Related papers: Euler Characteristic Curves and Profiles: a stable…
We provide a short introduction to the field of topological data analysis and discuss its possible relevance for the study of complex systems. Topological data analysis provides a set of tools to characterise the shape of data, in terms of…
Persistent homology is a cornerstone of topological data analysis, offering a multiscale summary of topology with robustness to nuisance transformations, such as rotations and small deformations. Persistent homology has seen broad use…
The existence of characteristic structure, or shape, in complex data sets has been recognized as increasingly important for mathematical data analysis. This realization has motivated the development of new tools such as persistent homology…
Under the banner of `Big Data', the detection and classification of structure in extremely large, high dimensional, data sets, is, one of the central statistical challenges of our times. Among the most intriguing approaches to this…
Topological data analysis offers a rich source of valuable information to study vision problems. Yet, so far we lack a theoretically sound connection to popular kernel-based learning techniques, such as kernel SVMs or kernel PCA. In this…
The goal of this numerical study is to get insight into singular solutions of the two-dimensional (2D) Euler equations for non-smooth initial data, in particular for vortex sheets. To this end high resolution computations of vortex layers…
Graph pattern matching is often defined in terms of subgraph isomorphism, an NP-complete problem. To lower its complexity, various extensions of graph simulation have been considered instead. These extensions allow pattern matching to be…
We present here a survey of recent results concerning the mathematical analysis of instabilities of the interface between two incompressible, non viscous, fluids of constant density and vorticity concentrated on the interface. This…
With the explosive growth of multi-source data, multi-view clustering has attracted great attention in recent years. Most existing multi-view methods operate in raw feature space and heavily depend on the quality of original feature…
Topological data analysis is an emerging mathematical concept for characterizing shapes in multi-scale data. In this field, persistence diagrams are widely used as a descriptor of the input data, and can distinguish robust and noisy…
In the study of oceanic flows at the geophysical scale, the phenomenon of density stratification plays a central role in the dynamics of the system. Two categories of mathematical models are commonly used to describe the role played by the…
In this work we use the persistent homology method, a technique in topological data analysis (TDA), to extract essential topological features from the data space and combine them with deep learning features for classification tasks. In TDA,…
The persistent Betti numbers are used in topological data analysis to infer the scales at which topological features appear and disappear in the filtration of a topological space. Most commonly by means of the corresponding barcode or…
Hypergraph data appear and are hidden in many places in the modern age. They are data structure that can be used to model many real data examples since their structures contain information about higher order relations among data points. One…
Topological Data Analysis (TDA) provides tools to describe the shape of data, but integrating topological features into deep learning pipelines remains challenging, especially when preserving local geometric structure rather than…
This paper resolves the characteristic initial data problem for the three-dimensional compressible Euler equations - an open problem analogous to Christodoulou's characteristic initial value formulation for the vacuum Einstein field…
In many applications concerning the comparison of data expressed by $\mathbb{R}^m$-valued functions defined on a topological space $X$, the invariance with respect to a given group $G$ of self-homeomorphisms of $X$ is required. While…
Latent space matching, which consists of matching distributions of features in latent space, is a crucial component for tasks such as adversarial attacks and defenses, domain adaptation, and generative modelling. Metrics for probability…
Optimization, a key tool in machine learning and statistics, relies on regularization to reduce overfitting. Traditional regularization methods control a norm of the solution to ensure its smoothness. Recently, topological methods have…
We construct a new class of self-similar implosion profiles for the multi-dimensional compressible Euler equations. These profiles are smooth, genuinely non-isentropic, radially/spherically symmetric, and have explicit (closed-form)…