Topological Data Analysis

Also known as TDA

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4 papers in the last four weeks, with none the four weeks before. 0.0% of all new papers.

Jul 13Week of Sep 28

Latest papers 42

Oct 6, 2026stat.ME

Where Do Two Populations of Persistence Diagrams Differ? Calibrated Local Inference at a Fixed Budget

Many two-sample tests for populations of persistence diagrams assess global differences without identifying the regions of the birth-death plane that contribute to them. We study simultaneous inference for local mean contrasts when the number of available diagrams is fixed. They are differences in expected weighted feature mass within

\ell_\infty

neighborhoods at several centers and radii. We estimate these contrasts using additive landmark responses. A Gaussian multiplier bootstrap calibrates simultaneous confidence intervals while allowing unequal group covariances. The neighborhoods whose intervals exclude zero form a map with approximate family-wise error control, and selecting a subset of original intervals for display preserves their joint coverage guarantee. On the simultaneous coverage event, every reported neighborhood lies within twice its radius of the support of the mean-measure difference. A geometric result gives sufficient radius conditions for a displaced feature to produce a nonzero contrast. A comparison of sufficient detection thresholds quantifies the tradeoff between reducing the number of tested coordinates and reserving observations for an independent pilot. In simulations with 40 to 120 diagrams per class, the bands achieved 94%-98% simultaneous coverage under both the strict null and equal means with unequal covariances. In the latter setting, a permutation maximum and the pooled-t implementation of the two-stage persistence-image test of Moon and Lazar rejected in up to 32% and 26% of runs, respectively. In the fixed-budget simulations, spending a third of the observations on a pilot to choose landmarks or radii located changes less often than a prespecified grid at a single radius. On the MUTAG benchmark, the localized region concentrates on rings of fused-ring systems, an exploratory reading.

Pramita Bagchi, Edward Bae, Atish Mitra +3

The George Washington University · Montana Technological University · University of Ljubljana

Two-Sample Testing Confidence Region Estimation Multiple Hypothesis Testing Topological Data Analysis

Oct 4, 2026math.AT

The sublevel Flood bifiltration: towards scalable 2-parameter persistent homology

Multiparameter persistent homology is a rapidly developing branch of topological data analysis that improves the robustness of single-parameter persistent homology to outliers, while still capturing the metric characteristics of the data. However, a notable limitation is its lack of scalability. In this paper, we introduce a novel approach for efficiently computing 2-parameter persistent homology on large point sets. Our work extends the Flood filtration, originally developed for single-parameter persistence. Our construction, called the sublevel Flood bifiltration, offers a scalable approximation of the sublevel offset bifiltration. We show that it benefits from theoretical stability properties and describe how to compute it efficiently. We demonstrate the performance of our approach in classification tasks on low-dimensional synthetic datasets, where density awareness is critical, as well as on real-world time series datasets.

Mattéo Clémot, Julie Digne, Julien Tierny

Université Lyon 1 · CNRS, Université Lyon 1 · CNRS, Sorbonne Université

Persistent Homology Topological Data Analysis

Oct 1, 2026cs.LG

Higher-Order Positional Encodings for Graph Representation Learning

Many real-world systems exhibit higher-order interactions among groups of entities that cannot be captured by pairwise relationships alone. Graph Transformers and Graph Neural Networks increasingly rely on positional encodings to enrich graph representations, yet existing positional encodings are computed solely from the original graph and therefore cannot directly capture observed higher-order interactions. Topological Deep Learning addresses this limitation by lifting graphs to simplicial complexes, but typically requires performing message passing or attention on higher-order neural network representations. We introduce a representation learning paradigm that enriches graph representations with higher-order topology through positional encodings, enabling standard graph learning models to exploit lifted incidence structure without modifying the backbone. We derive a theoretical characterization of the expressivity of higher-order positional encodings, proving that node-level operators induced by higher-order lifts can mix graph Laplacian frequencies in ways that scalar graph spectral filters cannot. Guided by this theory, we instantiate higher-order positional encodings using Hodge Laplacians derived from clique complexes. Experiments with Graph Transformers on ZINC and controlled synthetic benchmarks demonstrate improvements in predictive performance, while a fixed-1-skeleton experiment shows that the pipeline can transmit higher-order information when cells are supplied independently of the graph. Together, our results establish higher-order positional encodings as a principled bridge between graph positional encodings and topological deep learning.

Caleb Stam, Aagrim Hoysal, Sanjukta Krishnagopal

University of California, Santa Barbara

Graph Representation Learning Positional Encoding Graph Transformer Topological Representation Learning Topological Data Analysis

Sep 21, 2026cs.LG

Topological Signal Processing With Unoriented Operators

Topological signal processing (TSP) processes signals on simplicial complexes with oriented boundary operators, which is the natural choice for flow signals or when the topological invariants play a role for the task at hand. However, many higher-order signals carry no orientation, and applying oriented operators to them is not well-defined since it introduces an arbitrary choice of simplex orientation. We study an unoriented TSP (UTSP) framework that replaces oriented boundaries with unoriented incidence matrices. First, we show that unoriented incidence and Laplacian matrices between arbitrary simplicial levels admit graph-like spectral properties. Second, since dropping orientation removes the Hodge decomposition, we introduce an unoriented counterpart, termed interaction-order decomposition, which quantifies how much of a higher-order signal is explained by aggregating lower-order signals. Third, we use this decomposition to derive regularizers for signal reconstruction that penalize each interaction order separately. Experiments on real-world data show that the order-aware regularizers outperform oriented baselines, with the largest gains when the signal energy is unevenly distributed across orders.

