Persistent Cross Entropy

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 similarity function with persistence weighting to define an induced probability. The induced probability reflects information from one diagram on the event space of the other diagram and assigns unexplained probability mass to the unexplained event. Using the induced probability, we extend cross entropy to persistence diagrams, called persistent cross entropy (PCE). We establish the main properties of both the induced probability and PCE and prove stability theorems for both. Through three numerical studies, we show that PCE distinguishes diagrams with the same persistent entropy, separates causal directions in dynamical systems without constructing a joint persistent diagram, and can be used as a directional topology loss for knowledge distillation.

Magnitude Homology Is the Associated Graded of the Length Filtration

Magnitude homology is graded by length and knows nothing of persistence. Its persistent refinement knows nothing of where its bars begin and end. We show that the two are one construction: filtering the length nerve by sublevel sets of the length yields the persistence module, and the associated graded of that filtration is the magnitude complex. A long exact sequence exchanges them, and each side gains what it lacked. Magnitude homology locates the critical values of the barcode, so a graded computation lists the lengths at which an endpoint can occur, and the barcode acquires a stability estimate of $(n+1)δ$ in degree $n$ under a perturbation of size $δ$, while a computed perturbation moves a barcode by more than $δ$, so the factor cannot be dropped. We apply this to quantitative equational theories, whose free algebras are metric spaces built from syntax: an inclusion of theories induces a morphism of the presenting monads and a comparison of barcodes with an explicit bound, so the invariant measures axiomatic strength. Four examples are computed, one in every degree.

Transformer Geometry Observatory TGO-IV: Developmental Topology Observatory

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 network as a whole, while the developmental evolution of individual representations and its manifolds across transformer layers remains underexplored. With this work, we aim at providing a comprehensive analysis of the evolution of representations as the representation point cloud transforms across the layers; thereby attempting to isolate layers or establish a trend which comes closer to justifying how and when raw input representations evolve into task-relevant feature representations. Thus, Transformer Geometry Observatory-TGO-IV introduces a topological framework for analysing the evolution of Transformer representations through the lens of Persistent Homology. Rather than studying local geometric properties alone, TGO-IV constructs Vietoris--Rips simplicial complexes from token-level representation point clouds and investigates the evolution of their persistent topological signatures across Transformer layers. The proposed framework comprises complementary topological observatories including Persistence Diagrams, Barcode Diagrams, Betti Curves, Persistence Landscapes, Bottleneck Distance, and Wasserstein Distance, enabling a comprehensive analysis of how the global topology of representation point clouds develops throughout the forward pass.

Topological Simplification in Predictive Coding Networks

We study the topology of learned representations in predictive coding networks (PCNs), a neuro-inspired bidirectional architecture, using a quantitative layer-wise persistent homology analysis. We train well-performing PCNs on a synthetic classification dataset ($\geq 99.9\%$ test accuracy) and on MNIST ($\geq 95\%$ test accuracy), and measure how topological features change across layers for different architectures and activation functions. We find that smaller PCNs collapse connected components across layers earlier than larger models (Spearman $\unicode{x1D70C} \in [0.72, 0.79]$ across activations), with model size measured as the sum of hidden-layer widths. We also observe a strong negative correlation ($\unicode{x1D70C} = -0.58$) between the depth at which simplification occurs and reconstruction error; i.e., architectures that simplify later reconstruct better. Finally, a seed-level bootstrap comparison across architectures and activations shows that PCNs consistently collapse connected components later than matched MLPs, with an average difference of $3.6$ layers. These results suggest that persistent homology offers a useful quantitative lens on the compression--reconstruction tradeoff in PCNs, and that both model capacity and the recurrent, bidirectional dynamics of predictive coding inference shape when this tradeoff is resolved across layers.

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.

