Topological data analysis (TDA) is an area of data science that focuses on using invariants from algebraic topology to provide multiscale shape descriptors for geometric data sets, such as graphs and point clouds.

The problem of (point) forecasting univariate time series is considered.

In medical image analysis, feature engineering plays an important role in the design and performance of machine learning models. Persistent homology (PH), from the field of topological data analysis (TDA), demonstrates robustness and stability to data perturbations and addresses the limitation from traditional feature extraction approaches where a small change in input results in a large change in feature representation. Using PH, we store persistent topological and geometrical features in the form of the persistence barcode whereby large bars represent global topological features and small bars encapsulate geometrical information of the data. When multiple barcodes are computed from 2D or 3D medical images, two approaches can be used to construct the final topological feature vector in each dimension: aggregating persistence barcodes followed by featurization or concatenating topological feature vectors derived from each barcode. In this study, we conduct a comprehensive analysis across diverse medical imaging datasets to compare the effects of the two aforementioned approaches on the performance of classification models. The results of this analysis indicate that feature concatenation preserves detailed topological information from individual barcodes, yields better classification performance and is therefore a preferred approach when conducting similar experiments.

The recent adoption of artificial intelligence (AI) 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. To date, 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 explores how topological features contribute to explainable safety regions in social navigation. Instead of using behavioral parameters, we leverage topological data analysis to classify and characterize different simulation behaviors. First, we apply global rule-based classification to distinguish between safe (collision-free) and unsafe scenarios based on topological properties. Then, we define safety regions, $S_\varepsilon$, in the topological feature space, ensuring a maximum classification error of $\varepsilon$. These regions are built with adjustable SVM classifiers and order statistics, providing robust decision boundaries. Local rules extracted from these regions enhance interpretability, keeping the decision-making process transparent. Our approach initially separates simulations with and without collisions, outperforming methods that not incorporate topological features. It offers a deeper understanding of robot interactions within a navigable space. 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.

We propose a new topological tool for computer vision - Scalar Function Topology Divergence (SFTD), which measures the dissimilarity of multi-scale topology between sublevel sets of two functions having a common domain. Functions can be defined on an undirected graph or Euclidean space of any dimensionality. Most of the existing methods for comparing topology are based on Wasserstein distance between persistence barcodes and they don't take into account the localization of topological features. On the other hand, the minimization of SFTD ensures that the corresponding topological features of scalar functions are located in the same places. The proposed tool provides useful visualizations depicting areas where functions have topological dissimilarities. We provide applications of the proposed method to 3D computer vision. In particular, experiments demonstrate that SFTD improves the reconstruction of cellular 3D shapes from 2D fluorescence microscopy images, and helps to identify topological errors in 3D segmentation.

Topological data analysis (TDA) is an area of data science that focuses on using invariants from algebraic topology to provide multiscale shape descriptors for geometric data sets such as point clouds. One of the most important such descriptors is {\em persistent homology}, which encodes the change in shape as a filtration parameter changes; a typical parameter is the feature scale. For many data sets, it is useful to simultaneously vary multiple filtration parameters, for example feature scale and density. While the theoretical properties of single parameter persistent homology are well understood, less is known about the multiparameter case. In particular, a central question is the problem of representing multiparameter persistent homology by elements of a vector space for integration with standard machine learning algorithms. Existing approaches to this problem either ignore most of the multiparameter information to reduce to the one-parameter case or are heuristic and potentially unstable in the face of noise. In this article, we introduce a new general representation framework that leverages recent results on {\em decompositions} of multiparameter persistent homology. This framework is rich in information, fast to compute, and encompasses previous approaches. Moreover, we establish theoretical stability guarantees under this framework as well as efficient algorithms for practical computation, making this framework an applicable and versatile tool for analyzing geometric and point cloud data. We validate our stability results and algorithms with numerical experiments that demonstrate statistical convergence, prediction accuracy, and fast running times on several real data sets.

We propose a method for learning topology-preserving data representations (dimensionality reduction). The method aims to provide topological similarity between the data manifold and its latent representation via enforcing the similarity in topological features (clusters, loops, 2D voids, etc.) and their localization. The core of the method is the minimization of the Representation Topology Divergence (RTD) between original high-dimensional data and low-dimensional representation in latent space. RTD minimization provides closeness in topological features with strong theoretical guarantees. We develop a scheme for RTD differentiation and apply it as a loss term for the autoencoder. The proposed method "RTD-AE" better preserves the global structure and topology of the data manifold than state-of-the-art competitors as measured by linear correlation, triplet distance ranking accuracy, and Wasserstein distance between persistence barcodes.

Image segmentation is a largely researched field where neural networks find vast applications in many facets of technology. Some of the most popular approaches to train segmentation networks employ loss functions optimizing pixel-overlap, an objective that is insufficient for many segmentation tasks. In recent years, their limitations fueled a growing interest in topology-aware methods, which aim to recover the correct topology of the segmented structures. However, so far, none of the existing approaches achieve a spatially correct matching between the topological features of ground truth and prediction. In this work, we propose the first topologically and feature-wise accurate metric and loss function for supervised image segmentation, which we term Betti matching. We show how induced matchings guarantee the spatially correct matching between barcodes in a segmentation setting. Furthermore, we propose an efficient algorithm to compute the Betti matching of images. We show that the Betti matching error is an interpretable metric to evaluate the topological correctness of segmentations, which is more sensitive than the well-established Betti number error. Moreover, the differentiability of the Betti matching loss enables its use as a loss function. It improves the topological performance of segmentation networks across six diverse datasets while preserving the volumetric performance. Our code is available in https://github.com/nstucki/Betti-matching.