相关论文: Persistent de Rham-Hodge Laplacians in Eulerian re…

Topological data analysis combines machine learning with methods from algebraic topology. Persistent homology, a method to characterize topological features occurring in data at multiple scales is of particular interest. A major obstacle to…

Point clouds, as a form of Lagrangian representation, allow for powerful and flexible applications in a large number of computational disciplines. We propose a novel deep-learning method to learn stable and temporally coherent feature…

Understanding the stoichiometry and associated stability of virus-like particles (VLPs) is crucial for optimizing their assembly efficiency and immunogenic properties, which are essential for advancing biotechnology, vaccine design, and…

Understanding the decision-making processes of large language models is critical given their widespread applications. To achieve this, we aim to connect a formal mathematical framework - zigzag persistence from topological data analysis -…

A structure-preserving kernel ridge regression method is presented that allows the recovery of nonlinear Hamiltonian functions out of datasets made of noisy observations of Hamiltonian vector fields. The method proposes a closed-form…

Convolutional neural networks (CNNs) are a standard tool for computer vision tasks such as image classification. However, typical model architectures may result in the loss of topological information. In specific domains such as…

Prediction and discovery of new materials with desired properties are at the forefront of quantum science and technology research. A major bottleneck in this field is the computational resources and time complexity related to finding new…

Through the use of examples, we explain one way in which applied topology has evolved since the birth of persistent homology in the early 2000s. The first applications of topology to data emphasized the global shape of a dataset, such as…

Understanding and predicting thermal transport in disordered materials remains a significant challenge due to the absence of periodicity and the complex nature of medium-range structural motifs. In this work, we investigate amorphous…

We introduce a homothetic extension of classical Weyl integrable geometry by generalizing the usual linear gauge transformations to affine homothetic transformations centered at a distinguished harmonic, scale-invariant form $\alpha_d$.…

In this paper, we consider the harmonic extension problem, which is widely used in many applications of machine learning. We find that the transitional method of graph Laplacian fails to produce a good approximation of the classical…

Learning task-specific representations of persistence diagrams is an important problem in topological data analysis and machine learning. However, current state of the art methods are restricted in terms of their expressivity as they are…

Measuring the similarity between data points often requires domain knowledge, which can in parts be compensated by relying on unsupervised methods such as latent-variable models, where similarity/distance is estimated in a more compact…

Computational topologists recently developed a method, called persistent homology to analyze data presented in terms of similarity or dissimilarity. Indeed, persistent homology studies the evolution of topological features in terms of a…

Vectorization methods for \emph{Persistent Homology} (PH), such as the \emph{Persistence Image} (PI), encode persistence diagrams into finite dimensional vector spaces while preserving stability. In parallel, the \emph{Persistent Laplacian}…

Finding an optimal parameter of a black-box function is important for searching stable material structures and finding optimal neural network structures, and Bayesian optimization algorithms are widely used for the purpose. However, most of…

Laplacian-based methods are popular for the dimensionality reduction of data lying in $\mathbb{R}^N$. Several theoretical results for these algorithms depend on the fact that the Euclidean distance locally approximates the geodesic distance…

Deep learning methods have demonstrated outstanding performances on classification and regression tasks on homogeneous data types (e.g., image, audio, and text data). However, tabular data still pose a challenge, with classic machine…

Latent space geometry provides a rigorous and empirically valuable framework for interacting with the latent variables of deep generative models. This approach reinterprets Euclidean latent spaces as Riemannian through a pull-back metric,…

Persistent homology is a branch of computational algebraic topology that studies shapes and extracts features over multiple scales. In this paper, we present an unsupervised approach that uses persistent homology to study divergent behavior…