Calibrated Alzheimer's Conversion Risk in Mild Cognitive Impairment: Persistent Homology of Clinical Trajectories with Conformal Guarantees

Background. Predicting conversion from mild cognitive impairment (MCI) to Alzheimer's disease (AD) is central to trial enrichment and care planning, yet existing models provide no individual-level uncertainty estimates and rarely include transparent leakage audits. We introduce the first application of persistent homology to longitudinal clinical trajectory point clouds for this task, and the first split-conformal individual risk guarantee for any AD-conversion model. Methods. We analysed 741 MCI subjects (240 converters, 32.4%) from ADNI with a uniform 4-year follow-up cap. Five leakage sources were corrected; without them a naive pipeline achieved AUC=0.934, inflated by +0.075. Vietoris-Rips persistent homology and sublevel-set proxies were combined with trajectory slopes and engineered features (76 total) in a stacking ensemble evaluated by 5-fold cross-validation. Results. Cox and Random Survival Forest models with TDA features achieved concordance C=0.799 and C=0.826 versus C=0.753 and C=0.812 without (+0.045 and +0.014). The primary nested AUC is 0.840 (same-fold bound 0.866); external AUC was 0.879 on a zero-overlap ADNI-2/GO/3 cohort. H0 persistence entropy was the top SHAP feature and significantly associated with APOE4 dosage (Spearman r=-0.191, p<0.0001, Bonferroni-corrected). Cross-conformal coverage was 90.4%+-2.2% (target 90%); empirical external coverage 96.9%. Maximum fairness gap in false-negative rate across seven subgroups was 0.092. Conclusions. We propose H0 persistence entropy as a topological biomarker of cognitive decline and demonstrate that a leakage-audited, conformally calibrated pipeline reaches competitive accuracy with individual-level uncertainty quantification not previously available for this task.

Discrete Ricci Curvature on Protein Contact Graphs for Lightweight Fold Classification

Protein fold classification can be approached via sequence-based representations or structural descriptors, but direct comparisons between lightweight handcrafted descriptors and pretrained protein language model embeddings remain limited. We investigate discrete Ricci curvature on Calpha contact graphs as a lightweight structural descriptor for fold classification. Each protein domain is represented by a 22-dimensional fixed-length feature derived from summary statistics and quantiles of Ollivier-Ricci and Forman-Ricci edge curvature distributions. We evaluate on CATH top-10 Topology classification and on the ASTRAL 40%-identity SCOPe top-10 Fold benchmark, comparing against geometry, contact-graph statistics, persistent homology, and mean-pooled ESM-2 (150M) baselines. On both datasets, lightweight structural descriptors substantially outperform mean-pooled ESM-2 embeddings, with a larger performance gap on the ASTRAL 40% SCOPe benchmark. Ricci alone uses 22 dimensions, or 3.4% of the ESM-2 baseline dimensionality, and already outperforms mean-pooled ESM-2 on both datasets. Combining Ricci with persistent homology yields the strongest performance, achieving macro-F1 of 0.71 on CATH and 0.68 on SCOPe with a 112-dimensional feature vector. These results identify a regime where lightweight interpretable graph descriptors offer a practical alternative to pretrained protein language model embeddings.

Information-Geometric Superposed Vowel Evaluation: Part 1. Moraic Syllabary (Japanese)

This paper explains the principles and provides examples of a new method for distinguishing between FAKE human speech synthesized by generative AI and natural speech. Since synthetic speech is generated based on information from a limited set of training spectra, the variety of vowels - which are key to identifying individuals - is limited. In contrast, natural speech exhibits a more diverse distribution of vowel spectra due to the flexibility of the human articulatory organ. In this paper, using Japanese - a Syllabary limited to five vowel phonemes, each of which corresponds one-to-one with a specific sound - as an example, we outline a method for distinguishing between synthetic and natural speech reading the same text by analyzing the spectral distributions. If we normalize the spectra of speech sounds and regard them as probability density functions for the frequency bands received by the hair cells of the human cochlea, and evaluate the distance between spectra using the Wasserstein metric, the Wasserstein distances between the vowels of synthetic speech are short. By preserving this distance and performing a topological mapping using persistent homology, the spectral probability density functions of synthetic and natural speech can be decomposed into clusters.

