applied algebraic topology

Addressing the challenges of modeling topological structures in high-dimensional time series data (e.g., meteorological and financial) and the limited robustness and interpretability of conventional statistical and machine learning methods, this paper proposes a persistent discrete homology framework. It constructs filtration sequences of graphs from pairwise dependencies—measured via Pearson correlation or mutual information—and pioneers the integration of discrete cubical homology with persistent homology to enable efficient topological feature extraction from graph-structured data. This approach establishes a novel “correlation-driven graph representation” paradigm, circumventing issues associated with high-dimensional embedding and parameter sensitivity. Evaluated on real-world meteorological and financial datasets, the method significantly enhances robustness and interpretability in anomaly detection and periodic pattern identification, outperforming classical topological data analysis (TDA) techniques and supervised learning baselines.

Existing tensor eigenvalue analysis for multimodal data fusion lacks topological interpretability and robustness. Method: Breaking from conventional matrix-analogy paradigms, this work pioneers the integration of algebraic topology—particularly homological invariants such as Betti numbers—into tensor eigenvalue theory, establishing a rigorous mathematical link between eigenvalues and the underlying topological structure of data. We propose the first topological characterization framework unifying tensor algebra and homological computation, and develop a systematic theorem system connecting algebraic spectral properties with homological features. Contribution/Results: On cross-modal fusion tasks, our method improves feature discriminability by 12.7% and enhances noise robustness by 31.5% over baseline methods. This work provides novel theoretical foundations for interpretable AI and introduces a structural-aware modeling paradigm grounded in topology.

This work proposes a pedagogical framework for introducing topological data analysis to students of mathematics and computer science, balancing mathematical rigor with accessibility. Departing from conventional metric-space-based approaches, the framework models data as information-carrying functions and foregrounds the role of the observer along with symmetry constraints. It naturally bridges persistent homology and symmetry-aware modeling in machine learning through group equivariant non-expansive operators (GENEOs). By integrating persistent homology, algebraic topology, and monodromy theory from two-parameter persistence, the approach forms a self-contained instructional system that significantly enhances conceptual clarity and cross-disciplinary applicability, making it well-suited for advanced undergraduate and graduate instruction.

This work addresses the limitations of traditional graph models in capturing non-binary higher-order relationships and the lack of statistical frameworks for random signals in existing topological signal analysis. It establishes, for the first time, a theory of stationarity for random signals defined on simplicial complexes, generalizing classical stationarity by characterizing stationary signals as outputs of white noise passed through topological filters. The paper rigorously defines the topological power spectral density (PSD) and constructs a comprehensive spectral analysis and filtering framework by integrating algebraic topology, Hodge and Dirac theory, and spectral graph methods. Experimental results demonstrate that the proposed notion of topological stationarity significantly enhances signal modeling and processing performance on both synthetic and real-world datasets.

This work addresses anomaly detection in topological signals defined on simplicial complexes, focusing on deviations arising from structural violations—such as breaches of irrotationality or solenoidality constraints imposed by Hodge/Dirac spectral subspaces. We propose a Neyman–Pearson-optimal matched subspace detection framework, the first to support both single-layer (e.g., edge signals) and multi-layer joint topological subspace testing. The method is robust to missing values, admits closed-form performance analysis, and explicitly incorporates higher-order topological priors encoded in the simplicial complex. Leveraging spectral theory of the Hodge Laplacian and Dirac operator, we design an energy-based projection test statistic. Empirical evaluation on real-world data—including foreign exchange arbitrage networks—demonstrates significant gains over graph-signal baselines and precise identification of anomalous transaction flows violating no-arbitrage conditions.

This work addresses the lack of a rigorous mathematical foundation in existing distributed data fabric architectures, which struggle to simultaneously ensure consistency, data provenance, and scalability. We propose the first unified framework integrating category theory, hypergraphs, and geometric analogies from Hurwitz spaces. Data sets, metadata, transformations, and policies are uniformly modeled as hypergraphs, with data treated as objects and transformations as morphisms. Modular tensor categories and braided monoidal structures are introduced to capture relational symmetries. Within this formalism, we rigorously prove the NP-hardness of key tasks and develop a fault-tolerant operational mechanism—leveraging sparse incidence matrices, spectral methods, and symmetry-aware alignment algorithms—that guarantees consistency, completeness, and causality while adhering to the CAP and CAL theorems. This framework provides a scalable mathematical foundation for large-scale federated learning and data integration.

Existing approaches to persistent homology struggle to characterize the evolution, reorganization, and memory effects of homological features in parameter-dependent topological data. This work proposes a unified framework based on the zero modes of the combinatorial Hodge Laplacian to track homological feature evolution within a shared chain space. For the first time, it incorporates curvature and holonomy from differential geometry to capture local reorganization dynamics and cyclically accumulated memory, respectively. By transcending the representational limitations of traditional persistence diagrams, the method successfully identifies instability in feature tracking in time-varying point cloud experiments, distinguishes systems whose persistence diagrams are highly similar, and reveals higher-order cyclic memory structures that pairwise matching approaches fail to detect.

Commercial and economic data often exhibit nonlinear, multiscale structures that linear methods fail to capture effectively. To address this, we introduce topological data analysis (TDA), constructing simplicial complexes via persistent homology and integrating SAX/eSAX symbolic representation with multiscale distance metrics to extract robust topological features. Our key contribution is the Topological Stability Index (TSI), a novel interpretable metric quantifying structural variability and providing actionable insights into systemic fluctuations. We validate the framework across three real-world domains—consumer behavior, stock market dynamics, and foreign exchange time series—demonstrating its reproducible ability to uncover clustering structures and temporal patterns missed by conventional statistical approaches. Results show that TDA substantially enhances pattern discovery in complex business data, while TSI exhibits strong discriminative stability and domain-relevant interpretability.

This study addresses the complexity in point-based constructions arising from varying morphism definitions in modal logic by establishing a Stone-type duality between an algebraic category equipped with paired modal operators and a category of topological spaces endowed with binary relations. By introducing a semi-continuity condition on relations, the work reveals a direct correspondence between modal axioms and relational properties of the underlying spaces, thereby significantly simplifying point-set manipulations in traditional dualities. This approach not only unifies several existing dualities between modal frameworks and topological semantics but also provides a precise bridge between algebraic and relational semantics, offering new categorical tools for the systematic study of modal logics.

This work addresses the joint modeling of causal relationships and nonlinear dynamic dependencies among nodes, along with topology inference, in dynamic graph scenarios—such as brain networks, transportation systems, and financial markets—where the underlying graph structure is unknown. To overcome the limitations of conventional linear time-invariant models in capturing time-varying, nonlinear, and directed dependencies, we propose a unified framework based on kernel dictionary selection. The framework seamlessly integrates structural priors including sparsity, acyclicity, low-rankness, and graph smoothness, supporting both batch and online learning, and naturally extending to tensor representations. It unifies covariance selection, structural equation modeling, nonlinear vector autoregression, kernelized modeling, tensor decomposition, and convex optimization. Theoretically guaranteed convergence is established. Experiments demonstrate significant improvements in leveraging higher-order statistical information, enabling high-accuracy and interpretable inference of dynamic graph topologies.