Andrea Cavallo, Varun Sarathchandran, Geert Leus +1

Delft University of Technology, Delft, Netherlands

Graph Signal Processing Spectral Methods Topological Data Analysis

Sep 14, 2026cs.CG

Direct Topology Tracking in Continuous Implicit Models

We present a framework for tracking topological features directly within continuous implicit models. Such models, including implicit neural representations (INRs) and multivariate functional approximations (MFAs), are increasingly adopted to represent scientific data without the resolution constraints of discrete grids. They offer compact, smooth, and differentiable representations of complex fields, enabling new opportunities for high-performance data storage, reconstruction, and analysis. Given a continuous implicit model, our method tracks the evolution of critical points by querying the model and its derivatives, thereby eliminating the need to resample onto a grid. This approach enables faithful feature tracking while avoiding discretization-induced artifacts such as aliasing. We demonstrate the generality of our framework across a range of implicit representations, including analytic functions, MFAs, and INRs, and show that it produces smooth, coherent critical point trajectories. By enabling feature tracking directly on continuous representations, our method supports a new class of feature-driven visualization workflows centered on implicit models.

Guanqun Ma, David Lenz, Kaiyuan Tang +4

University of Utah · The Ohio State University

Topological Data Analysis

Aug 6, 2026stat.ML

Stochastic Dynamics on Persistence Diagram Space via Reinforcement Learning

Persistence diagrams (PDs) provide stable and interpretable summaries of multiscale topological structure. While substantial progress has been made in the statistical analysis of PDs, existing literature often treats diagrams as static objects and provide limited frameworks for probabilistic modeling and stochastic evolution on PD space. We introduce a reinforcement learning framework for stochastic dynamics on PD space, where diagrams evolve through topology aware local edit operations. The dynamics define controlled Markov processes on spaces of finite PDs with variable cardinality. We establish conditions under which the induced Markov chains are irreducible, aperiodic, and geometrically ergodic, implying the existence of unique stationary probability laws on PD space. To guide the dynamics toward scientifically relevant topological targets, we formulate objectives that encompass distribution matching, task specific topological statistics, and structure-preserving compression. The resulting rewards balance task specific distributional targets, diagram fidelity, and complexity reduction, and yield a framework for adaptive topological simplification and probabilistic modeling. Experiments on synthetic and neuroimaging PDs demonstrate that the proposed framework can preserve dominant topological structure while reducing diagram complexity.

Farzana Nasrin

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

RL for Generative Models Persistent Homology Topological Data Analysis

Jul 31, 2026stat.ML

Persistent Convolution: A Topological Framework for AI Alignment Testing and Semantic Space Characterization

Modern opaque AI models prize performance over interpretability, which makes testing difficult. However, formal statistical tests conducted on a model's embedding space can provide robust characterizations of semantic structure, concept separation, and knowledge graph alignment. Model developers would benefit from a model comparison technique that leverages human-curated knowledge structures to test alignment. The scale of the input space for even relatively simple tasks motivates the need for alignment checks that augment standard outcome reasoning. This work develops and demonstrates a topology-based multi-modal alignment test to make deployment, selection, and comparison of opaque models more interpretable. These methods also offer an intuitive connection to possibility theory and a unified decision theoretic framework from data to deployment.

Tyler Ashoff, Jordan Rodu

Department of Statistics, University of Virginia. · Department of Statistics, University of Virginia

AI Alignment Cross-Modal Alignment Multimodal Model Interpretability Topological Data Analysis

Jul 26, 2026cs.LG

Topological Data Analysis and Graph-Theoretic Approaches for Tennis Match Prediction

We present two approaches for predicting tennis match outcomes using topological data analysis and graph theory on ATP singles matches from 2000-2025. The first method applies lower-star filtration to player competitive networks, extracting topological features through persistent homology using four summary methods (VAB, HNAV, HWNAV, OW-HNPV) combined with Modified Band Depth analysis. Algorithmic optimizations including ego graph approximations and triangle elimination enable analysis of about 66k matches. Our Random Forest model achieves 66.2% accuracy (AUC = 0.719) using topological, graph-theoretic, and ranking features. Feature importance analysis reveals that rankings contribute 36.3%, centralities 25.5%, and TDA features 24.0%, with topological features providing complementary signal. When rankings are unavailable, the topology-only model maintains 63.56% accuracy, demonstrating that network-derived features alone capture meaningful competitive structure. The second method uses a modified Katz similarity index with temporal edge weighting, achieving 62.48% accuracy on held-out test data. This work represents the first application of lower-star filtration to tennis prediction, provides systematic comparison of four topological summary methods in sports analytics, and demonstrates that TDA can achieve above-chance prediction using network topology alone while providing additional value when combined with traditional features.

Jake Schwaderer, Alexander Bastien, Omid Khormali +3

Department of Mathematics University of Evansville Evansville, Indiana 47722, USA.

Sports Outcome Prediction Topological Data Analysis

Jul 15, 2026quant-ph

Quantum Topological Data Encoding

Many datasets encountered across a wide range of domains possess rich geometric and topological structure that is difficult to capture using conventional vector-based representations. Quantum machine learning offers the possibility of processing high-dimensional data in Hilbert spaces, but its practical success depends critically on how classical data is encoded into quantum states. We introduce \emph{quantum topological data encoding} (QTDE), a general framework for encoding topological information into quantum states via topology-driven quantum evolution. Our method generalises an existing topology-driven quantum encoding framework to higher-dimensional data. We test the proposed method on clique-complexes classification tasks, and provide preliminary evidence that topology-driven quantum representations can capture discriminative information beyond that available through direct comparisons of classical topological descriptors. The proposed quantum representations consistently outperform a baseline based on direct comparisons of the combinatorial Laplacians describing the underlying topological structure. We indicate several areas of application where the framework can be used to provide a more efficient and reliable data representation.