Complexity of Normalized Persistence Problems for Topological Data Analysis and Local Hamiltonians

Topological data analysis (TDA) is a machine learning technique that uses topology to extract patterns from data and has shown the potential to exhibit quantum advantage. A key concept in TDA is persistent homology, which measures the robustness of topological information at different lengthscales. In this paper, we introduce and study the problem of normalized persistence, a practically motivated and easily interpretable version of persistent homology that counts the fraction of holes that persist at different lengthscales. We prove that a variant of normalized persistence is $\mathsf{DQC}_1$-hard and contained in $\mathsf{BQP}$, giving evidence of an exponential quantum speedup for TDA under the standard assumption that $\mathsf{DQC}_1 \not\subseteq \mathsf{BPP}$. These are the first $\mathsf{DQC}_1$-hardness results that are directly applicable to TDA instances. We also find a close connection between normalized persistence and the complexity of estimating spectral quantities in the low-energy subspace of local Hamiltonians. We study a family of such problems, including a low-energy normalized subtrace and spectral density. We show that these are $\mathsf{DQC}_1$-hard for $O(1)$-local Hamiltonians, strengthening previous results that required log-local interactions. We also introduce a variant of $\mathsf{DQC}_1$ with perfect completeness ($\mathsf{SDQC}_1$) to characterize the hardness of problems normalized by an exact kernel. This includes normalized persistence for $O(1)$-local Hamiltonians, which we show is $\mathsf{SDQC}_1$-hard.

Comparing Chatbot Performance Enhanced with Persistent Homology

Chatbots have become increasingly prevalent across various domains, offering automated assistance in many areas, especially mental health support. The training is done using extremely large datasets, which are sometimes not available in very specific domains. Moreover, it would sometimes be ideal to train the chatbot with personal information about the patients, which, of course, cannot be done on shared servers since it would violate patient confidentiality. Hence, being able to improve the performance of a chatbot, possibly trained locally and on a restricted dataset, without having to increase the dataset itself, would be extremely beneficial. In this work, we will enhance the input datasets using persistent homology (PH) vectorizations computed from the raw datasets themselves. Then we will compare, across several metrics, the performance of multiple chatbot models with or without the PH enhancement. Our experiments suggest that, while at times the PH enhancement is not particularly beneficial, it sometimes brings remarkable advantages for virtually no cost.

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.

Learning Topology-Aware Representations via Test-Time Adaptation for Anomaly Segmentation

Test-time adaptation (TTA) has emerged as a promising paradigm for mitigating distribution shifts in deep models. However, existing TTA approaches for anomaly segmentation remain limited by their reliance on pixel-level heuristics, such as confidence thresholding or entropy minimisation, which fail to preserve structural consistency under noise and texture variation. Moreover, they typically treat anomaly maps as flat intensity fields, ignoring the higher-order spatial relationships that characterise complex defect geometries. We introduce TopoTTA (Topological Test-Time Adaptation), a novel framework that integrates persistent homology, a tool from topological data analysis, into the TTA pipeline to enforce geometric and structural coherence during adaptation. By applying multi-level cubical complex filtration to anomaly score maps, TopoTTA derives robust topological pseudo-labels that guide a lightweight test-time classifier, enhancing segmentation quality without retraining the backbone model. The approach avoids reliance on method-specific raw-score thresholding for mask binarisation, preserves connectivity, and generalises across both 2D and 3D modalities. Extensive experiments across six standard benchmarks (MVTec AD, VisA, Real-IAD, MVTec 3D-AD, AnomalyShapeNet, and MVTec LOCO) demonstrate an average 15% F1 improvement over state-of-the-art unsupervised anomaly detection and segmentation methods, with the largest gains on anomalies exhibiting complex geometric or structural variations. These findings suggest that integrating topological reasoning into test-time adaptation provides a principled route to structure-aware generalisation, bridging the gap between geometric learning and robust adaptation.

