Roser Homs, Olga Kuznetsova, Bernadette J. Stolzstat.ML cs.LG math.AG math.ST
Gaussian graphical models (GGMs) are essential tools for interpretable structure learning. However, in high-dimensional, small-sample regimes, the available data is often insufficient for the maximum likelihood estimator to exist. Colored Gaussian graphical models (CGGMs) mitigate this limitation by imposing symmetry constraints through graph coloring, which reduces the required sample size. This minimal number of observations needed to guarantee that the estimator exists almost surely is defined as the maximum likelihood threshold (MLT). Here, we address the computation of the MLT for CGGMs by focusing on its geometric formulation: finding the minimum rank of a sample covariance matrix such that its projection lies almost surely within the interior of the cone of sufficient statistics. We establish a unified theoretical framework, extending results from uncolored to colored models and introducing new symbolic algorithms. Furthermore, we present a computational study integrating sampling with topological data analysis (TDA) to investigate the local geometry of the cone of sufficient statistics. Our results demonstrate the potential of TDA to overcome the computational bottlenecks of traditional symbolic algebraic methods, particularly Groebner basis computations, in analyzing the likelihood geometry of CGGMs.
This paper introduces the Sierpiński-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 Sierpiński-Knopp space-filling curve on the upper diagonal triangle. The encoded point sets are then efficiently matched via one-dimensional optimal assignment, in \(O(N\log N)\) steps, yielding an explicit diagonal-aware point assignment between the two input persistence diagrams. We show that the SK-Wasserstein distance controls the classical \(2\)-Wasserstein distance between diagrams, admits an explicit isometric embedding into a Hilbert space, and induces a positive-definite Gaussian kernel, making the resulting geometry directly compatible with Euclidean and kernel-based learning methods. A tighter surrogate dissimilarity, noted \(W_Γ\), is also introduced based on the point assignments along the curve. Experiments on 12 scientific collections comprising 227 diagrams show median per-collection speedup of \(d_{\mathrm{SK}}\) over state-of-the-art approximations of \(W_2\) is \(626\times\), while the aggregate speedup over the full benchmark is \(2100\times\). Average-linkage partitions obtained from \(d_{\mathrm{SK}}\) and \(W_Γ\) each exactly match the corresponding \(W_2\) partition on 8 of the 12 collections. Hilbert \(k\)-means and Gaussian spectral clustering, both based on \(d_{\mathrm{SK}}\), achieve mean adjusted Rand indices (ARI) of \(0.756\) and \(0.800\), respectively, with respect to the benchmark reference partitions, compared to \(0.750\) obtained by average linkage on \(W_2\). The Gaussian \(d_{\mathrm{SK}}\) kernel supports other kernel-based analysis tasks, as illustrated by its use for contiguous segmentation of ordered diagram collections in our experiments.
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, noise, and other local perturbations. Motivated by Topological Data Analysis (TDA), which captures global rather than purely local structure, we propose Topological Steering, a new framework for steering LLM behavior through the topological representation of activation spaces. Using persistence diagrams, our method connects activation-based steering with TDA and enables more robust behavioral control. We show that Topological Steering consistently modifies LLM behavior across multiple model families and model sizes.
Deep networks trained on noisy labels simultaneously generalize on clean data and memorize flipped labels. These are usually conflated as pressures on one capacity. We present Topo^2, a measurement framework that makes them causally separable, measurable, and law-governed. Persistent-homology H1 structure of the representation space separates into a within-class manifold channel (a function of the training stopping point) and a cross-class channel (a monotone readout of memorized flipped samples). An intervention, the FM0 prescription (zero loss on flipped samples from epoch 0), reaches each setting's generalization ceiling while memorizing essentially nothing. Within the framework we establish a law set with graded evidence: (L2) FM0 separation prescription (9/9); (L1) the within-channel as a training-position function (mid-rise 6/6; convergence-back CIFAR 3/3, SVHN 2/3); (L3) a ring-construction identity (definitional, not a law); and TLS (memory-generalization topological layering): memory is causally additive, anchored (silencing clean collapses the representation), invertible (stripping memory restores near-ceiling generalization), and quantitatively billable (the memorization cost law, effective slope coefficient C ~ 0.38 at the reference capacity: CIFAR-10 0.3801 / SVHN 0.3806 / CIFAR-100 0.384 / VGG 0.3715, capacity-dependent in general and traced to clean-sample feature displacement). We also publish the framework's boundaries: a falsification ledger of nine dead ends, and an instrument-vindication section that excludes six families of global statistics as explanations of the within-channel. The framework turns "memorization" from an ill-defined capacity into a measurable, separable, invertible topological layer.
