We introduce the \emph{Topological Stability Index} (TSI), a variance-based scalar measure for persistence barcodes that quantifies the dispersion of persistence lifetimes. Unlike persistent entropy, which depends only on normalized weights, the TSI captures absolute variability and is sensitive to heterogeneous feature scales. We establish fundamental properties of the TSI, including its scaling behavior, invariance under lifetime translation and explicit update formulas under insertion and deletion of bars. We also consider a complementary first-moment-type quantity, the Topological Signal Index (TSigI), which captures the typical scale of persistence lifetimes and provides additional interpretability alongside the TSI. We further introduce a normalized version, $cv\text{TSI}$, which is scale invariant and admits an explicit algebraic relation to the Rényi entropy of order two. In particular, $cv\text{TSI}$ is an affine function of the collision probability $\sum_i p_i^2$, and therefore a monotone reparametrization of the Rényi entropy, providing a direct link between variance-based and entropy-based summaries in topological data analysis. Numerical experiments on synthetic data and stochastic time series demonstrate that the TSI captures structural variability complementary to entropy: it is relatively insensitive to deterministic trends, while responding strongly to stochastic fluctuations and variations in persistence magnitude.

We present a unified pipeline for univariate time series classification via complex networks and persistent homology. A time series is mapped to a graph through one of five constructions across three families--visibility (natural and horizontal visibility graphs), transition, and proximity--and the graph is converted to a dissimilarity matrix from which a Vietoris-Rips filtration yields persistence diagrams. These diagrams are vectorized into fixed-length features through persistence landscapes and topological summary statistics. By standardizing the downstream processing, differences in classification performance are attributable to the network construction and distance metric alone. Experiments on twelve UCR benchmarks show that (i) no single construction dominates: the optimal graph type depends on the signal's discriminative structure; (ii) the graph distance metric is a first-order design choice, with diffusion distance uniformly outperforming shortest-path alternatives; and (iii) persistence-based features degrade gracefully under noise, consistent with the classical stability theorem of persistent homology.

Machine learning for point clouds has been attracting much attention, with many applications in various fields, such as shape recognition and material science. For enhancing the accuracy of such machine learning methods, it is often effective to incorporate global topological features, which are typically extracted by persistent homology. In the calculation of persistent homology for a point cloud, we choose a filtration for the point cloud, an increasing sequence of spaces. Since the performance of machine learning methods combined with persistent homology is highly affected by the choice of a filtration, we need to tune it depending on data and tasks. In this paper, we propose a framework that learns a filtration adaptively with the use of neural networks. In order to make the resulting persistent homology isometry-invariant, we develop a neural network architecture with such invariance. Additionally, we show a theoretical result on a finite-dimensional approximation of filtration functions, which justifies the proposed network architecture. Experimental results demonstrated the efficacy of our framework in several classification tasks.

We propose a robust and reliable evaluation metric for generative models called Topological Precision and Recall (TopP&R, pronounced "topper"), which systematically estimates supports by retaining only topologically and statistically significant features with a certain level of confidence. Existing metrics, such as Inception Score (IS), Fréchet Inception Distance (FID), and various Precision and Recall(P&R) variants, rely heavily on support estimates derived from sample features. However, the reliability of these estimates has been overlooked, even though the quality of the evaluation hinges entirely on their accuracy. In this paper, we demonstrate that current methods not only fail to accurately assess sample quality when support estimation is unreliable, but also yield inconsistent results. In contrast, TopP&R reliably evaluates the sample quality and ensures statistical consistency in its results. Our theoretical and experimental findings reveal that TopP&R provides a robust evaluation, accurately capturing the true trend of change in samples, even in the presence of outliers and non-independent and identically distributed (Non-IID) perturbations where other methods result in inaccurate support estimations. To our knowledge, TopP&Ris the first evaluation metric specifically focused on the robust estimation of supports, offering statistical consistency under noise conditions.