CANNs: A Toolkit for Research on Continuous Attractor Neural Networks

Continuous attractor neural networks (CANNs) are the canonical computational framework for how the brain encodes continuous variables such as spatial position, head direction, and movement direction, and explain the activity of hippocampal place cells, entorhinal grid cells, and head-direction cells. CANN research, however, is fragmented: most results rest on lab-specific implementations, general-purpose simulators lack CANN-specific abstractions, and the path from spike trains to attractor geometry in real recordings lacks a standardized toolkit. Here, we present a comprehensive open-source toolkit that unifies the full CANN research workflow. It combines three tightly integrated components: 1) canns, a Python library on BrainPy/JAX that provides standardized 1D/2D CANNs, spike-frequency-adaptation variants, grid cell networks, hierarchical path-integration models, and brain-inspired attractor architectures, together with curated datasets, task generators, an analyzer module and trainer modules for biologically plausible plasticity; 2) canns-lib, a Rust acceleration backend delivering hundreds-of-times speedups for spatial-navigation workloads and modest gains for Ripser-based persistent homology; 3) ASA (Attractor Structure Analyzer), a PySide6 pipeline applying persistent homology and cohomology to experimental neural recordings to detect ring-like and toroidal attractor signatures in real data. The toolkit ships with full-detail reproducible pipelines that recover recent CANN results including SFA-driven anticipative tracking, theta sweeps in head-direction/place/grid systems, and hierarchical path integration.

TopoCast: A Topological Fidelity Framework for Evaluating Transformer-Based Time Series Forecasting

Deep learning-based models have achieved state-of-the-art performance in Time Series Forecasting (TSF), yet their evaluation remains dominated by pointwise error metrics such as Mean Squared Error (MSE), which quantify numerical accuracy but overlook structural properties of the forecast signal, including recurrent dynamics, oscillatory behavior, and phase alignment. As a result, forecasts exhibiting over-smoothing, phase shifts, or frequency distortions may achieve favorable error scores despite substantial structural degradation. To address this limitation, we propose TopoCast, a topology-driven framework for evaluating structural fidelity in TSF. TopoCast reconstructs phase-space representations of forecast and ground-truth sequences using Takens delay embedding and applies persistent homology to characterize their intrinsic dynamics. We derive four complementary topological fidelity measures from persistence diagrams and aggregate them into a Topological Fidelity Score (TFS). We further introduce dominant cycle overlap, a novel metric that maps persistent topological features to the temporal domain to assess whether dominant oscillatory patterns occur at the correct time points. Combined with TFS, this yields the Localized Topological Fidelity Score (LTFS), a phase-aware measure that captures temporal localization errors invisible to existing evaluation metrics. Experiments on five Transformer architectures across three real-world benchmark datasets demonstrate that models with similar forecasting errors can exhibit markedly different structural fidelity profiles, revealing failure modes overlooked by conventional evaluation and highlighting the value of topology-aware forecast assessment.

Tracking Representation Dynamics in Large Language Models with Persistent Homology

Large language models are commonly aligned through supervised fine-tuning, yet little is known about how their internal representations evolve during this process. We study alignment dynamics using persistent homology by tracking the topology of activation spaces throughout fine-tuning. Across four transformer language models ranging from 1B to 7B parameters and three alignment objectives corresponding to helpful, harmless, and mixed training data, we find that the majority of topological reorganization occurs during the earliest stages of training. A dense checkpoint analysis reveals a transient peak in topological activity followed by rapid stabilization. We further show that different alignment objectives induce distinguishable topological trajectories, while instruction-tuned and pretrained models exhibit qualitatively different patterns of evolution. Our results suggest that persistent homology provides a complementary perspective on alignment, revealing representation-level changes that are not apparent from behavioral metrics alone.

Non-negative Matrix Factorisation with Topological Regularisation

We investigate the learning of interpretable bases in non-negative matrix factorisation (NMF) by regularising the topology of the learned basis functions. Our approach is motivated by the observation that many data modalities can be viewed as non-negative functions on a structured domain, where the quality of a basis is intrinsically linked to its topology. However, naive methods for incorporating the topology of the support are often hindered by discreteness and threshold dependence, rendering them unsuitable for continuous optimisation. We address these challenges by employing persistent homology as a stable, threshold-free topological quantifier and by designing topological scores that integrate into the NMF objective as regularisers. The resulting framework encompasses spatially coherent image components, periodic time-series structures, and clique-like graph signals within a unified modelling language.