Adam Wesołowski, Dimitrios Thanos, Daniel Leykam +1

Royal Holloway University of London, Department of Computer Science, UK · Leiden Institute of Advanced Computer Science (LIACS), Leiden University, Leiden, The Netherlands · Science, Mathematics and Technology Cluster, Singapore University of Technology and Design, Singapore +1

Quantum Machine Learning Topological Data Analysis

Jul 10, 2026quant-ph

Depth-Efficient Quantum Topological Data Analysis for Regime-Specific Detection of Financial Stress

We present, to our knowledge, the first adaptation of Pauli Correlation Encoding (PCE) to quantum topological data analysis, reformulating Betti number estimation as a depth-efficient variational optimization over a compressed qubit register. From a Takens embedding and Vietoris--Rips filtration of S&P500 returns, we extract combinatorial Laplacians and recast null-space counting as a continuous-PCE Rayleigh-quotient minimization with variational deflation, encoding $n_k$ simplex indices into $O(n_k^{1/κ})$ qubits with shallow, ancilla-free circuits. Because the resulting loss is rational rather than bilinear in the correlators, the barren-plateau bound of\cite{Sciorilli25} does not transfer; empirically the gradient variance decays only polynomially, with no exponential barren plateau, over

n=4

--

12

qubits. The classical stage matches ripser~\cite{bauer2021ripser} on all 190 sliding windows (2007-2009). On the real market Laplacians (

β_1=1

--

22

), warm-starting from a classical null-space surrogate allows PCE-VQE to recover

β_1

exactly at every scale, placing the obstacle in the optimisation landscape rather than the encoding. Chronologically split classification gives in-regime ROC AUC

0.818

, but out-of-distribution evaluation on the 2020 COVID shock and 2022 rate cycle (AUC

0.009

,

0.515

) shows the calibration does not generalize across crisis regimes.

Arul Rhik Mazumder, Shreyan Ronit Mazumder

Carnegie Mellon University Pittsburgh, PA, USA · Cambridge Rindge and Latin School Cambridge, MA, USA

Quantitative Finance Variational Quantum Algorithms Topological Data Analysis

Jul 6, 2026cs.CV

Shape Over Intensity: Directional Topological Encoding for False Positive Reduction in Intracranial Aneurysm Detection

Automated detection of intracranial aneurysms (IAs) from CT angiography (CTA) is severely hindered by high false-positive rates. Convolutional neural networks (CNNs) rely on local pixel intensities, causing systematic confusion between saccular aneurysms and vascular bifurcations - a problem especially acute for small lesions (<3 mm), where detection sensitivity falls below 60%. We propose a plug-and-play, topology-aware false-positive reduction framework evaluating the Smooth Euler Characteristic Transform (SECT) - a directional representation encoding global 3D vascular geometry independently of intensity - against persistence-based summaries (Persistence Images and Landscapes), tested on a stratified subset of the RSNA 2025 dataset. SECT achieves an AUC of 0.943, substantially outperforming direction-agnostic methods (AUC ~0.68), and exhibits a clinical performance inversion: it excels on the sub-3 mm cohort, maintaining 0.943 AUC and 78.5% sensitivity at 95% specificity. The representation is also scanner-agnostic, achieving 0.927 mean AUC under leave-one-scanner-out (LOGO) validation across four manufacturers. By capturing asymmetric geometric invariants rather than intensity profiles, SECT reliably resolves the primary structural confounder in IA detection, positioning it as a robust downstream filter for hybrid deep-learning diagnostic pipelines.

Akshay Gokhale, Mansi Dhamne

Computer Science and Engineering Sardar Patel Institute of Technology Mumbai, India · Computer Engineering Sardar Patel Institute of Technology Mumbai, India

FDR Control Medical Image Anomaly Detection Topological Data Analysis

Jun 29, 2026cs.CV

TopoAgent: An Agentic Framework for Automated Topology Learning in Medical Imaging

Topological data analysis (TDA), particularly persistent homology (PH), captures geometric structural properties in medical images (e.g., connected components, loops, shape characteristics), which conventional pixel-level deep learning approaches often neglect. While many topological descriptors are known for converting persistence diagrams (PDs) or raw images into topological feature vectors, existing methods mostly default to a single fixed descriptor (e.g., persistence images), leaving the diversity of topological representations largely unexplored. To the best of our knowledge, there is no known large language model (LLM)-based agentic framework that can automatically determine the most suitable topological descriptors for a given image dataset and produce the corresponding topological feature vectors for downstream tasks. To fill this gap, we propose \textbf{TopoAgent}, an LLM-based agentic framework that automates topology learning for medical image analysis.TopoAgent operates through a Perception--Reasoning--Action--Reflection loop supported by 21 domain-specific tools and dual memory that accumulates experience across runs. Its skill set is distilled from systematic evaluation of 15 topological descriptors across 26 datasets with six classifiers. TopoAgent analyzes input images and their topological characteristics, reasons about which topological descriptors best suit the input, and determines the optimal descriptor and its configuration, all without task-specific training.