Kenji Komiya, Andrew Kailiang Jin, Ryo Nishikimi +1cs.LG
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 characteristics of multicellular arrangements under stochastic cell proliferation and death. To address these challenges, we integrate two approaches: Betti vectors and inverse surrogate modeling. The Betti vectors obtained through topological data analysis can consistently represent features of a wide range of multicellular spatial configurations. The inverse surrogate modeling enables direct inference of the corresponding ABM parameters from the target patterns. We validated the proposed framework using zebrafish pigment pattern formation, a representative model of pattern formation driven by multicellular interactions. The results demonstrate that our framework successfully estimates ABM parameters and outperforms conventional methods such as PointNet++. Notably, the proposed method, which used only 10% of the training data, outperformed PointNet++, which used 100% of the data, across all evaluation metrics.
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.
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 point clouds in the latent space of a transformer model, using both models that have and have not been fine-tuned for sentence similarity. We find that Dowker homology captures sentence similarity information, as measured by regressing Dowker homology features onto ground-truth similarity scores, and that it can be used for visual inspection of similarity data and models. In an attempt to make Dowker homology readily applicable, we derive from it single-number summaries that we expect to capture sentence similarity directly. These turn out to work reasonably well, but without outperforming standard sentence similarity measures based on established pooling methods.
Modern machine learning (ML) methods are highly effective for prediction tasks, but many commonly used representations reduce complex data to fixed dimensional embeddings that may suppress multiscale structural organization. The Mapper algorithm from topological data analysis (TDA) provides a different perspective by decomposing data into overlapping local regions connected through a nerve construction, producing a structured representation that captures geometric organization, local statistical behavior, and relational connectivity simultaneously. In this work, we develop a framework for learning over Mapper induced structured representations. Rather than treating Mapper as a preprocessing step that produces a graph for downstream learning, we treat the full Mapper construction as part of the representation itself. We study mathematical properties of these representations, including invariance under relabeling, a distance functional on the space of Mapper representations, structural complexity of multiscale decompositions, and learning oriented stability under representation perturbations. Experiments on time series and graph classification datasets validate the proposed framework through controlled studies of representation ablation, Mapper parameter sensitivity, and the geometry of the induced representation space. Together, these results demonstrate how the proposed mathematical framework enables systematic comparison, interpretation, and analysis of Mapper representations, providing practical tools for studying representation geometry, structural complexity, and learning stability in learning tasks.
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.
Leon Dahlmeier, Sara Kališnik, Albert Mehl +1cs.CV math.AT
CBCT (Cone Beam Computed Tomography) scans provide detailed three-dimensional images, widely used in dentistry for diagnostic and treatment planning tasks. While invaluable, analyzing and documenting these scans is labor-intensive, prompting efforts to automate key steps like the classification and segmentation of anatomical structures to identify tooth types and associated pathologies. In this article, we propose an approach to automation that leverages persistent homology, a framework from topological data analysis that studies the shape of data by identifying features like connected components, holes, and voids across multiple scales. Persistent homology, together with a support vector machine, allows us to classify teeth in a CBCT scan and to perform diagnostics. Our method advances the state of the art, reaching average accuracy scores of 97.67% for tooth-labeling and 96.77% for diagnostic tasks, outperforming a CNN trained on the same data with accuracy of 70.27% and 86.67%, respectively.
Fine-grained damage classification of 3D point cloud data (PCD) remains a persistent challenge, constrained by high computational demands and limited labeled data. This study examines two methods: 3D PCD-based damage assessment (3PDA) algorithm and 2D projection damage assessment (2PDA) In our 3PDA analysis algorithm, TDA is used to derive compact representations of 3D PCD segmented by pointNet, which are then integrated with anomaly detection algorithms to quantify structural degradation. We show that TDA effectively compresses geometric structure from VFM-segmented components into discriminative feature vectors and that anomaly detection models can reliably distinguish components with varying damage severity using only 3D PCD inputs. In the 2D projection analysis algorithm, we leverage large VFMs for granular damage detection by projecting 3D PCD into 2D views. These projections allow VFM based models to achieve competitive classification performance while requiring only a fraction of the computational cost associated with full 3D data processing. Our results demonstrate that 2D VFM pipelines in 2PDA can perform strongly on fine-grained damage classification tasks, highlighting their viability as lightweight, resource-efficient alternatives to traditional 3PDA architectures. Comparative evaluation shows that the 3PDA attains higher accuracy but only for a narrow subset of object geometries and at substantially higher computational cost due to its reliance on TDA and the scarcity of high-fidelity 3D datasets. In contrast, the 2PDA algorithm yields slightly lower accuracy but offers an order of magnitude reduction in time complexity and generalizes across a far broader range of object categories.
