Topological and Geometric Data Analysis

Latest papers

• Designing for Interpretation Uncertainty: Architecture and Principles for Topological Learning Analytics Dashboards

  Hitoshi Inoue, Koichi Yasutake · 2 October 2026

  Topological Data Analysis (TDA) offers novel methods for understanding temporal dynamics in complex systems, yet its application in information systems design faces a fundamental challenge: how should systems present analytical outputs when interpretation frameworks are still developing? This paper …

• Participation-Sensitive Convergence and the Fragment First, Converge Later Pattern in Asynchronous Online Learning: A Topological Analysis Across 22 OULAD Courses

  Hitoshi Inoue, Koichi Yasutake · 2 October 2026

  Asynchronous online learning offers temporal flexibility at a structural cost: learning communities tend to fragment rather than cohere. $\beta_0$, the number of disconnected behavioral clusters from Zigzag Persistent Homology, serves as a cohort-level indicator of this structure. Two questions rema…

• Period Segmentation in Transition Network Analysis: A Topological Data Analysis Approach

  Hitoshi Inoue, Koichi Yasutake · 29 September 2026

  Temporal dynamics in learning behavior can be revealed through period segmentation in Transition Network Analysis (TNA). Cristea et al. demonstrated that segmenting courses into halves and quarters reveals how learning strategies evolve and relate to academic performance. Building on this approach, …

• LandscapeSHAP: Which Persistent Homology Class Gets the Credit?

  Nikola Mili\'cevi\'c · 28 September 2026

  Shapley values, a solution concept from cooperative game theory, have recently become a standard tool for feature credit allocation in machine learning. They provide an axiomatically justified method to fairly distribute a model's prediction among the data features. Shapley values have not yet been …

• Persistent Homology of Time Series through Complex Networks

  \.Ismail G\"uzel · 28 September 2026

  We present a unified pipeline for univariate time series classification via complex networks and persistent homology. A time series is mapped to a graph through one of five constructions across three families (visibility (natural and horizontal visibility graphs), transition, and proximity) and the …

• A New Non-archimedean Metric on Persistent Homology

  \.Ismail G\"uzel, Atabey Kaygun · 28 September 2026

  In this article, we define a new non-archimedean metric structure, called cophenetic metric, on persistent homology classes of all degrees. We then show that zeroth persistent homology together with the cophenetic metric and hierarchical clustering algorithms with a number of different metrics do de…

• TopoFuse: Topology-Aware Tri-Planar Fusion for 3D Cryo-Electron Tomography Segmentation

  Rohit Kumar Salla, Neelesh Gupta, Xingjian Li, Min Xu · 25 September 2026

  Automated segmentation of cryo-electron tomograms routinely produces masks that are voxel-accurate but topologically broken: membranes fragment, organelles merge into one another, and enclosed cavities collapse. Existing topology-aware losses reduce these violations but cannot eliminate them, becaus…

• Topological Signal Processing With Unoriented Operators

  Andrea Cavallo, Varun Sarathchandran, Geert Leus, Elvin Isufi · 23 September 2026

  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 orie…

• COMPLEX: A Closed-Form Certified Embedding of Multiparameter Persistence Modules

  Sushovan Majhi, Atish Mitra, \v{Z}iga Virk, Pramita Bagchi · 21 September 2026

  Every multiparameter persistence vectorization we know of carries a one-sided Lipschitz upper bound and nothing below it: without a lower gauge there is no sense in which the features are faithful, and no per-prediction guarantee can be built on them. This paper supplies the missing side. COMPLEX is…

• TOPO-Bench: An Open-Source Topological Mapping Evaluation Framework with Quantifiable Perceptual Aliasing

  Jiaming Wang, Jizhuo Chen, Diwen Liu, Jiaxuan Da, Jiamo Hu, Zhiwei Xue, Linh K\"astner, Harold Soh · 16 September 2026

  Topological mapping offers a compact and robust representation for navigation, but progress in the field is hindered by the lack of standardized evaluation metrics, datasets, and protocols. Existing systems are assessed using different environments and criteria, preventing fair and reproducible comp…

• Positive Topology and Feasible Refinement: Forcing Matrices, Positivity, and Information

  Mirco A. Mannucci, Giovanni Sambin · 15 September 2026

  We develop a conceptual and operational account of Positive Topology starting from a basic relation between points or models and observable properties. From this relation, two complementary structures emerge. The first captures universal refinement and cover: what must hold across all relevant cases…

• Direct Topology Tracking in Continuous Implicit Models

  Guanqun Ma, David Lenz, Kaiyuan Tang, Hanqi Guo, Chaoli Wang, Tom Peterka, Bei Wang · 14 September 2026

  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 constrain…

• Recovering topological information of light by topological learning

  Benquan Wang, Trishita Das, Yuhan Peng, Tatjana Kleine, Shanshan Chang, Jinhui Chen, Nilo Mata-Cervera, Chunyu Li, Kelin Xia, Andrew Forbes, Yijie Shen · 9 September 2026

  The evolution of modern-day communication networks towards optical solutions with enhanced capacity and robustness is driving interest in topological light waves, exploiting their stability against perturbations through a topological invariant, e.g., the skyrmion number. However, detecting the under…