Guangyu Meng, Pengfei Gu, Xueyang Li +3

Dept. of Computer Science and Engineering, University of Notre Dame, Notre Dame, IN, USA · Dept. of Computer Science, The University of Texas Rio Grande Valley, Edinburg, TX, USA

Medical Image Analysis Persistent Homology Automated Feature Engineering Topological Representation Learning Topological Data Analysis

Jun 24, 2026cs.LG

Topology-Informed Neural Networks for Flood Detection in Optical and Synthetic Aperture Radar Imagery

Floods frequently impact regions around the world. Rapid and accurate flood detection is crucial for emergency response and timely mitigation of human and economic loss. The expanding availability of satellite data and advances in artificial intelligence have enhanced monitoring of environmental hazards, but many flood events remain challenging to detect because cloud cover obscures optical satellite imagery. Rambour et al. introduced the SEN12-FLOOD dataset and extracted per-image features using a ResNet-50 convolutional neural network backbone, then fed these features into a gated recurrent unit network to show that temporal information can substantially improve accuracy compared to single-image baselines. More recently, Chamatidis et al. showed that a vision transformer can achieve strong performance with popular convolutional architectures. However, these models typically function as opaque black boxes, making it difficult to interpret their decision boundaries, learned features, and internal reasoning, especially in safety-critical domains like remote sensing. In contrast, topological data analysis (TDA) provides a mathematically grounded framework for capturing global structural features of data. TDA has emerged as a powerful tool for analyzing complex imagery, especially imagery with geometrically interpretable structures, of which floods are a prime candidate. In this work, we systematically evaluate topological descriptors for flood detection using the open-source SEN12-FLOOD dataset. By extracting topological features from each image and incorporating them into neural networks, we demonstrate that topological descriptors carry meaningful flood signals independently and complement existing networks to yield more robust and interpretable flood detection systems.

Sophia Li, Max Zhao, Raghu G. Raj +1

aU.S. Naval Research Laboratory, Washington, DC, USA · bStanford University, Stanford, CA, USA

Remote Sensing Image Understanding Disaster Response Topological Data Analysis

Jun 22, 2026cs.LG

Collapsed Effective Operators for Higher-order Structures

Higher-order structures are powerful relational modeling tools, yet existing spectral operators decompose the topology into separate ranks, leaving practitioners to fuse the information back to vertices through ad hoc choices. We introduce Collapsed Effective Operators, which condense higher-order degrees of freedom into a single vertex-level operator via Schur complementation of a graded Laplacian. This yields a (generally dense) operator that encodes long-range interactions mediated by topology and is applicable to arbitrary higher-order constructs. We show it preserves positive semi-definiteness with a spectral upper bound relative to the rank-0 Hodge Laplacian, effectively lowering system energy under higher-order connectivity. Empirically, our operator improves spectral clustering, signal smoothing, and enables the inclusion of topological features in neural network architectures via positional encoding. The project page can be found http://circle-group.github.io/research/CollapsedEffectiveOperators

Maximilian Krahn, Lennart Bastian, Vikas Garg +2

Department of Computing, Imperial College London, London, UK · Chair of Health Informatics, Technical University of Munich, Germany · Munich Center for Machine Learning, Germany +2

Graph Clustering Spectral Methods Topological Data Analysis

Jun 20, 2026cs.CV

Topological summaries of fingerprint ridge patterns carry identity information

Fingerprints are the most widely deployed biometric. Verifying whether two impressions come from the same finger typically relies on minutiae, small landmarks such as skin ridge endings and bifurcations. These landmarks are extracted through a multi-stage pipeline of image enhancement, skeletonization, minutiae detection, and alignment. We investigate an alternative: using topological data analysis to represent the full pattern of skin ridges and valleys directly, bypassing minutiae detection and the downstream matching pipeline. We apply persistent homology, a topological tool that tracks how loops in the ridge pattern form and fill in across spatial scales, producing multi-scale summaries of ridge geometry. We develop and compare a range of verification methods on a standard benchmark dataset, FVC2000 DB1. Even the simplest topological summaries, with no trained parameters, substantially outperform geometry-only baselines. A trained method achieves an AUC of 0.91, while an optimal-transport method excels at the strictest false-accept thresholds, suggesting they capture different aspects of the ridge pattern. Fusing these two approaches yields the best performance at every low false-accept threshold we examine. Our results establish that these topological summaries capture substantial fingerprint identity information, far more effective for verification than raw pixel-level geometry. Because the entire pipeline is openly specified, it offers a transparent complement to minutiae-based systems, and we provide a modular framework for constructing, evaluating, and combining topological verification methods.

Chad M. Topaz, Niny Arcila-Maya, Elizabeth Munch +2

Williams College, Williamstown, MA 01267 · QSIDE Institute, Williamstown, MA 01267 · University of Colorado Boulder, Boulder, CO 80309 +3

Biometric Identification Persistent Homology Topological Data Analysis

Jun 18, 2026eess.SY

Topological Data Analysis for High-Dimensional Dynamic Process Monitoring

Real-time process monitoring requires methods that extract actionable information from high-dimensional time-series data. In this work, we present a new approach for process monitoring that combines tools of topological data analysis (TDA) and machine learning. In the proposed approach, we represent multivariate time-series data as manifolds and use topological descriptors to summarize the structure of such data; we then use a neural ordinary differential equation to learn the dynamic evolution of the topological structure of the system. Using real data from an industrial process, we show that this trajectory-based event detection approach is effective at detecting diverse types of events. We contrast this approach against reconstruction-based approaches such as principal component analysis and autoencoders and against a trajectory-based approach that uses Koopman autoencoders.

