Topological Data Analysis

Topological neural networks have emerged as effective tools for modeling higher-order relational structures beyond pairwise graphs, including hypergraphs, simplicial complexes, and cell complexes. However, existing Weisfeiler-Leman type expressivity analyses are typically developed on different structural domains and rely on domain-specific neighborhood systems, making their expressive powers difficult to compare within a common formalism. In this paper, we introduce the Combinatorial Complex Weisfeiler-Leman (CCWL) framework, a unified expressive power refinement defined on combinatorial complexes. By exploiting the ability of combinatorial complexes to represent both set-type relations and part-whole hierarchies, CCWL performs topological color refinement through four structural neighborhoods: boundary, co-boundary, lower adjacency, and upper adjacency. We show that, under specified lifting maps, CCWL can simulate several domain-specific WL-type refinements, thereby providing a common theoretical baseline for analyzing topological message passing. We further study the neighborhood sufficiency problem and prove that, under explicit coverage conditions, a reduced refinement using only lower- and upper-adjacent bridge information preserves the distinguishing power of the full four-neighborhood CCWL refinement. Guided by this theoretical result, we instantiate the reduced refinement as the Combinatorial Complex Isomorphism Network (CCIN). Experiments on synthetic and real-world benchmarks demonstrate that CCIN achieves competitive performance against representative graph and topological neural network baselines. Ablation studies and resource-efficiency analyses further support the effectiveness of the proposed lower/upper-neighborhood design.

Representational collapse, where embeddings become anisotropic and lose multi-scale structure, can erode downstream performance long before performance metrics react. We propose an online, topology-aware monitor for evolving neural representations that couples Modular Morse Homology Maintenance (MMHM) with a composite Collapse Index (CI). Instead of rebuilding complexes each epoch, we apply sparse edits at a fixed scale and maintain a discrete Morse matching, yielding fast, incremental updates. Across LLM fine-tuning and temporal KGE training, CI provides a low-latency early-warning signal suitable for in-training interventions. Code and experimental scripts will be released publicly

Dimensionality reduction methods such as UMAP and t-SNE are central tools for visualising high-dimensional data, but their local-neighborhood objectives can preserve sampling noise while distorting global topology. We show that standard local metrics reward this noise memorisation: top-performing embeddings invent cycles and disconnected islands absent from the data. We introduce a topology-faithfulness benchmark based on noisy manifolds with known homology, tune DiRe against it, and find Pareto-optimal configurations that match or beat GPU-accelerated UMAP on classification while recovering exact first Betti numbers on stress tests. On 723K arXiv paper embeddings, DiRe preserves 3-4 times more topological structure than UMAP at comparable wall-clock.

Recent advances in pretraining 3D point cloud encoders (e.g., Point-BERT, Point-MAE) have produced powerful models, whose abilities are typically evaluated on geometric or semantic tasks. At the same time, topological descriptors have been shown to provide informative summaries of a shape's multiscale structure. In this paper we pose the question whether topological information can be derived from features produced by 3D encoders. To address this question, we first introduce DONUT, a synthetic benchmark with controlled topological complexity, and propose FILTR (Filtration Transformer), a learnable framework to predict persistence diagrams directly from frozen encoders. FILTR adapts a transformer decoder to treat diagram generation as a set prediction task. Our analysis on DONUT reveals that existing encoders retain only limited global topological signals, yet FILTR successfully leverages information produced by these encoders to approximate persistence diagrams. Our approach enables, for the first time, data-driven extraction of persistence diagrams from raw point clouds through an efficient learnable feed-forward mechanism.

Persistent homology, a method from topological data analysis, extracts robust, multi-scale features from data. It produces stable representations of time series by applying varying thresholds to their values (a process known as a \textit{filtration}). We develop novel filtrations for time series and introduce topological methods for the analysis of eye-tracking data, by interpreting fixation sequences as time series, and constructing ``hybrid models'' that combine topological features with traditional statistical features. We empirically evaluate our method by applying it to the task of dyslexia detection from eye-tracking-while-reading data using the Copenhagen Corpus, which contains scanpaths from dyslexic and non-dyslexic L1 and L2 readers. Our hybrid models outperform existing approaches that rely solely on traditional features, showing that persistent homology captures complementary information encoded in fixation sequences. The strength of these topological features is further underscored by their achieving performance comparable to established baseline methods. Importantly, our proposed filtrations outperform existing ones.

Persistent homology (PH) encodes global information, such as cycles, and is thus increasingly integrated into graph neural networks (GNNs). PH methods in GNNs typically traverse an increasing sequence of subgraphs. In this work, we first expose limitations of this inclusion procedure. To remedy these shortcomings, we analyze contractions as a principled topological operation, in particular, for graph representation learning. We study the persistence of contraction sequences, which we call Contraction Homology (CH). We establish that forward PH and CH differ in expressivity. We then introduce Hourglass Persistence, a class of topological descriptors that interleave a sequence of inclusions and contractions to boost expressivity, learnability, and stability. We also study related families parametrized by two paradigms. We also discuss how our framework extends to simplicial and cellular networks. We further design efficient algorithms that are pluggable into end-to-end differentiable GNN pipelines, enabling consistent empirical improvements over many PH methods across standard real-world graph datasets. Code is available at this https URL.

