Related papers: Learning 3-Manifold Triangulations
Understanding low-dimensional structures within high-dimensional data is crucial for visualization, interpretation, and denoising in complex datasets. Despite the advancements in manifold learning techniques, key challenges-such as limited…
Supervised manifold learning methods learn data representations by preserving the geometric structure of data while enhancing the separation between data samples from different classes. In this work, we propose a theoretical study of…
We introduce the notion of round surgery diagrams in $S^3$ for representing 3-manifolds similar to Dehn surgery diagrams. We give a correspondence between a certain class of round surgery diagrams and Dehn surgery diagrams for 3-manifolds.…
3D scanning as a technique to digitize objects in reality and create their 3D models, is used in many fields and areas. Though the quality of 3D scans depends on the technical characteristics of the 3D scanner, the common drawback is the…
We consider the natural problem of counting isotopy classes of essential surfaces in 3-manifolds, focusing on closed essential surfaces in a broad class of hyperbolic 3-manifolds. Our main result is that the count of (possibly disconnected)…
Signed graphs are powerful models for representing complex relations with both positive and negative connections. Recently, Signed Graph Neural Networks (SGNNs) have emerged as potent tools for analyzing such graphs. To our knowledge, no…
Semi-supervised learning algorithms typically construct a weighted graph of data points to represent a manifold. However, an explicit graph representation is problematic for neural networks operating in the online setting. Here, we propose…
Sign language is a gesture-based symbolic communication medium among speech and hearing impaired people. It also serves as a communication bridge between non-impaired and impaired populations. Unfortunately, in most situations, a…
Most attempts to represent 3D shapes for deep learning have focused on volumetric grids, multi-view images and point clouds. In this paper we look at the most popular representation of 3D shapes in computer graphics - a triangular mesh -…
Meier and Zupan showed that every surface in the four-sphere admits a bridge trisection and can therefore be represented by three simple tangles. This raises the possibility of applying methods from link homology to knotted surfaces. We use…
Heegaard splittings provide a natural representation of closed 3-manifolds by gluing two handlebodies along a common surface. These splittings can be equivalently given by two finite sets of meridians lying on the surface, which define a…
Statistical shape modeling is an important tool to characterize variation in anatomical morphology. Typical shapes of interest are measured using 3D imaging and a subsequent pipeline of registration, segmentation, and some extraction of…
3D printing of surfaces has become an established method for prototyping and visualisation. However, surfaces often contain certain degenerations, such as self-intersecting faces or non-manifold parts, which pose problems in obtaining a 3D…
Graph Neural Networks (GNNs) have gained popularity in various learning tasks, with successful applications in fields like molecular biology, transportation systems, and electrical grids. These fields naturally use graph data, benefiting…
Autoencoding is a popular method in representation learning. Conventional autoencoders employ symmetric encoding-decoding procedures and a simple Euclidean latent space to detect hidden low-dimensional structures in an unsupervised way.…
Deep convolutional neural networks (CNNs) are state-of-the-art for semantic image segmentation, but typically require many labeled training samples. Obtaining 3D segmentations of medical images for supervised training is difficult and labor…
Neural networks have been shown to have a remarkable ability to uncover low dimensional structure in data: the space of possible reconstructed images form a reduced model manifold in image space. We explore this idea directly by analyzing…
It is well known that a three dimensional (closed, connected and compact) manifold is obtained by identifying boundary faces from a polyhedron P. The study of (\partial P)/~, the boundary \partial P with the polygonal faces identified in…
Nonlinear manifold learning from unorganized data points is a very challenging unsupervised learning and data visualization problem with a great variety of applications. In this paper we present a new algorithm for manifold learning and…
Manifold hypotheses are typically used for tasks such as dimensionality reduction, interpolation, or improving classification performance. In the less common problem of manifold estimation, the task is to characterize the geometric…