PHINN: Persistent Homology Inspired Neural Network for Rare-Event Time Series Generation

Rare events in time series are critical to model but hard to learn due to data scarcity. Current generative models struggle with extreme values. We observe that rare events leave distinct topological fingerprints - transitions in Betti numbers from point-cloud embeddings - that are more stable and discriminative than statistical moments. We introduce PHINN, a flow-matching framework using dynamic Betti curves as conditioning signals and a persistence landscape loss for homology consistency. It scales to multivariate data, includes a natural-language interface to set Betti targets, supports cross-domain meta-learning and few-shot generation, and provides certified adversarial robustness. On financial, epidemiological, and multi-modal benchmarks, PHINN outperforms statistical and diffusion baselines in topological fidelity (beta-RMSE down 41-63%, transition accuracy up 84%) and matches jump-diffusion models in tail coverage while exceeding them in shape fidelity. All results have 95% confidence intervals.

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.

A Closed-Form Persistence-Landmark Pipeline for Certified Point-Cloud and Graph Classification

We introduce PLACE (Persistence-Landmark Analytic Classification Engine), a closed-form pipeline for classifying point clouds and graphs through their persistent-homology signatures. Three quantitative guarantees -- a margin-based excess-risk rate, a closed-form descriptor-selection rule, and a per-prediction certificate -- are derived from training labels alone, with no learned weights or held-out calibration. The embedding sums Mitra-Virk single-point coordinate functions over a sparse landmark grid; closed-form weights maximize a structural distortion constant $λ(ν)$ (a Lipschitz lower bound on $\mathcal{D}_n$ under non-interference). (i) An $O(kR/(Δ\sqrt{m_{\min}}))$ margin bound, driven by class-mean separation $Δ$ and embedding radius $R$, matched by a sample-starved minimax lower bound. (ii) The Mahalanobis margin under Ledoit-Wolf-shrunk covariance is the strongest closed-form descriptor selector on a heterogeneous 64-descriptor chemical-graph pool (mean Spearman $ρ\approx +0.54$ across 10 benchmarks, positive on 9 of 10); the isotropic surrogate $Δ/\sqrt\ell$ admits a closed-form selection-consistency rate on homogeneous (14-15 descriptor) protein/social pools. (iii) A training-time-decided certificate with no per-prediction overhead, in non-asymptotic Pinelis and asymptotic Gaussian plug-in forms. Empirically, PLACE is the strongest diagram-based method on Orbit5k and matches the strongest topology-based baseline within statistical noise on MUTAG and COX2. The remaining gaps fall into two diagnosable regimes: descriptor blindness on NCI1/NCI109, and pool-coverage limits elsewhere. Both radii exceed the firing threshold $\hatΔ/2$ on every benchmark at our training-set sizes, dominated by the $\sqrt\ell$ scaling of the multivariate-norm bound; the per-prediction certificate is constructive but not yet operational at these sizes.

Topology-Aware Representation Alignment for Semi-Supervised Vision-Language Learning

Vision-language models have shown strong performance, but they often generalize poorly to specialized domains. While semi-supervised vision-language learning mitigates this limitation by leveraging a small set of labeled image-text pairs together with abundant unlabeled images, existing methods remain fundamentally pairwise and fail to model the global structure of multimodal representation manifolds. Existing topology-based alignment methods rely on persistence diagram matching, which neither guarantees geometric alignment nor utilizes the image-text pairing information central to vision-language learning. We propose Topology-Aware Multimodal Representation Alignment (ToMA), a framework that uses persistent homology to identify topologically salient edges and aligns them across modalities through available cross-modal correspondences. ToMA leverages both H_0-death edges and lightweight H_1-birth edges, allowing it to capture both connectivity and cycle structure without constructing 2-simplices. Experiments show that ToMA yields stable gains, with clear improvements on remote sensing and modest but consistent benefits on fashion retrieval. Additional analysis shows that ToMA is more stable than alternative topology-based objectives and that lightweight H_1-birth edges provide useful higher-order structural signals.