Angan Mukherjee, Tyler A. Soderstrom, Michael J. Kurtz +1

Department of Chemical & Biological Engineering, University of Wisconsin-Madison, 1415 Engineering Drive, Madison, WI 53706, USA · ExxonMobil Technology and Engineering, 22777 Springwoods Village Pkwy, Spring, TX 77389, USA

Process Supervision Topological Representation Learning Topological Data Analysis

Jun 10, 2026stat.ML

From Persistence to Survival: Hypothesis Testing, Effect Sizes and Vectorisation for Topological Features

Persistence diagrams are common representations in topological data analysis, but they do not naturally live in a vector space, and the statistical tools developed for comparing them have largely evolved separately from those used for downstream prediction. We introduce STRAND (Survival Topological Representation ANalysis of Diagrams), which treats (collections of) PDs as survival data: each topological feature with persistence value

p = d - b

is a fully observed time-to-event, and the persistence survival function

S(t) = \mathbb{P}(p > t)

is the central object for comparing diagrams. From this single representation we derive (i) a non-parametric two-sample test with calibrated Type I error and high power from a small number of diagrams; (ii) interpretable effect sizes; and (iii) a 1-Wasserstein-stable feature vector for downstream machine learning. We validate calibration and power on synthetic manifolds with controlled topology, demonstrate competitive vectorisation across 14 graph and 3D point cloud benchmarks, and apply the method to study functional brain connectivity in fMRI/neuroscience data. To our knowledge, STRAND is the first method to provide hypothesis testing and vectorisation for persistence diagrams from a single coherent and interpretable representation.

Juliette Murris, Bernadette Stolz, Karsten Borgwardt

Department of Machine Learning and Systems Biology, Max Planck Institute of Biochemistry, Martinsried, Germany

Two-Sample Testing Persistent Homology Statistical Hypothesis Testing Topological Data Analysis

Jun 9, 2026cs.LG

Encoding the Euler Characteristic Transform

The Euler Characteristic Curve (ECC) records the Euler characteristic of a linearly embedded cell complex as a function of filtration height in a given direction, and the Euler Characteristic Transform (ECT) is the injective shape descriptor obtained by collecting ECCs over many directions. How the ECT is encoded for a neural network is itself an inductive bias, conventionally fixed by discretizing each ECC. We introduce a continuous encoding: for each direction and each vertex it records the net Euler-characteristic change attributed to that vertex, producing a per-direction token sequence that a small transformer maps to a feature vector. We separate the resulting pipeline into two stages on orthogonal axes: an ECC encoder that acts within each direction, mapping its curve to a fixed-length vector, and an ECT representation that acts across directions, aggregating the per-direction vectors into one. We study six ECT representation architectures spanning a range of inductive biases, from a structure-agnostic feedforward baseline to convolutional and complex-valued models that preserve equivariance under planar rotations. Across six classification benchmarks covering point clouds, graphs, cubical complexes, and meshes, the continuous encoding improves accuracy on five of six datasets, and control experiments attribute the gain to the tokenization itself rather than to the added transformer capacity. The representation architecture matters less than the encoding, and the payoff from its inductive biases depends on the encoding: a feedforward network performs best under continuous encoding but is less robust under discretization than convolutional architectures.

Nello Blaser, Odin Hoff Gardaa, Lars M. Salbu +2

Department of Informatics, University of Bergen, Norway · Department of Informatics, University of Fribourg, Switzerland

Representation Learning Geometric Representation Learning Topological Representation Learning Topological Data Analysis

Jun 6, 2026cs.GR

MS-COOT: Comparing Morse-Smale Complexes with Co-Optimal Transport

Understanding and comparing structures in scalar fields is a central challenge in scientific visualization, with applications ranging from feature analysis to temporal and structural comparison. The Morse-Smale (MS) complex provides a natural representation by decomposing a scalar field into regions induced by gradient flow. However, existing approaches typically rely on graph-based representations, capturing relationships between critical points while discarding region-level structure. In this work, we represent the MS complex as a hypergraph, where critical points form nodes and regions define hyperedges. We introduce MS-COOT, a co-optimal transport distance that jointly computes correspondences between critical points and regions. This formulation enables explicit region-to-region matching within a distance-based framework, allowing identification of region-level events such as splitting and merging. We instantiate this framework with domain-specific components, including a hypernetwork function encoding critical point-region relationships, persistence-based probability measures that emphasize topologically significant features, and a sample cost term that incorporates critical point attributes. We evaluate MS-COOT on five datasets spanning 2D simulations, 3D surface meshes, and volumetric data. Our results show that MS-COOT captures region-level structural changes that are not reflected by graph-based distances, while achieving strong performance in downstream tasks such as classification and resolution discrimination.

Guangyu Meng, Mingzhe Li, Erin Wolf Chambers

Department of Computer Science and Engineering, University of Notre Dame, Notre Dame, Indiana, United States

Optimal Transport Topological Data Analysis

Jun 4, 2026stat.ML

Symmetric Divergence and Normalized Similarity: A Unified Topological Framework for Representation Analysis

Topological Data Analysis (TDA) offers a principled, intrinsic lens for comparing neural representations. However, existing paired topological divergences (e.g., RTD) are limited by heuristic asymmetry and, more critically, unbounded scores that depend on sample size, hindering reliable cross-scenario benchmarking. To address these challenges, we develop a unified topological toolkit serving two complementary needs: fine-grained structural diagnosis and robust, standardized evaluation. First, we complete the RTD framework by introducing Symmetric Representation Topology Divergence (SRTD) and its efficient variant SRTD-lite. Beyond resolving the theoretical asymmetry of prior variants, SRTD consolidates diagnostic information into a single, comprehensive cross-barcode signature. This allows for precise localization of structural discrepancies and serves as an effective optimization objective without the overhead of dual directional computations. Second, to enable reliable benchmarking across heterogeneous settings, we propose Normalized Topological Similarity (NTS). By measuring the rank correlation of hierarchical merge orders, NTS yields a scale-invariant metric bounded between -1 and 1, effectively overcoming the scale and sample-dependence of unnormalized divergences. Experiments across synthetic and real-world deep learning settings demonstrate that our toolkit captures functional shifts in CNNs missed by geometric measures and robustly maps LLM genealogy even under distance saturation, offering a rigorous, topology-aware perspective that complements measures like CKA.

