology tracks the birth and death of homology classes across this filtration, yielding barcodes or persistence diagrams. Those diagrams are finally vectorized or summarized into fixed-length features that can be passed to classifiers, regressors, or boosting models (Smith et al., 2020, Tauzin et al., 2020).

This architecture appears in several distinct formulations. In "Time Series Featurization via Topological Data Analysis" (Kim et al., 2018), the pipeline is delay embedding $\rightarrow$ PCA denoising $\rightarrow$ Rips filtration $\rightarrow$ persistent homology $\rightarrow$ persistence landscapes or silhouettes. In "Topological Data Analysis of Time Series Data for B2B Customer Relationship Management" (Rivera-Castro et al., 2019), it is RFM time series $\rightarrow$ sliding windows $\rightarrow$ delay embeddings $\rightarrow$ Rips filtration $f$0 barcodes $f$1 clustering features. In "Time-Series Forecasting via Topological Information Supervised Framework with Efficient Topological Feature Learning" (Lin et al., 31 Mar 2025), the deterministic branch is Takens embedding $f$2 simplicial or cubical complex $f$3 filtration $f$4 persistent homology $f$5 persistence diagrams $f$6 persistence landscapes $f$7 statistical vectorization. This suggests that the defining property of a TDA feature engineering pipeline is not a particular model class, but the insertion of topology-preserving multiscale summaries between representation learning and prediction.

2. Data representations and filtrations

For time series, the dominant representation is time-delay embedding. A univariate series $f$8 is converted into vectors

$f$9

or, equivalently in sliding-window form,

$M^{(-\infty,b)} = \{x \in M : f(x) \le b\}.$0

depending on the notation of the source. The resulting trajectory matrix or point cloud approximates a quasi-attractor of the underlying dynamical system. Kim, Kim, and Rinaldo then project this cloud by PCA before constructing Vietoris–Rips complexes, whereas the CRM Topological RFM pipeline applies the same embedding logic separately to Recency, Frequency, and Monetary series (Kim et al., 2018, Rivera-Castro et al., 2019).

For images and scalar fields, the representation is often cubical rather than simplicial. "A Topological 'Reading' Lesson: Classification of MNIST using TDA" (Garin et al., 2019) represents each pixel as a cube in a cubical complex and defines a natural grayscale filtration through sublevel sets of image intensity. The same paper also uses binary filtration, height filtrations in multiple directions, radial filtrations from multiple centers, density filtrations with radii $M^{(-\infty,b)} = \{x \in M : f(x) \le b\}.$1, dilation filtration, erosion filtration, signed distance filtration, and a Vietoris–Rips filtration on a point-cloud view of pixel coordinates. In "TDA-Net: Fusion of Persistent Homology and Deep Learning Features for COVID-19 Detection in Chest X-Ray Images" (Hajij et al., 2021), the image is instead converted to a 1-dimensional simplicial complex with one vertex per pixel and edges defined by 8-neighborhood connectivity, and a lower-star filtration is induced by pixel intensity.

For volumetric CT, the recent patch-based formulation replaces direct 3D cubical filtration by a point-cloud construction. Each $M^{(-\infty,b)} = \{x \in M : f(x) \le b\}.$2 patch is encoded by a Morton code for spatial location and either PCA coordinates or patch statistics such as mean, median, mode, standard deviation, interquartile range, entropy, range, $M^{(-\infty,b)} = \{x \in M : f(x) \le b\}.$3, and $M^{(-\infty,b)} = \{x \in M : f(x) \le b\}.$4. The set of encoded patches becomes a point cloud in $M^{(-\infty,b)} = \{x \in M : f(x) \le b\}.$5 or $M^{(-\infty,b)} = \{x \in M : f(x) \le b\}.$6, and persistent homology is then computed on an alpha complex filtration

$M^{(-\infty,b)} = \{x \in M : f(x) \le b\}.$7

This patch-to-point conversion is a decisive change of representation: it trades a voxel-scale cubical complex for a topological summary of localized 3D texture and position (Ali et al., 13 Dec 2025).

The choice of filtration is therefore tied to modality. Vietoris–Rips, weighted Vietoris–Rips with Distance-to-Measure weights, Čech, cubical, directed flag, lower-star, and alpha constructions all occur in the cited literature, and each induces a different notion of scale and adjacency (Tauzin et al., 2020, Lin et al., 31 Mar 2025).

3. Persistent homology and feature vector construction

Persistent homology formalizes multiscale topology by tracking homology groups $M^{(-\infty,b)} = \{x \in M : f(x) \le b\}.$8 through a filtration. In the standard notation used across the literature,

$M^{(-\infty,b)} = \{x \in M : f(x) \le b\}.$9

and the Betti number

$\rightarrow$0

counts connected components, loops, voids, and higher-dimensional cavities. The persistent diagram in dimension $\rightarrow$1 is a multiset

$\rightarrow$2

where $\rightarrow$3 and $\rightarrow$4 are the birth and death scales of a topological feature (Smith et al., 2020, Lin et al., 31 Mar 2025).

