Related papers: Learning 3-Manifold Triangulations
This paper introduces the geodesics of triangulated image object shapes. Both rectilinear and curvilinear triangulations of shapes are considered. The triangulation of image object shapes leads to collections of what are known as nerve…
Supervised machine learning can be used to predict properties of string geometries with previously unknown features. Using the complete intersection Calabi-Yau (CICY) threefold dataset as a theoretical laboratory for this investigation, we…
In image set classification, a considerable advance has been made by modeling the original image sets by second order statistics or linear subspace, which typically lie on the Riemannian manifold. Specifically, they are Symmetric Positive…
Achieving precise medical image segmentation is vital for effective treatment planning and accurate disease diagnosis. Traditional fully-supervised deep learning methods, though highly precise, are heavily reliant on large volumes of…
Sparsity-based representations have recently led to notable results in various visual recognition tasks. In a separate line of research, Riemannian manifolds have been shown useful for dealing with features and models that do not lie in…
Representing images and videos with Symmetric Positive Definite (SPD) matrices and considering the Riemannian geometry of the resulting space has proven beneficial for many recognition tasks. Unfortunately, computation on the Riemannian…
Despite high-dimensionality of images, the sets of images of 3D objects have long been hypothesized to form low-dimensional manifolds. What is the nature of such manifolds? How do they differ across objects and object classes? Answering…
We call a cusped hyperbolic 3-manifold tetrahedral if it can be decomposed into regular ideal tetrahedra. Following an earlier publication by three of the authors, we give a census of all tetrahedral manifolds and all of their combinatorial…
With the rapid expansion of applied 3D computational vision, shape descriptors have become increasingly important for a wide variety of applications and objects from molecules to planets. Appropriate shape descriptors are critical for…
Invariant object recognition is one of the most fundamental cognitive tasks performed by the brain. In the neural state space, different objects with stimulus variabilities are represented as different manifolds. In this geometrical…
We construct algorithms and topological invariants that allow us to distinguish the topological type of a surface, as well as functions and vector fields for their topological equivalence. In the first part (arXiv:2501.15657), we discused…
Many machine learning techniques incorporate identity-preserving transformations into their models to generalize their performance to previously unseen data. These transformations are typically selected from a set of functions that are…
We propose a learning paradigm for numerical approximation of differential invariants of planar curves. Deep neural-networks' (DNNs) universal approximation properties are utilized to estimate geometric measures. The proposed framework is…
Manifold learning techniques for nonlinear dimension reduction assume that high-dimensional feature vectors lie on a low-dimensional manifold, then attempt to exploit manifold structure to obtain useful low-dimensional Euclidean…
Analyzing large volumes of high-dimensional data requires dimensionality reduction: finding meaningful low-dimensional structures hidden in their high-dimensional observations. Such practice is needed in atomistic simulations of complex…
A method is developed here for building differentiable three-dimensional manifolds on multicube structures. This method constructs a sequence of reference metrics that determine differentiable structures on the cubic regions that serve as…
Autoencoders are commonly used in representation learning. They consist of an encoder and a decoder, which provide a straightforward way to map n-dimensional data in input space to a lower m-dimensional representation space and back. The…
Despite the widespread application of recurrent neural networks (RNNs) across a variety of tasks, a unified understanding of how RNNs solve these tasks remains elusive. In particular, it is unclear what dynamical patterns arise in trained…
Meshes are ubiquitous in visual computing and simulation, yet most existing machine learning techniques represent meshes only indirectly, e.g. as the level set of a scalar field or deformation of a template, or as a disordered triangle soup…
Dimension 4 is the first dimension in which exotic smooth manifold pairs appear -- manifolds which are topologically the same but for which there is no smooth deformation of one into the other. Whilst smooth and triangulated 4-manifolds do…