Related papers: Manifold learning with arbitrary norms
We present a new technique that enables manifold learning to accurately embed data manifolds that contain holes, without discarding any topological information. Manifold learning aims to embed high dimensional data into a lower dimensional…
Using the embedded gradient vector field method (see P. Birtea, D. Comanescu, Hessian operators on constraint manifolds, J. Nonlinear Science 25, 2015), we present a general formula for the Laplace-Beltrami operator defined on a constraint…
In this paper, we present a method of embedding physics data manifolds with metric structure into lower dimensional spaces with simpler metrics, such as Euclidean and Hyperbolic spaces. We then demonstrate that it can be a powerful step in…
Manifold learning-based encoders have been playing important roles in nonlinear dimensionality reduction (NLDR) for data exploration. However, existing methods can often fail to preserve geometric, topological and/or distributional…
A general invariant manifold theorem is needed to study the topological classes of smooth dynamical systems. These classes are often invariant under renormalization. The classical invariant manifold theorem cannot be applied, because the…
Spatial networks are networks whose graph topology is constrained by their embedded spatial space. Understanding the coupled spatial-graph properties is crucial for extracting powerful representations from spatial networks. Therefore,…
The problem of recovering the configuration of points from their partial pairwise distances, referred to as the Euclidean Distance Matrix Completion (EDMC) problem, arises in a broad range of applications, including sensor network…
Learning a graph with a specific structure is essential for interpretability and identification of the relationships among data. It is well known that structured graph learning from observed samples is an NP-hard combinatorial problem. In…
In image set classification, a considerable progress has been made by representing original image sets on Grassmann manifolds. In order to extend the advantages of the Euclidean based dimensionality reduction methods to the Grassmann…
Modeling non-Euclidean data is drawing extensive attention along with the unprecedented successes of deep neural networks in diverse fields. Particularly, a symmetric positive definite matrix is being actively studied in computer vision,…
Learning representations on Grassmann manifolds is popular in quite a few visual recognition tasks. In order to enable deep learning on Grassmann manifolds, this paper proposes a deep network architecture by generalizing the Euclidean…
We study the problem of recovering a globally consistent Euclidean embedding of data, given only a local distance graph and propose a method that optimally represents these distances. The method operates solely on a neighborhood graph…
Fitting an unknown number of hyperplanes to data is a fundamental yet challenging problem in machine learning, characterized by its non-convexity, non-differentiability, and unknown model order. Existing approaches often struggle with local…
Riemannian geometry provides the fundamental framework for optimization on nonlinear spaces such as matrix manifolds, which arise in machine learning, signal processing, and robotics. While the underlying theory is classical, existing…
Non-linear manifold learning enables high-dimensional data analysis, but requires out-of-sample-extension methods to process new data points. In this paper, we propose a manifold learning algorithm based on deep learning to create an…
The problem of identifying geometric structure in data is a cornerstone of (unsupervised) learning. As a result, Geometric Representation Learning has been widely applied across scientific and engineering domains. In this work, we…
This paper studies the relationship between a graph neural network (GNN) and a manifold neural network (MNN) when the graph is constructed from a set of points sampled from the manifold, thus encoding geometric information. We consider…
Manifold learning and dimensionality reduction techniques are ubiquitous in science and engineering, but can be computationally expensive procedures when applied to large data sets or when similarities are expensive to compute. To date,…
Options have been shown to be an effective tool in reinforcement learning, facilitating improved exploration and learning. In this paper, we present an approach based on spectral graph theory and derive an algorithm that systematically…
Pretrained (language) embeddings are versatile, task-agnostic feature representations of entities, like words, that are central to many machine learning applications. These representations can be enriched through retrofitting, a class of…