A central problem in feature engineering is that persistence diagrams are multisets of variable cardinality, whereas downstream models generally require fixed-length vectors. The literature therefore relies on several families of vectorization.

The first family is functional. Kim, Kim, and Rinaldo define persistence landscapes through tent functions $\rightarrow$5 and take the $\rightarrow$6-th largest envelope

$\rightarrow$7

while silhouettes compute a weighted average

$\rightarrow$8

These are sampled on a grid and flattened into vectors (Kim et al., 2018). TDA-Net uses a Betti curve sampled to a vector in $\rightarrow$9, where $\rightarrow$0 counts how many connected components are alive at threshold $\rightarrow$1 (Hajij et al., 2021). giotto-tda exposes persistence landscape, Betti curves, silhouette, heat representation, and persistence image as scikit-learn-compatible transformers (Tauzin et al., 2020).

The second family is image-like or kernel-like. Persistence images place weighted Gaussian kernels on diagram points and discretize the resulting persistence surface on a grid. The broader TDA review also lists Wasserstein and bottleneck distances, heat kernels, and persistence images as stable representations of diagrams (Smith et al., 2020, Tauzin et al., 2020).

The third family is statistical. "Time-Series Forecasting via Topological Information Supervised Framework with Efficient Topological Feature Learning" (Lin et al., 31 Mar 2025) constructs a 12-dimensional feature vector per variable from number of birth–death pairs, max lifetime, mean lifetime, and persistence landscape mean, variance, and median. The Topological RFM pipeline extracts count of features, total persistence, max persistence, mean persistence, and optionally persistence entropy from $\rightarrow$2 and $\rightarrow$3 diagrams before applying K-means (Rivera-Castro et al., 2019). The patch-based CT pipeline summarizes births, deaths, lifespans, and midpoints by mean, median, standard deviation, full range, interquartile range, 10th, 25th, 75th, and 90th percentiles, together with bar count and persistence entropy, then concatenates these summaries across $\rightarrow$4, $\rightarrow$5, and $\rightarrow$6 (Ali et al., 13 Dec 2025). This suggests a general distinction between function-valued vectorizations, image-valued vectorizations, and low-order statistical descriptors.

4. Representative pipeline variants across domains

In noisy time-series analysis, "Time Series Featurization via Topological Data Analysis" (Kim et al., 2018) is a canonical example. It begins with a sliding-window or Takens embedding, projects the trajectory matrix onto the first $\rightarrow$7 principal components, computes persistent homology of the Rips filtration up to $\rightarrow$8, and then uses persistence landscapes or silhouettes as features. The paper proves stability bounds for the bottleneck distance and for the resulting landscapes and silhouettes, and reports lower error rates for TDA + 2LP and TDA + CNN than for the corresponding non-topological baselines.

In B2B customer analytics, "Topological Data Analysis of Time Series Data for B2B Customer Relationship Management" (Rivera-Castro et al., 2019) replaces a static RFM snapshot with Recency, Frequency, and Monetary time series, computes separate point clouds for each, extracts topological statistics from barcodes, and clusters customers with K-means. The resulting cluster labels are embedded as categorical features in CatBoost. Performance is explicitly dataset-dependent: on the Cloud dataset, Topological RFM achieved RMSE $\rightarrow$9, outperforming No RFM ($\rightarrow$0), RFM ($\rightarrow$1), and Time Series RFM ($\rightarrow$2); on Hospitality, Topological RFM yielded RMSE $\rightarrow$3, worse than Time Series RFM at $\rightarrow$4.

For grayscale image classification, "A Topological 'Reading' Lesson: Classification of MNIST using TDA" (Garin et al., 2019) uses multiple filtrations rather than a single one. Persistence diagrams in dimensions 0 and 1 are vectorized by amplitudes, Betti curves, persistence landscapes, heat kernel summaries, and persistent entropy. The complete feature set contains 728 topological features; random-forest importance ranking followed by removal of features with Pearson correlation $\rightarrow$5 reduces this to 84 selected topological features; and 28 TDA features achieve accuracy comparable to roughly 700 pixel features. The paper thereby treats TDA as a dimensionality-reduction and feature-selection pipeline, not merely as a descriptor generator.

For image-model fusion, TDA-Net constructs a lower-star filtration on a simplicial complex built from pixels and 8-neighborhood edges, computes only the 0-dimensional persistence diagram, transforms it into a 100-dimensional Betti-curve vector, and feeds that vector into a topological MLP stream fused with a CNN stream (Hajij et al., 2021). The reported validation accuracies are 0.87 for the base CNN, 0.89 for TDA-Net$\rightarrow$6, 0.92 for TDA-Net$\rightarrow$7, and 0.93 for TDA-Net$\rightarrow$8, indicating that the topological stream is useful both alone and in fusion.

For robustness-oriented image classification, "LightGBM robust optimization algorithm based on topological data analysis" (Yang et al., 2024) explicitly partitions feature engineering into a pixel feature stream and a topological feature stream, concatenates the resulting vectors, and uses the merged representation as input to LightGBM. Under noisy conditions, the paper reports a 3% accuracy improvement over LightGBM on the SOCOFing dataset across five classification tasks, a 6% improvement on Ultrasound Breast Images for Breast Cancer, and a 15% improvement on Masked CASIA WebFace; on CIFAR-10, by contrast, TDA-LightGBM does not outperform plain LightGBM.