Yan Wang, Tianyang Hu

School of Data Science The Chinese University of Hong Kong, Shenzhen

Representational Similarity Analysis Neural Representation Geometry Persistent Homology Topological Data Analysis

Jun 4, 2026q-bio.QM

$p$ -adic Bi-Filtrations for Topological Machine Learning on Genomic Sequences

We introduce pVR, a topological machine learning framework for alignment-free genomic sequence classification that combines

p

-adic numbers with topological data analysis. Each DNA sequence is encoded along two complementary axes: a

p

-adic distance on

k

-mer prefixes, which captures hierarchical positional structure, and a compositional

L_1

distance on

k

-mer frequencies, which captures local sequence content. The two distances jointly parameterise a bi-filtered Vietoris--Rips complex, and per-sequence topological summaries from this bi-filtration serve as features for standard machine learning classifiers. We establish theoretical guarantees for the construction: stability under metric perturbations and invariance to the choice of prime, alongside a result that explains why a single

p

-adic axis is topologically uninformative and why the bi-filtration recovers nontrivial homology. On twelve genomic benchmarks (

28

to

500

sequences,

3

to

7

classes), pVR outperforms four established alignment-free baselines on three of six low-sample datasets, with gains of up to

21

percentage points; it underperforms only on a SARS-CoV-2 variant benchmark whose point-mutation divergence violates the hierarchical assumption, and all methods saturate in the large-sample regime. pVR also outperforms zero-shot frozen embeddings from the 500M-parameter Nucleotide Transformer v2 by

6.7

to

11.4

percentage points on three low-sample benchmarks. The pVR codebase is publicly available at https://github.com/MAHI-Group/pVR.

Tirtharaj Dash, Gunja Sachdeva

Department of CS & IS, BITS Pilani, K K Birla Goa Campus, Zuarinagar, Goa 403726, India. · Department of Mathematics, BITS Pilani, K K Birla Goa Campus, Zuarinagar, Goa 403726, India.

Genomics Persistent Homology Sequence Modeling Topological Data Analysis

Jun 3, 2026cs.CV

Fast Cubical Persistent Homology on 2D and 3D Images via Union-Find, Pruning, and Lookup Tables

We present Flash Cubical, a highly efficient computation of cubical persistence on a V-filtration for 2D and 3D images over

\mathbb{F}_2

. The implementation is built around three core ideas. First, cubical complexes satisfy properties that allow for the computation of persistence of the highest dimension via union-find and duality. Second, pruning of certain edges allows for a fast and efficient implementation of union-find. Third, the use of a lookup table, which exploits the regularity of cubical complexes to pre-compute local information. This avoids the need to compute local information at run time. To the best of our knowledge, this is the most efficient implementation of cubical persistence with a V-filtration, both in terms of time and memory costs. Although the paper focuses on persistence for V-filtration cubical complexes, the underlying ideas generalise naturally to T-filtrations on cubical complexes and suggest promising directions for other complexes.

Titouan Le Breton, Karol Szustakowski, Marie Piraud

1Helmholtz AI, Helmholtz Munich, Neuherberg, Germany · 2École des Ponts ParisTech, ENS Paris-Saclay · Institute of AI for Health, Helmholtz Munich, Neuherberg, Germany

Persistent Homology Topological Data Analysis

May 31, 2026stat.ME

Topological Ignorability for Structural Causal Effects Beyond Means

Many interventions alter the structure of an outcome distribution rather than its mean: they can split a population into disconnected regimes, create loops or holes, generate branches, or reorganize an outcome cloud while leaving the average response nearly unchanged. In such settings, mean-based causal estimands such as the average treatment effect may miss important structural effects. We introduce topological-geometrical causal metrics based on summaries of interventional outcome laws, including density-superlevel Betti summaries, Euler signatures, and persistent-homology summaries. These metrics quantify structural differences between treated and untreated outcome laws beyond averages. We also study the assumptions needed for causal interpretation. We introduce topological ignorability, a topological analogue of conditional ignorability that requires invariance of the chosen structural feature rather than the full counterfactual distribution. When the chosen summary is injective, this condition coincides with weak ignorability; for noninjective summaries, it can identify the structural feature of interest without identifying the full interventional law. We define a covariate-standardized topological-geometrical causal effect and develop practical estimators. We validate the framework in two hidden-confounding benchmarks: a fully synthetic exact benchmark and a real-covariate semi-synthetic benchmark using Wisconsin breast-cancer covariates. In both, weak ignorability fails and balancing observed covariates nearly eliminates standardized mean differences, yet the coordinate-mean average treatment effect remains biased. By contrast, selected finite density-superlevel Betti and Euler contrasts remain stable across oracle, observational, and weighted analyses.

Usef Faghihi

∗Université du Québec à Trois-Rivières (UQTR).