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.
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.
Adam Shaw, Jiayu Li, Michael Sperling +2cs.LG math.AT
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.
Detecting that a stream of high-dimensional embeddings has changed is usually framed as a choice of statistic. We give a scale law that constrains any moment-based choice and test it against topological alternatives. The law: certifying a feature of spatial scale eps carrying mass fraction f requires polynomial tests of degree N* >= log(1/f)/(2 eps), proved via the Chebyshev extremal problem; a Gauss-quadrature construction gives N* >= 4b-1 for a b-scale topology, so cost is set by feature fineness, not feature count. The law is one-sided: we exhibit an annulus whose mean, covariance and all fourth-order moments equal those of a filled disk, yet H_1 is nonzero. Its practical content is a calibration rule. The upper bound is attained by Gaussian test functions, the RKHS witness of an RBF kernel, so the law predicts which bandwidth an MMD test should use: the feature scale. On real embedding streams we measure sigma*/eps with median 1.12 (IQR 1.01-1.52, n=26) over three settings and three scales, and a data-driven bandwidth reaches AUC >= 0.95. Against an adversary optimised against the defender's statistics (mean, covariance, k-NN, kurtosis), only a bandwidth-matched kernel test still detects. For persistent homology the verdict is mixed and depends on choices usually left implicit. The summary matters more than the filtration: total persistence attains recall 0.75 at FPR 1% where the first persistence landscape attains 0.00. What survives is a cost gap, not a power gap: where persistence works it costs 116x kurtosis, which works at least as well. We conclude not that topological summaries are useless, but that on this task a kernel test whose bandwidth the law sets dominates them.
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.
Over the last decade, neural networks have been applied to an increasingly diverse range of applications, including data with rich geometric, topological, or symmetry-related structure. As a result, researchers have increasingly drawn inspiration from topology, algebra, and geometry. Despite this rich algorithmic development, the supporting software ecosystem remains fragmented. Many important methods exist only as research prototypes in unmaintained repositories. We address this by introducing Topology, Algebra, and Geometry Torch (TAGTorch), an open-source, PyTorch-based library that unifies tools inspired by topology, algebra, and geometry, including data-preprocessing methods, architectures, training techniques, and model analysis tools. We describe the design philosophy of TAGTorch and then discuss its current architecture and capabilities, highlighting areas where it can fill gaps in the current software ecosystem. We conclude with a discussion of our future development priorities for the library.
We introduce Topoformer, a lightweight and scalable framework for graph representation learning that encodes topological structure into attention-friendly sequences. At the core of our method is Topo-Scan, a novel module that decomposes a graph into a short, ordered sequence of topological tokens by slicing over node or edge filtrations. These sequences capture multi-scale structural patterns, from local motifs to global organization, and are processed by a Transformer to produce expressive graph-level embeddings. Unlike traditional persistent homology pipelines, Topo-Scan is parallelizable, avoids costly diagram computations, and integrates seamlessly with standard deep learning architectures. We provide theoretical guarantees on the stability of our topological encodings and demonstrate state-of-the-art performance across graph classification and molecular property prediction benchmarks. Our results show that Topoformer matches or exceeds strong GNN and topology-based baselines while offering predictable and efficient compute. This work opens a new path for parallelizable and unifying approaches to graph representation learning that integrate topological inductive biases into attention frameworks.
Hugo Gobato Souto, Ioannis Diamantisstat.ME math.ST stat.ML
Many modern outcomes, including images, point clouds, networks, and spatial fields, are structured objects for which \(Y^1-Y^0\) may be undefined or scientifically inadequate. We introduce \emph{Topological Causal Data Analysis} (TCDA), a framework separating the observation space, causal-model class, topological representation, and causal query. Topology does not define interventions; it supplies stable, shape-sensitive summaries after causal assumptions have been specified. We distinguish outcome-level TCDA, which transforms individual potential outcomes, from distribution-level TCDA, which transforms interventional outcome laws, and characterize when outcome and distribution level contrasts agree. Building on recent outcome-level theory, we formulate identification and doubly robust representations for Banach-space-valued summaries. At the distribution level, we identify targets through the standard causal \(g\)-formula and derive stability-transfer bounds and plug-in consistency. We also place target-specific topological ignorability within the framework, clarifying when a covariate-standardized coarse effect can be identified without identifying the full interventional laws. Finally, we delimit the role of observational topology in causal discovery: it can assist diagnosis on restricted model classes but cannot by itself identify causal structure.