Persistent homology (PH) encodes global information, such as cycles, and is thus increasingly integrated into graph neural networks (GNNs). PH methods in GNNs typically traverse an increasing sequence of subgraphs. In this work, we first expose limitations of this inclusion procedure. To remedy these shortcomings, we analyze contractions as a principled topological operation, in particular, for graph representation learning. We study the persistence of contraction sequences, which we call Contraction Homology (CH). We establish that forward PH and CH differ in expressivity. We then introduce Hourglass Persistence, a class of topological descriptors that interleave a sequence of inclusions and contractions to boost expressivity, learnability, and stability. We also study related families parametrized by two paradigms. We also discuss how our framework extends to simplicial and cellular networks. We further design efficient algorithms that are pluggable into end-to-end differentiable GNN pipelines, enabling consistent empirical improvements over many PH methods across standard real-world graph datasets. Code is available at \href{https://github.com/Aalto-QuML/Hourglass}{this https URL}.

We prove that the persistence diagram of the sublevel set filtration of the quadratic form f(x) = x^T M x restricted to the unit sphere S^{n-1} is analytically determined by the eigenvalues of the symmetric matrix M. By Morse theory, the diagram has exactly n-1 finite bars, with the k-th bar living in homological dimension k-1 and having length equal to the k-th eigenvalue spacing s_k = λ_{k+1} - λ_k. This identification transfers random matrix theory (RMT) universality to persistence diagram universality: for matrices drawn from the Gaussian Orthogonal Ensemble (GOE), we derive the closed-form persistence entropy PE = log(8n/π) - 1, and verify numerically that the coefficient of variation of persistence statistics decays as n^{-0.6}. Different random matrix ensembles (GOE, GUE, Wishart) produce distinct universal persistence diagrams, providing topological fingerprints of RMT universality classes. As a practical consequence, we show that persistence entropy outperforms the standard level spacing ratio \langle r \rangle for discriminating GOE from GUE matrices (AUC 0.978 vs. 0.952 at n = 100, non-overlapping bootstrap 95% CIs), and detects global spectral perturbations in the Rosenzweig-Porter model to which \langle r \rangle is blind. These results establish persistence entropy as a new spectral diagnostic that captures complementary information to existing RMT tools.

Computational topology provides a tool, persistent homology, to extract quantitative descriptors from structured objects (images, graphs, point clouds, etc). These descriptors can then be involved in optimization problems, typically as a way to incorporate topological priors or to regularize machine learning models. This is usually achieved by minimizing adequate, topologically-informed losses based on these descriptors, which, in turn, naturally raises theoretical and practical questions about the possibility of optimizing such loss functions using gradient-based algorithms. This has been an active research field in the topological data analysis community over the last decade, and various techniques have been developed to enable optimization of persistence-based loss functions with gradient descent schemes. This survey presents the current state of this field, covering its theoretical foundations, the algorithmic aspects, and showcasing practical uses in several applications. It includes a detailed introduction to persistence theory and, as such, aims at being accessible to mathematicians and data scientists newcomers to the field. It is accompanied by an open-source library which implements the different approaches covered in this survey, providing a convenient playground for researchers to get familiar with the field.

We improve and extend persistence spheres, introduced in~\cite{pegoraro2025persistence}. Persistence spheres map an integrable measure $μ$ on the upper half-plane, including persistence diagrams (PDs) as counting measures, to a function $S(μ)\in C(\mathbb{S}^2)$, and the map is stable with respect to 1-Wasserstein partial transport distance $\mathrm{POT}_1$. Moreover, to the best of our knowledge, persistence spheres are the first explicit representation used in topological machine learning for which continuity of the inverse on the image is established at every compactly supported target. Recent bounded-cardinality bi-Lipschitz embedding results in partial transport spaces, despite being powerful, are not given by the kind of explicit summary map considered here. Our construction is rooted in convex geometry: for positive measures, the defining ReLU integral is the support function of the lift zonoid. Building on~\cite{pegoraro2025persistence}, we refine the definition to better match the $\mathrm{POT}_1$ deletion mechanism, encoding partial transport via a signed diagonal augmentation. In particular, for integrable $μ$, the uniform norm between $S(0)$ and $S(μ)$ depends only on the persistence of $μ$, without any need of ad-hoc re-weightings, reflecting optimal transport to the diagonal at persistence cost. This yields a parameter-free representation at the level of measures (up to numerical discretization), while accommodating future extensions where $μ$ is a smoothed measure derived from PDs (e.g., persistence intensity functions~\citep{wu2024estimation}). Across clustering, regression, and classification tasks involving functional data, time series, graphs, meshes, and point clouds, the updated persistence spheres are competitive and often improve upon persistence images, persistence landscapes, persistence splines, and sliced Wasserstein kernel baselines.