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Self-attention modules have demonstrated remarkable capabilities in capturing long-range relationships and improving the performance of point cloud tasks. However, point cloud objects are typically characterized by complex, disordered, and…
For manifold learning, it is assumed that high-dimensional sample/data points are embedded on a low-dimensional manifold. Usually, distances among samples are computed to capture an underlying data structure. Here we propose a metric…
Recent advances in image generation have led to remarkable improvements in synthesizing perspective images. However, these models still struggle with panoramic image generation due to unique challenges, including varying levels of geometric…
Topological data analysis (TDA) is a rapidly developing collection of methods for studying the shape of point cloud and other data types. One popular approach, designed to be robust to noise and outliers, is to first use a smoothing…
Understanding the topological characteristics of data is important to many areas of research. Recent work has demonstrated that synthetic 4D image-type data can be useful to train 4D convolutional neural network models to see topological…
Unfolding, in the context of high-energy particle physics, refers to the process of removing detector distortions in experimental data. The resulting unfolded measurements are straightforward to use for direct comparisons between…
This work is concerned with a representation of shapes that disentangles fine, local and possibly repeating geometry, from global, coarse structures. Achieving such disentanglement leads to two unrelated advantages: i) a significant…
Given data, deep generative models, such as variational autoencoders (VAE) and generative adversarial networks (GAN), train a lower dimensional latent representation of the data space. The linear Euclidean geometry of data space pulls back…
Since Kolmogorov proposed his phenomenological theory of hydrodynamic turbulence in 1941, the description of mechanism leading to the energy cascade and anomalous scaling remains an open problem in fluid mechanics. Soon after, in 1949…
Gaussian process regression is widely used because of its ability to provide well-calibrated uncertainty estimates and handle small or sparse datasets. However, it struggles with high-dimensional data. One possible way to scale this…
Representing 3D shape is a fundamental problem in artificial intelligence, which has numerous applications within computer vision and graphics. One avenue that has recently begun to be explored is the use of latent representations of…
We prove that the irreducible desingularization of a singularity given by the Grauert blow down of a negative holomorphic vector bundle over a compact complex manifold is unique up to isomorphism, and as an application, we show that two…
We propose to model the topology of three-dimensional (3D) continua by Yin sets, regular open semianalytic sets with bounded boundary. Our model differs from manifold-based models in that singular points of a 3D continuum, i.e., boundary…
Deep generative models like variational autoencoders approximate the intrinsic geometry of high dimensional data manifolds by learning low-dimensional latent-space variables and an embedding function. The geometric properties of these…
In machine learning, data is usually represented in a (flat) Euclidean space where distances between points are along straight lines. Researchers have recently considered more exotic (non-Euclidean) Riemannian manifolds such as hyperbolic…
Real world data often lie on low-dimensional Riemannian manifolds embedded in high-dimensional spaces. This motivates learning degenerate normalizing flows that map between the ambient space and a low-dimensional latent space. However, if…
Data-driven Riemannian geometry has emerged as a powerful tool for interpretable representation learning, offering improved efficiency in downstream tasks. Moving forward, it is crucial to balance cheap manifold mappings with efficient…
We introduce a novel computational framework for digital geometry processing, based upon the derivation of a nonlinear operator associated to the total variation functional. Such operator admits a generalized notion of spectral…
simpcomp is an extension to GAP, the well known system for computational discrete algebra. It allows the user to work with simplicial complexes. In the latest version, support for simplicial blowups and discrete normal surfaces was added,…
Learning the distribution of data on Riemannian manifolds is crucial for modeling data from non-Euclidean space, which is required by many applications in diverse scientific fields. Yet, existing generative models on manifolds suffer from…