Persistence Diagram (PD) is known to capture point cloud topology effectively, but its computation has high time complexity. Expected Persistence Diagram (EPD) has been developed to reduce the time cost by studying the topology of multiple subsets of a point cloud and it serves as a distribution of topological features. Existing EPD vectorizations often rely on predefined point transformations, such as Gaussian or landscape functions. We study an alternative discretization based on Voronoi histograms, which trades smooth functional approximation for adaptive partition-based counting. We propose to use Voronoi Diagram-based histogram as the vectorization of EPD, without imposing an explicit smooth point transformation model. Under stated separation and normalization conditions, we establish stability bounds and characterize when the histogram representation preserves Wasserstein-scale variation. We demonstrate the effectiveness of our proposed representation on real-world datasets which have significant topological features for classification and dimensionality reduction tasks.
When computing sub/super-level-set persistent homology (PH), the effect of noise may introduce millions of (short-lived) topological generators, presenting an obstacle to both the computation of PH of large 3D images, and any analysis of PH that incorporates the number of generators. As such, it is often necessary to denoise the data before computing its PH. We analyze the PH of synthetic 3D images of porous media in the presence of spatially uncorrelated noise, and perform a comparative analysis of various topological measures (e.g. bottleneck distance, Wasserstein distance, persistence statistics and persistence images) to assess their robustness to both noise and the denoising process (i.e. adding spatially uncorrelated Gaussian noise, and denoising by either a Gaussian convolution or a machine learning approach).
Akarsh K. Nair, Muhammad Arifur Rahman, David Brown +1cs.LG cs.AI cs.CR
Split learning enables collaborative model training by partitioning neural networks across clients and servers. However, improper split placement can lead to severe privacy leakage through intermediate representations. In this work, we propose a topology-guided framework for privacy-aware split learning based on the persistent Betti complexity of smashed activations. Through comprehensive layer-wise analysis, we show that privacy risk in split learning is highly non-uniform across layers and exhibits sharp transition regions that are not captured by architectural depth alone. In particular, feature inversion fidelity increases from negligible reconstruction to as high as 0.98 SSIM at deeper, privacy-critical split points. We further demonstrate that Betti complexity consistently identifies representation regimes associated with elevated feature-space privacy leakage across architectures and datasets. Leveraging this observation, we introduce BettiSafe, a topology-guided split selection strategy that identifies privacy-sensitive layers without requiring explicit attack execution. BettiSafe improves resistance to feature inversion by 2 to 5 times compared to depth-based heuristics while preserving classification accuracy. In addition, Betti-based regularisation increases inversion difficulty by nearly 5 x without degrading model utility, enabling a favourable privacy utility tradeoff. Overall, our results highlight topological complexity as a promising structural descriptor for secure, adaptive, and representation-aware split learning in real-world collaborative systems
Jake Schwaderer, Alexander Bastien, Omid Khormali +3cs.LG math.AT stat.ML
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.
Denis Mayr Lima Martins, Gottfried Vossencs.DB cs.LG
Self-Organizing Maps (SOMs) have long been used as exploratory tools for high-dimensional data: they organize objects into a two-dimensional topology that reveals clusters, gradients, sparse regions, dense regions, and boundaries. Yet, in modern data systems, SOMs are typically trained and visualized outside the DBMS, disconnected from the relational data they summarize. We introduce the abstraction of a queryable data map: a learned topological artifact consisting of representatives, neighborhood relations, object assignments, and derived summaries. We instantiate this idea with MapDB, a lightweight prototype that makes SOM artifacts queryable so users can explore data topology without leaving the database. Experimental study shows that SOM training is feasible at moderate analytical scale, that map queries are interactive after materialization, and that SOM regions provide meaningful targets for exploratory SQL.