Causal Effect Estimation Causal Identifiability Topological Data Analysis

May 26, 2026cs.LG

Periodic Topological Deep Learning for Polymer Design and Discovery

Polymers underpin applications across energy, healthcare, and materials science, yet their vast chemical space makes systematic discovery challenging. Most machine learning approaches represent polymers as molecular graphs of a single repeating unit, thereby missing both the periodicity of polymer chains and many-body interactions beyond pairwise bonds. We introduce Periodic-TDL, a deep learning framework built on periodic Vietoris-Rips complexes that capture many-body interactions across multiple spatial scales, followed by a hierarchical simplicial message-passing (HSMP) encoder that propagates information from long-range interactions to covalent bonds, yielding representations enriched by higher-order topological features. Periodic-TDL outperforms all state-of-the-art models across polymer property prediction tasks spanning electronic, optical, physical, and thermal targets. Furthermore, we quantitatively validate how ester-to-amide substitution and

α

-methylation enhance thermal stability. Using a computationally synthesized dataset of 48,208 structures-generated via systematic substitution of acrylate and acrylamide polymers-we observed a mean

T_g

increase of

\sim 55^\circ

C for ester-to-amide substitutions and

\sim 14^\circ

C for backbone

α

-methylation across matched polymer pairs. To verify these predicted trends, we use our Periodic-TDL model to analyze six novel polymer pairs from independent experimental measurements, including three newly synthesized polymers previously unreported in the literature. The experimental data successfully confirmed the model's predictions. Ultimately, these findings demonstrate that Periodic-TDL captures the underlying physical effects of specific functional group modifications, rather than merely optimizing predictive performance on benchmark datasets.

Yasharth Yadav, Tze Kwang Gerald Er, Atsushi Goto +1

School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371 · School of Chemistry, Chemical Engineering and Biotechnology (CCEB), Nanyang Technological University, Singapore 637371

Materials Property Prediction Molecular Representation Learning Topological Data Analysis

May 18, 2026eess.SP

Topological Signal Processing: An Application-Oriented Tutorial

Many modern datasets are large and carry complex structural relationships. Graph-based methods have traditionally been used to represent networked data, modeling individual elements as nodes and pairwise interactions as edges. Furthermore, Graph Signal Processing (GSP) has been developed to analyze signals on graph nodes, such as temperature measurements (node signals) across different regions of a country represented as a graph. Topological Signal Processing (TSP) is an emerging field that generalizes GSP, enabling the analysis of signals defined not only on nodes but also on edges, triangles, and higher-dimensional network elements, modeled as simplicial complexes and related topological structures. This makes TSP naturally well-suited for studying higher-order interactions in complex systems by extending classical signal processing concepts, such as filtering and Fourier transforms, to the topological level. Despite its versatility, TSP remains challenging for many practitioners. Therefore, we present an accessible overview of TSP foundations while drawing connections with application-oriented settings. We focus on processing techniques based on the combinatorial Hodge Laplacian, which generalizes the graph Laplacian to simplicial complexes. In particular, we review key TSP concepts, relate them to real-world examples, and discuss how higher-order structures and signals can be derived from datasets. For instance, we introduce an edge-level signal capturing lagged interactions between nodal signals, and demonstrate its use in a case study on TSP-based analysis of brain imaging data, revealing nontrivial interactions between sets of brain regions. Overall, we aim to promote a broader adoption of TSP by bridging methodological developments with applications, fostering its use among a wide community of theoretical and applied researchers.

Flavia Petruso, Maria Giulia Preti, Dimitri Van De Ville

Topological Data Analysis

May 14, 2026cs.LG

ToMAToMP: Robust and Multi-Parameter Topological Clustering

Topological clustering, and its main algorithm ToMATo, is a clustering method from Topological Data Analysis (TDA) which has been applied successfully in several applications during the last few years. This is due to its high versatility, as clusters are detected from the persistent components in the sublevel sets of any user-defined function (gene expression, pixel values, etc), and efficiency, as topological clustering enjoys robustness guarantees. However, ToMATo is also limited in several ways. First, a graph on the data points needs to be provided as a hyper-parameter of the method (whose fine-tuning is left to the user). Second, ToMATo is known to be very sensitive to outlier values in the function range. Finally, and most importantly, ToMATo can only handle one function at a time, whereas it is critical to use several functions in various applications. In this article, we introduce ToMAToMP: the first topological clustering method able to handle several functions at the same time with theoretical guarantees. More specifically, we leverage a recent tool from multi-parameter persistent homology, called MMA decomposition, to design our clustering algorithm, and prove that it enjoys robustness properties. As corollaries, we show that it can be used to make ToMATo independent of graph tuning, and robust to outliers. Finally, we provide a set of numerical experiments showcasing the efficiency and quality of the clusterings produced by ToMAToMP, by showing strong improvement over non-topological and topological baselines for various datasets.

Ludo Andrianirina, Mathieu Carrière

DataShape, Centre Inria d’Université Côte d’Azur, Sophia-Antipolis, France

Clustering Persistent Homology Topological Data Analysis

May 12, 2026cs.LG

scShapeBench: Discovering geometry from high dimensional scRNAseq data

High-dimensional point cloud data arise across many scientific domains, especially single-cell biology. The shapes or topologies of these datasets determine the types of information that can be extracted. For example, clustered data supports cell-type identification, trajectory structures support transition analysis, and archetypal structures capture continua of cellular behaviors. Existing analysis pipelines often assume a specific shape. The standard Seurat pipeline combines UMAP visualization with Louvain clustering and therefore assumes clustered data, while tools such as Monocle and SPADE assume tree-like structures, and flow-based models such as MIOFlow and Conditional Flow Matching target trajectories. Choosing which pipeline to apply is therefore often left to bioinformaticians who visually inspect datasets before selecting an analysis strategy. With the rise of agentic AI scientists, automating shape detection is increasingly important for selecting downstream analysis pipelines. To address this problem, we introduce scShapeBench, a benchmark dataset for shape detection containing both synthetic and expert-annotated single-cell datasets. Synthetic datasets are sampled from ground-truth skeleton graphs with controlled variance. Real single-cell datasets are curated from diverse sources and annotated by experts into four categories: clusters, single trajectory, multi-branching, and archetypal. We additionally introduce scReebTower, a baseline method that uses diffusion geometry to extract Reeb graphs and connect visualization with pipeline selection. We provide topology-aware evaluation metrics and compare scReebTower against PAGA and Mapper on synthetic and real data. Our results indicate that scReebTower outperforms existing baselines. Overall, our contributions span benchmarks, evaluation metrics, and a baseline for automated shape detection in single-cell data.

