arXiv:2608. 06276v1 Announce Type: cross Abstract: Persistence diagrams (PDs) provide stable and interpretable summaries of multiscale topological structure.
arXiv:2606. 11911v1 Announce Type: cross Abstract: Persistence diagrams are common representations in topological data analysis, but they do not naturally live in a vector space, and the statistical tools developed for comparing them have largely evolved separately from those used for downstream prediction.
arXiv:2507. 07156v2 Announce Type: replace-cross Abstract: Supervised machine learning pipelines trained on features derived from persistent homology have been experimentally observed to ignore much of the information contained in a persistence diagram.
The paper demonstrates that a stochastic hybrid system (SHS), which combines continuous dynamics governed by a stochastic differential equation (SDE) with discrete resets triggered by a Markov kernel, can be approximated by a single SDE in a higher‑dimensional latent space. By encoding the reset branches with auxiliary variables, the resets become deterministic, allowing the system’s manifold to be glued and embedded into Euclidean space. This embedding eliminates the need for explicit reset terms in the hybrid Fokker‑Planck equation, and the authors propose a loss function that matches evolving state distributions, enabling the latent SDE to recover the SHS’s probability evolution without mode labeling, trajectory segmentation, or event‑based simulations.
arXiv:2606. 00427v1 Announce Type: new Abstract: State abstraction in reinforcement learning is usually formulated as a partition of states based on reward and transition similarity.
arXiv:2606. 29763v1 Announce Type: cross Abstract: Topological data analysis (TDA), particularly persistent homology (PH), captures geometric structural properties in medical images (e.
arXiv:2608. 04334v1 Announce Type: cross Abstract: Contemporary model-free reinforcement learning algorithms can achieve very high performance, but have low sample efficiency and are not robust to changes in the environment.
arXiv:2606. 14737v1 Announce Type: cross Abstract: Molecular dynamics (MD) simulations generate trajectories in a high-dimensional configuration space whose analysis critically depends on molecular descriptors, typically handcrafted observables or learned kinetic embeddings.
The paper introduces T-ARC, a clustering algorithm that integrates topological information into the K‑means objective by coupling a data‑fidelity term with a graph‑cut penalty. The latent graph is modeled as a random realization from a Stochastic Block Model, whose parameter is optimized via Distributionally Robust Optimization, using a persistence‑based similarity matrix derived from zero‑dimensional persistent homology. Experiments on synthetic non‑convex data and Fashion‑MNIST subsets demonstrate that T‑ARC recovers latent topological structures and outperforms K‑means on curved and interleaved clusters while remaining competitive and more stable on real data.
arXiv:2606. 21295v2 Announce Type: replace-cross Abstract: Existing sequence models, including RNNs, LSTMs, continuous-time networks, and Transformers, share a common structural principle: layer-wise dynamics, where all neurons in the same layer co-evolve through a shared parameterized operator, leaving individual neurons no freedom to evolve independently.
Topological Steering is a new framework that steers large language model (LLM) behavior by leveraging topological representations of activation spaces, specifically using persistence diagrams from Topological Data Analysis (TDA). Unlike traditional methods that intervene directly in activation or feature space, this approach captures global structure, making it more robust to outliers, distributional shifts, noise, and local perturbations. Experiments demonstrate that Topological Steering consistently modifies LLM behavior across multiple model families and sizes.
arXiv:2606. 09806v1 Announce Type: cross Abstract: We introduce Topological Neural Operators (TNOs), a principled framework for operator learning on cell complexes that lifts neural operators (NOs) from functions on points and/or edges to topological domains.
arXiv:2512.23348v3 Announce Type: replace-cross Abstract: We introduce a data-analysis framework based on filtrations of finite topological spaces. Starting from a finite metric data set, we construc...