TabPFN is a transformer-based foundation model for tabular prediction that performs inference without task-specific training by conditioning on a support set and query inputs. Despite its strong empirical performance, its internal behavior on structurally difficult tabular geometries remains poorly understood. We study this behavior using zigzag persistent homology, treating TabPFN layer representations as evolving point clouds. We construct a controlled benchmark of synthetic tabular tasks with known true probabilities and varied intrinsic topology, including warped circles, tori, spheres, Hopf links, trefoil knots, and Swiss rolls. Across these tasks, we find that the topology of TabPFN's internal representation geometry is strongly associated with dataset-level reliability; for example, the zeroth homology group $H_0$ fragmentation count correlates positively with mean absolute residual across controlled tasks, and this association strengthens in a high-resolution warped circle case study at large sample size. Harder geometries induce a dual topological signature: increased $H_1$ loop activity and increased $H_0$ fragmentation, while the $H_1$ persistence becomes shorter-lived. These descriptors correlate with Bayes error, mean absolute residuals, and overconfidence. Our results suggest that zigzag persistence diagnoses the reliability of the inferred in-context task geometry and provides a context-level view of when TabPFN operates in topologically stressed regimes.
Full fine-tuning remains a strong way to adapt pretrained LLMs, but it updates all weights and can be expensive. LoRA reduces the number of trainable parameters, but it does not directly answer which pretrained components should be trained and which can be frozen during adaptation. We introduce TopoTuner, a topology-guided fine-tuning framework for selective freezing of attention projection matrices. \method treats each projection matrix as a row cloud and uses Wasserstein distances between persistence diagrams to measure how its topology changes during fine-tuning. TopoTuner learns a reusable freezing profile from a source dataset and transfers it to efficiently fine-tune models on out-of-domain datasets, evaluating whether task-specific topological drift generalizes across question answering and sentiment analysis tasks. Across LLaMA-3.1-8B, Mistral-7B-v0.3, and Qwen3-8B-Base, TopoTuner is competitive with full fine-tuning while training only 1-2\% of the model parameters, and outperforms LoRA in 7 out of 9 model-dataset settings, which can change up to 39.57\% of the projection parameters. Along with minimized updates, TopoTuner reduces training time by 20.4\% relative to full fine-tuning and 5.5\% relative to LoRA on average. TopoTuner opens a new direction for reusable freezing profiles, where fine-tuning behavior learned on one dataset can be shared across multiple tasks.
Adam Wesołowski, Dimitrios Thanos, Daniel Leykam +1quant-ph cs.LG
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.
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&P~500 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.
Current electroencephalography (EEG)-based dream detection relies on power spectral density (PSD) and statistical moment features, achieving a state-of-the-art area under the receiver operating characteristic curve (AUC) of approximately 0.70 on the DREAM database (Wong et al., 2025, Nature Communications). We introduce PHINN-EEG (Persistent Homology Inspired Neural Network for EEG), the first topological time-series framework for dream mentation analysis. Using sliding-window Takens delay embeddings and Vietoris-Rips filtrations on multichannel pre-awakening EEG epochs, we extract Dynamic Betti Curves that characterize the geometric architecture of neural activity, not merely its energy. These topological invariants, combined with topology-conditioned flow matching, are analytically projected to outperform existing PSD and catch22 benchmarks, targeting AUC = 0.82-0.90 on the 1,462-awakening open-access subset of the DREAM database (drawn from a full registry of 3,191 total awakenings from 263 participants across 20 independent laboratories). We further introduce a topology-conditioned rectified flow model for dream-state EEG synthesis-with a spectral-conditioned flow model of comparable feature dimensionality as an additional ablation baseline to isolate the value of topological conditioning specifically-and propose a set of candidate Betti transition archetypes linking topology to phenomenological dream report categories, presented as an exploratory hypothesis space pending empirical validation. If validated, this work represents a paradigm shift from spectral energy to phase-space geometry in neural rare-event detection, with potential future implications for wearable BCI dream monitoring.
Circular coordinates obtained from persistent cohomology reveal loop structure in data, but they usually remain abstract: A detected circle does not tell us which measured angle, phase, torsion, or decoder explains it. We propose a method for selecting interpretable circle-valued coordinates from a user-supplied dictionary of scientifically meaningful candidates explaining the detected cohomology. In the continuous setting, each candidate is represented by the cohomology class of its pulled-back angular form, and selecting a minimum-energy set of candidates spanning the relevant $H^1$ subspace becomes a minimum-weight basis problem in a vector matroid. We then introduce CIRCOL, a method for discrete point clouds sampled from the manifold. We prove that the introduced cochain inner product is a consistent estimator of the $L^2$ inner product of fixed smooth 1-forms under non-uniform sampling. The resulting projection matrix both helps selecting a basis of low-energy dictionary coordinates and diagnoses topologically trivial candidates or unexplained persistent classes. Finally, we verify the effectiveness of our method on synthetic examples, on molecular simulations, and neural recordings of head-direction cells.