Andrew J Steindl, João Felipe Rocha, Brian Tshilengi Di Bassinga +13

Yale University · Mila / Université de Montréal · Garvan Institute of Medical Research +2

Single-Cell RNA Sequencing Topological Data Analysis

May 9, 2026cs.CG

Towards Scalable Persistence-Based Topological Optimization

Persistence-based topological optimization deforms a point cloud

X \subset \mathbb{R}^d

by minimizing objectives of the form

L(X) = \ell(\mathrm{Dgm}(X))

, where

\mathrm{Dgm}(X)

is a persistence diagram. In practice, optimization is limited by two coupled issues: persistent homology is typically computed on subsamples, and the resulting topological gradients are highly sparse, with only a few anchor points receiving nonzero updates. Motivated by diffeomorphic interpolation, which extends sparse gradients to smooth ambient vector fields via Reproducing Kernel Hilbert Space (RKHS) interpolation, we propose a more scalable pipeline that improves both subsampling and gradient extension. We introduce subsampling via random slicing, a lightweight scheme that promotes iteration-wise geometric coverage and mitigates density bias. We further replace the costly kernel solve with a fast Nadaraya-Watson (NW) Gaussian convolution, producing a globally defined smooth update field at a fraction of the computational cost, while being more suited for topological optimization tasks. We provide theoretical guarantees for NW smoothing, including anchor approximation bounds and global Lipschitz estimates. Experiments in

2

D and

3

D show that combining random slicing with NW smoothing yields consistent speedups and improved objective values over other baselines on common persistence losses.

Abderrahim Bendahi, Alexandre Duplessis, Arnaud Fickinger

ENS Ulm, PSL Paris, France · UC Berkeley Berkeley, CA, USA

Persistent Homology Topological Data Analysis

May 7, 2026cs.LG

No Triangulation Without Representation: Generalization in Topological Deep Learning

Despite an ever-increasing interest in topological deep learning models that target higher-order datasets, there is no consensus on how to evaluate such models. This is exacerbated by the fact that topological objects permit operations, such as structural refinements, that are not appropriate for graph data. In this work, we extend MANTRA, a benchmark dataset containing manifold triangulations, to a larger class of manifolds with more diverse homeomorphism types. We show that, unlike prior claims, both graph neural networks (GNNs) and higher-order message passing (HOMP) methods can saturate the benchmark. However, we find that this is contingent on the right representation and feature assignment, emphasizing their importance in baseline models. We thus provide a novel evaluation protocol based on representational diversity and triangulation refinement. Surprisingly, we find no indication that existing models are capable of generalizing beyond the combinatorial structure of the data. This points towards a research gap in developing models that understand topological structure independent of scale. Our work thus provides the necessary scaffolding to evaluate future models and enable the development of topology-aware inductive biases.

Johannes S. Schmidt, Martin Carrasco, Ernst Röell +3

Dept. of Math. & Statistics University of Montréal Montréal, Canada · Dept. of Informatics University of Fribourg Fribourg, Switzerland

Neural Network Generalization Graph Neural Networks Message Passing Neural Networks Topological Representation Learning Topological Data Analysis

May 4, 2026cs.LG

Global and Local Topology-Aware Attention with Persistent Homology and Euler Biases for Time-Series Forecasting

Scientific time series often encode predictive geometric structure, including connectivity, cycles, shell-like geometry, directional changes, and nonlinear neighborhoods, that standard dot-product attention does not explicitly represent. We introduce a topology-aware attention framework that adds such structure to attention logits using persistent homology (H0-H2), anchored Euler characteristic transforms, and kernel-Hilbert channels. A validation-gated local residual captures local topological signals, including a Zeng-style local H0 component, only when held-out validation data support the correction. Exact Vietoris-Rips computations and smooth topological surrogates are evaluated under a no-leakage protocol with train-only calibration, validation-only selection, and test-only reporting. We evaluate guarded topology-aware variants across three architecture families: lightweight attention/Ridge, PatchTSTForRegression, and TimeSeriesTransformerForPrediction. Experiments include synthetic benchmarks isolating higher-order topology and real datasets covering CO2, S&P 500 return-window geometry, and NASA IMS bearing degradation. The audit uses matched paired comparisons across seven dataset units, three random seeds, and three chronological splits, giving 63 paired units per architecture and 189 paired units overall. Topology-aware models show positive paired effects when geometry is predictive, with heterogeneous magnitude across datasets and architectures. Lightweight attention/Ridge improves in 46 of 63 units, with mean relative RMSE reduction of 12.5% and paired randomization p=7.2e-4; PatchTST improves in 33 units and retains the baseline in 20 units, with 23.5% reduction and p=3.5e-5; and TimeSeriesTransformer improves in 47 units, with 47.8% reduction and p<1e-4. The results support topology as a validation-selected, architecture-compatible inductive bias.

Usef Faghihi, Amir Saki

Department of Mathematics and computer science · University of Quebec Trois Rivieres

Time Series Forecasting Persistent Homology Topological Data Analysis