For volumetric medical imaging, "A Novel Patch-Based TDA Approach for Computed Tomography" (Ali et al., 13 Dec 2025) replaces direct 3D cubical PH with patch encoding and alpha complexes. Across all datasets, the patch-based method achieved average improvement of 10.38% in accuracy, 6.94% in AUC, 2.06% in sensitivity, 11.58% in specificity, and 8.51% in F1 score over the 3D cubical complex baseline, while also improving time-efficiency.

5. Integration with machine learning systems

The literature supports two complementary modes of integration: TDA as a preprocessing transformer for standard ML, and TDA as a learned or supervised component inside larger predictive systems.

The preprocessing mode is exemplified by giotto-tda, which integrates high-performance TDA with machine learning via a scikit-learn-compatible API. The library provides preprocessing for time series, images, graphs, and point clouds; persistence computation for Vietoris–Rips, Čech, alpha, cubical, and directed flag constructions; and vectorizations such as persistence landscape, Betti curves, silhouette, heat representation, persistence image, persistence entropy, number of points, and amplitude (Tauzin et al., 2020). This software view treats a TDA feature engineering pipeline as a composable estimator chain: fit, transform, fit_transform, cross-validation, and hyperparameter search operate exactly as in the rest of the scikit-learn ecosystem.

The learned-integration mode is represented by the "Topological Information Supervised" framework (Lin et al., 31 Mar 2025). Its deterministic TDA branch computes topological features from Takens embeddings, complexes, persistent homology, persistence diagrams, and persistence landscapes. A learned generator $\rightarrow$9 is then trained with reconstruction loss and GAN loss to emulate that deterministic mapping, yielding synthetic topological features quickly at inference. Those features are embedded by a small MLP and concatenated with raw historical input for TIS-BiGRU and TIS-Informer, while a combined loss adds topological consistency terms

$\rightarrow$0

to the standard prediction loss. In this formulation, TDA is no longer only a static feature extractor; it also becomes a supervisory signal and a learned critic of dynamical similarity.

A plausible implication is that TDA feature engineering has bifurcated into a classical branch, where persistent homology is computed explicitly and vectorized, and a surrogate branch, where explicit PH is approximated by trainable modules in order to reduce inference cost or enable end-to-end supervision.

6. Interpretability, limitations, and terminological ambiguity

One of the persistent attractions of TDA feature engineering is interpretability. The CRM study treats TDA-based clusters as customer personas and links strong $\rightarrow$1 features in Monetary or Frequency embeddings to regular, cyclical purchasing behavior (Rivera-Castro et al., 2019). The MNIST study shows that persistent entropy from height, radial, and density filtrations dominates the top-ranked features and provides geometric interpretations of misclassifications such as a 9 being read as a 4 because a loop is not fully closed (Garin et al., 2019). The chemical-engineering review goes further by coupling persistence-image weights with inverse analysis through volume-optimal cycles, mapping discriminative regions in persistence space back to physical structures such as loops in micrographs or voids in solvent density fields (Smith et al., 2020).

The same literature also emphasizes limits. Performance is not uniformly superior: Topological RFM underperforms on some datasets, TDA-LightGBM does not improve CIFAR-10, and the patch-based CT method still requires task-specific tuning of patch size and statistics (Rivera-Castro et al., 2019, Yang et al., 2024, Ali et al., 13 Dec 2025). Computational cost remains a structural issue for Vietoris–Rips and large cubical filtrations, motivating PCA denoising, patch encodings, sparse filtrations, and learned surrogates (Kim et al., 2018, Lin et al., 31 Mar 2025). Parameter sensitivity is another recurring constraint: embedding dimension $\rightarrow$2, delay $\rightarrow$3, patch size, homology dimension, filtration range, vectorization resolution, and choice of complex all materially affect the resulting features (Kim et al., 2018, Garin et al., 2019, Ali et al., 13 Dec 2025).

A further source of confusion is terminological. In the topological literature, TDA denotes Topological Data Analysis. However, "Tabular Data Adapters: Improving Outlier Detection for Unlabeled Private Data" uses TDA to denote Tabular Data Adapters, a distinct non-topological feature-engineering and weak-labeling pipeline based on dataset similarity, shared autoencoders, and crossover samples (Herurkar et al., 29 Apr 2025). This suggests that the phrase “TDA feature engineering pipeline” benefits from explicit disambiguation in contemporary writing.

Taken together, the cited work presents the TDA feature engineering pipeline as a modular, multiscale, and representation-dependent methodology. Its core operations—embedding, filtration, persistent homology, and vectorization—are stable across domains, but the most effective realizations differ sharply between time series, tabular CRM behavior, grayscale images, and volumetric CT. The field’s current trajectory combines classical persistent homology with scalable software abstractions, learned topological surrogates, and hybrid fusion architectures, while continuing to treat interpretability, computational complexity, and dataset dependence as first-order design constraints.