• On the Abundance of Critical Points of the t-SNE Energy

  Nakul Haridas, Ryan Murray · 7 September 2026

  This paper considers the energy landscape of the t-SNE algorithm. While this algorithm has enjoyed broad adoption, the non-convexity of the associated energy has made it difficult to rigorously understand what the algorithm captures in many settings. In particular, a number of well-known numerical e…

• Action abstractions for amortized sampling

  Oussama Boussif, L\'ena N\'ehale Ezzine, Joseph D Viviano, Micha{\l} Koziarski, Moksh Jain, Esmeralda S. Whitammer, Emmanuel Bengio, Rim Assouel, Yoshua Bengio · 3 September 2026

  As trajectories sampled by policies used by reinforcement learning (RL) and generative flow networks (GFlowNets) grow longer, credit assignment and exploration become more challenging, and the long planning horizon hinders mode discovery and generalization. The challenge is particularly pronounced i…

• Sierpi\'nski--Knopp Wasserstein Distance for Persistence Diagrams and Applications to 2-Wasserstein Approximation

  Sebastien Tchitchek, Julien Tierny · 2 September 2026

  This paper introduces the Sierpi\'nski-Knopp (SK) Wasserstein distance, a fast metric between persistence diagrams. The SK-Wasserstein distance, denoted $d_{\mathrm{SK}}$, maps diagram points and their diagonal projections to the unit interval via the Sierpi\'nski-Knopp space-filling curve on the up…

• Topological Steering

  Beno\^it Gu\'erand, Tan Minh Nguyen · 2 September 2026

  With the rapid rise of large language models (LLMs), controlling undesirable model behaviors has become increasingly important. Existing behavioral control methods typically intervene directly in activation or feature space, but such approaches can be sensitive to outliers, distributional shifts, no…

• Filling holes in science draws collective attention, but most higher-order holes remain unexplored

  Jiajie Luo, James A. Evans · 1 September 2026

  Much scientific discovery involves filling holes between ideas and arguments that unleash techno-scientific advance. Representing knowledge as high-dimensional concept embeddings, we use persistent homology to detect holes of increasing order, from gaps between disconnected ideas to higher-order cav…

• TI$^2$PS: A Topology-Informed Inverse Design Framework for Stochastic Multicellular Pattern Formation

  Kenji Komiya, Andrew Kailiang Jin, Ryo Nishikimi, Kunio Kashino · 31 August 2026

  This study proposes a novel framework to estimate parameters for reproducing target multicellular patterns using an agent-based model (ABM). Two major challenges in multicellular ABMs are estimating cell-level parameters (agent-specific variables) and quantitatively evaluating the topological charac…

• Persistent Cross Entropy

  Sijin Yeom, Jae-Hun Jung · 26 August 2026

  Persistent entropy is the Shannon entropy of a persistence-based probability measure defined on a persistence diagram. However, its cross-entropy version is not naturally defined because two persistence diagrams generally have different event spaces. To bridge these event spaces, we combine a simila…

• Beyond Uniform Local Isometry and Topology: FactoMap for Disentangled Representations

  Sohini Gupta, Bahareh Tolooshams · 26 August 2026

  Many disentanglement methods represent generative factors using Euclidean product coordinates, although the underlying factor spaces may wrap, collapse, or have position-dependent geometry. We introduce factor-space structure, combining factor domains, generator-induced identifications, and position…

• Exploring Dowker Homology for Sentence Similarity

  Marius Huber, Juri Opitz · 25 August 2026

  Dowker homology is a topological tool that may be used to analyze the relative position of two point clouds living in a common space. We investigate whether Dowker homology captures sentence similarity information by treating the embeddings of the tokens that constitute a sentence pair as a pair of …

• Learning Topological Features of $\widehat Z$-invariants

  Brandon Robinson, Shimal Harichurn, Fabian Ruehle, Sergei Gukov, Rak-Kyeong Seong, Miranda C. N. Cheng · 20 August 2026

  Machine learning and data analysis techniques have recently emerged as powerful tools for identifying patterns and formulating conjectures in mathematical research, most notably in the field of low-dimensional topology. In this paper, we initiate a systematic approach to handling mathematical data s…

• Look Before You Lift: Visual and Quantitative Diagnostics for Topological Deep Learning

  Mathilde Papillon, Guillermo Bernárdez, Álvaro Ballón Barreiro, Marco Montagna, Rémi Devaux, Antoine Jardin, Nina Miolane · 18 August 2026

  Topological deep learning (TDL) methods rely on lifting raw data into higher-order discrete domains such as simplicial complexes, cell complexes, and hypergraphs. In practice, this lifting step is often treated as a black box: practitioners select a lifting and then tune architectures, with limited …

• Transformer Geometry Observatory TGO-IV: Developmental Topology Observatory

  Kaustubh Kapil, Kishor P. Upla · 12 August 2026

  Transformers have had a profound impact on the world of language processing and computer vision. As efforts to answer the million-dollar question of ``How does a Transformer learn?" have been increasing, existing interpretability studies primarily analyze representations at isolated layers or the ne…