Related papers: Topologically-Informed Atlas Learning
Finding appropriate low dimensional representations of high-dimensional multi-modal data can be challenging, since each modality embodies unique deformations and interferences. In this paper, we address the problem using manifold learning,…
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…
For robots to work alongside humans and perform in unstructured environments, they must learn new motion skills and adapt them to unseen situations on the fly. This demands learning models that capture relevant motion patterns, while…
The Manifold Hypothesis is a widely accepted tenet of Machine Learning which asserts that nominally high-dimensional data are in fact concentrated near a low-dimensional manifold, embedded in high-dimensional space. This phenomenon is…
Graph embeddings, wherein the nodes of the graph are represented by points in a continuous space, are used in a broad range of Graph ML applications. The quality of such embeddings crucially depends on whether the geometry of the space…
Bayesian learning with Gaussian processes demonstrates encouraging regression and classification performances in solving computer vision tasks. However, Bayesian methods on 3D manifold-valued vision data, such as meshes and point clouds,…
We propose a method for learning embeddings for few-shot learning that is suitable for use with any number of ways and any number of shots (shot-free). Rather than fixing the class prototypes to be the Euclidean average of sample…
Recently, topological deep learning (TDL), which integrates algebraic topology with deep neural networks, has achieved tremendous success in processing point-cloud data, emerging as a promising paradigm in data science. However, TDL has not…
Convolutions are fundamental elements in deep learning architectures. Here, we present a theoretical framework for combining extrinsic and intrinsic approaches to manifold convolution through isometric embeddings into tori. In this way, we…
We propose a new method for embedding graphs while preserving directed edge information. Learning such continuous-space vector representations (or embeddings) of nodes in a graph is an important first step for using network information…
We introduce an unsupervised approach for constructing a global reference system by learning, in the ambient space, vector fields that span the tangent spaces of an unknown data manifold. In contrast to isometric objectives, which…
It is often of interest to infer lower-dimensional structure underlying complex data. As a flexible class of non-linear structures, it is common to focus on Riemannian manifolds. Most existing manifold learning algorithms replace the…
Learning faithful graph representations as sets of vertex embeddings has become a fundamental intermediary step in a wide range of machine learning applications. The quality of the embeddings is usually determined by how well the geometry…
This paper describes an approach for fitting an immersed submanifold of a finite-dimensional Euclidean space to random samples. The reconstruction mapping from the ambient space to the desired submanifold is implemented as a composition of…
Given a union of non-linear manifolds, non-linear subspace clustering or manifold clustering aims to cluster data points based on manifold structures and also learn to parameterize each manifold as a linear subspace in a feature space. Deep…
Samples from intimate (non-linear) mixtures are generally modeled as being drawn from a smooth manifold. Scenarios where the data contains multiple intimate mixtures with some constituent materials in common can be thought of as manifolds…
We propose a novel geometric approach for learning bilingual mappings given monolingual embeddings and a bilingual dictionary. Our approach decouples learning the transformation from the source language to the target language into (a)…
We propose a novel approach for preserving topological structures of the input space in latent representations of autoencoders. Using persistent homology, a technique from topological data analysis, we calculate topological signatures of…
Modern sample points in many applications no longer comprise real vectors in a real vector space but sample points of much more complex structures, which may be represented as points in a space with a certain underlying geometric structure,…
The correlation matrix is a central representation of functional brain networks in neuroimaging. Traditional analyses often treat pairwise interactions independently in a Euclidean setting, overlooking the intrinsic geometry of correlation…