The spectral kernel field equation R[k] = T[k] lacks a conservation-law analog. We prove (i) the fixed-point flow is strictly volume-expanding (tr DF > 0), precluding automatic conservation, and (ii) the conservation deficit per mode equals the Hessian stability margin exactly: D_m = -Delta'. Closing the deficit requires a scene-side compensating contribution, which we formalise as the distinction dynamics equation dc/dt = G[c, h_t], with MaxCal-optimal realisation G_opt. On fixed-topology 3D surface graphs we derive a conditional topology-preserving compression theorem: retaining k >= beta_0 + beta_1 modes (under a spectral-ordering assumption) preserves all Betti-number charges; we include a worked short-cycle counterexample (figure-eight) calibrating when the assumption fails. A triple necessary spectral diagnostic -- Fiedler-mode concentration, elevated curl energy, anomalous beta_1 -- is derived for planetary drainage networks at O(N) cost. Two internal real-data sequences serve as preliminary consistency checks; full benchmarks and adaptive-topology extensions are deferred.

Persistent entropy is a scalar summary of persistence barcodes widely used to detect regime changes, yet there is no account of when a structural change in a barcode must produce a detectable change in entropy. We establish a model-agnostic theorem supplying such conditions. Treating persistence diagrams as random objects indexed by a control parameter, we identify a dispersion-condensation mechanism in the normalized persistence weights and derive an explicit lower bound on the entropy difference between the two regimes, valid with high probability at finite sample size and insensitive to the absolute scale of bar lifetimes. We also give a procedure for verifying the hypotheses on empirical barcodes. Applied to convolutional networks, the criterion shows that the circular organization of learned filters reported by Gabrielsson and Carlsson emerges through a sharp topological phase transition, and locates its onset: within a few hundred iterations on MNIST, but an order of magnitude later on CIFAR-10. The same criterion detects the Kuramoto synchronization and Vicsek order-disorder transitions.

Ball Mapper summarizes a finite metric dataset by covering the sample with closed balls centered at selected landmarks and connecting landmarks whose balls share observations. Its construction therefore depends critically on repeated fixed radius range queries, yet the effect of replacing exact queries by approximate search has not been systematically characterized. We formulate Ball Mapper through an abstract range query procedure that separates the mathematical construction from the search backend used to realize it. Under fixed ordering, exact procedures preserve the landmark sequence, cover, graph, and membership-based colorings. For approximate procedures, we derive deterministic bounds on covering radius and landmark separation under additive and multiplicative query errors, prove inclusions for the induced nerve, characterize edge survival through witness redundancy for conservative approximations, and bound perturbations of mean vertex colorings. The accompanying implementation provides independent exact reference backends together with exhaustive and approximate search methods under a common closed ball convention. Experiments on Gaussian, mixture, and noisy curve data across three seeds show that approximation fidelity depends strongly on geometry and that edges supported by multiple witnesses are substantially more robust to missed memberships. At 20,000 observations, the approximate indexes did not outperform exhaustive FAISS Flat search. The results therefore establish a framework for controlled approximation rather than a universal speed advantage, and identify the geometric and combinatorial quantities that govern when approximate range search preserves the Ball Mapper summary.

We introduce a data-analysis framework based on filtrations of finite topological spaces. Starting from a finite metric data set, we construct a sequence of coarsening topologies on the same set of points. These topologies give persistence modules and barcodes in the usual way, but they also retain information that is lost when the filtration is reduced to homology. At each level one can examine, for example, which points are topologically indistinguishable, how their minimal neighbourhoods overlap, how connected components merge, and how these features change from one level to the next. We develop the basic theory of these filtrations, establish stability results under suitable hypotheses, and give practical constructions starting directly from a distance matrix. We then study what can be learned from the resulting finite topologies. On synthetic data with known clusters of different shapes, sizes, and densities, we examine how these regions appear among the finite-topological structures and how they merge as the topology coarsens. We also study what happens when points that become uncovered early in the construction are removed and the analysis is repeated. For one-dimensional homology, we use paths in the finite-topological structure to locate cycles and to examine how their appearance is related to the geometry of the data. We finally apply these ideas to two real data sets with quite different structures. On the Paul15 single-cell data, we use the evolving finite topology to examine fine cellular states, their overlaps and relations, their assembly into larger groups, and the effect of removing points that connect these structures. On COIL20, where images of an object are sampled through a full rotation, we study how the cyclic organization of the images is reflected in the finite-topological evolution and in the associated one-dimensional homology.

We propose persistent discrete homology as a tool for topological data analysis and discuss its advantages over the existing methods. In particular, we provide empirical evidence that persistent discrete homology is more noise-resistant than persistent homology of the Vietoris-Rips complex for data coming from non-metric settings.

In artificial-intelligence-aided signal processing, existing deep learning models often exhibit a black-box structure. Here, conceptually beyond spectral analysis, we demonstrate that topological methods not only effectively capture intrinsic and complex structural information but can also enhance neural networks. We provide a transparent methodology, TopCap, to capture topological features inherent in time series for basic machine learning. Compared to prior approaches, we obtain descriptors that probe finer information such as the vibration of a time series. Notably, in classifying voiced and voiceless consonants, TopCap achieves an accuracy consistently standing in comparison with neural network models. Moreover, by integrating TopCap features into those neural networks, our approach improves upon state-of-the-art methods in terms of robustness against noise, as well as accuracy, stability, convergence of loss function, and interpretability.