arXiv subjects

Victor Toscano-Duran

Publications and source records attributed to Victor Toscano-Duran.

4 recordsLinked to original sources

arXiv2026

Persistent Entropy Transform: An entropy-based descriptor for topological data analysis

Persistent entropy provides a compact summary of persistence diagrams, but discards geometric information inherent to the data. This limitation creates a gap between scalar summaries, which are computationally efficient but geometrically coarse, and directional topological transforms, which are expressive but high-dimensional and computationally demanding. In this work, we introduce the Persistent Entropy Transform (PET), a novel directional topological descriptor which can be interpreted as an entropy-based compression of directional topological transforms. We establish basic theoretical properties of PET. In particular, we prove translation invariance, scale invariance under positive uniform scalings, and orthogonal equivariance. Empirically, we use synthetic shapes to assess directional sensitivity, consistency with rotational equivariance, robustness under controlled perturbations, and dependence on directional sampling density. In addition, we test the novel tool on two real time-series benchmark datasets, the TwoLead- ECG and the MIT-BIH, to provide a proof of concept showing that PET embeddings can be used succesfully as compact feature vectors on real time-series benchmarks. These experiments support and validate PET as a compact and computationally tractable descriptor.

math.GN↗

arXiv2025

Barycentric Neural Networks and Length-Weighted Persistent Entropy Loss: A Green Geometric and Topological Framework for Function Approximation

While artificial neural networks are known as universal approximators for continuous functions, many modern approaches rely on overparameterized architectures with high computational cost. In this work, we introduce the Barycentric Neural Network (BNN): a compact shallow architecture that encodes both structure and parameters through a fixed set of base points and their associated barycentric coordinates. We show that the BNN enables the exact representation of continuous piecewise linear functions (CPLFs), ensuring strict continuity across segments. Given that any continuous function on a compact domain can be uniformly approximated by CPLFs, the BNN emerges as a flexible and interpretable tool for function approximation. To enhance geometric fidelity in low-resource scenarios, such as those with few base points to create BNNs or limited training epochs, we propose length-weighted persistent entropy (LWPE): a stable variant of persistent entropy. Our approach integrates the BNN with a loss function based on LWPE to optimize the base points that define the BNN, rather than its internal parameters. Experimental results show that our approach achieves superior and faster approximation performance compared to standard losses (MSE, RMSE, MAE and LogCosh), offering a computationally sustainable alternative for function approximation.

cs.LG↗

arXiv2025

Safe and Efficient Social Navigation through Explainable Safety Regions Based on Topological Features

The recent adoption of artificial intelligence in robotics has driven the development of algorithms that enable autonomous systems to adapt to complex social environments. In particular, safe and efficient social navigation is a key challenge, requiring AI not only to avoid collisions and deadlocks but also to interact intuitively and predictably with its surroundings. Methods based on probabilistic models and the generation of conformal safety regions have shown promising results in defining safety regions with a controlled margin of error, primarily relying on classification approaches and explicit rules to describe collision-free navigation conditions. This work extends the existing perspective by investigating how topological features can contribute to the creation of explainable safety regions in social navigation scenarios, enabling the classification and characterization of different simulation behaviors. Rather than relying on behaviors parameters to generate safety regions, we leverage topological features through topological data analysis. We first utilize global rule-based classification to provide interpretable characterizations of different simulation behaviors, distinguishing between safe and unsafe scenarios based on topological properties. Next, we define safety regions, $S_\varepsilon$, representing zones in the topological feature space where collisions are avoided with a maximum classification error of $\varepsilon$. These regions are constructed using adjustable SVM classifiers and order statistics, ensuring a robust and scalable decision boundary. Our approach initially separates simulations with and without collisions, outperforming methods that not incorporate topological features. We further refine safety regions to ensure deadlock-free simulations and integrate both aspects to define a compliant simulation space that guarantees safe and efficient navigation.

cs.RO↗

arXiv2025

Molecular Machine Learning Using Euler Characteristic Transforms

The shape of a molecule determines its physicochemical and biological properties. However, it is often underrepresented in standard molecular representation learning approaches. Here, we propose using the Euler Characteristic Transform (ECT) as a geometrical-topological descriptor. Computed directly on a molecular graph derived from handcrafted atomic features, the ECT enables the extraction of multiscale structural features, offering a novel way to represent and encode molecular shape in the feature space. We assess the predictive performance of this representation across nine benchmark regression datasets, all centered around predicting the inhibition constant $K_i$. In addition, we compare our proposed ECT-based representation against traditional molecular representations and methods, such as molecular fingerprints/descriptors and graph neural networks (GNNs). Our results show that our ECT-based representation achieves competitive performance, ranking among the best-performing methods on several datasets. More importantly, its combination with traditional representations, particularly with the AVALON fingerprint, significantly \emph{enhances predictive performance}, outperforming other methods on most datasets. These findings highlight the complementary value of multiscale topological information and its potential for being combined with established techniques. Our study suggests that hybrid approaches incorporating explicit shape information can lead to more informative and robust molecular representations, enhancing and opening new avenues in molecular machine learning tasks. To support reproducibility and foster open biomedical research, we provide open access to all experiments and code used in this work.

cs.LG↗