Related papers: Topologically-Informed Atlas Learning
While variational autoencoders have been successful in several tasks, the use of conventional priors are limited in their ability to encode the underlying structure of input data. We introduce an Encoded Prior Sliced Wasserstein AutoEncoder…
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…
Learning mappings of data on manifolds is an important topic in contemporary machine learning, with applications in astrophysics, geophysics, statistical physics, medical diagnosis, biochemistry, 3D object analysis. This paper studies the…
Many scientific fields study data with an underlying structure that is a non-Euclidean space. Some examples include social networks in computational social sciences, sensor networks in communications, functional networks in brain imaging,…
Topological data analysis (TDA) provides insight into data shape. The summaries obtained by these methods are principled global descriptions of multi-dimensional data whilst exhibiting stable properties such as robustness to deformation and…
We address the problem of estimating topological features from data in high dimensional Euclidean spaces under the manifold assumption. Our approach is based on the computation of persistent homology of the space of data points endowed with…
Manifold reconstruction has been extensively studied for the last decade or so, especially in two and three dimensions. Recently, significant improvements were made in higher dimensions, leading to new methods to reconstruct large classes…
The Euclidean scattering transform was introduced nearly a decade ago to improve the mathematical understanding of the success of convolutional neural networks (ConvNets) in image data analysis and other tasks. Inspired by recent interest…
Euclidean representations distort data with intrinsic non-Euclidean structure. While Riemannian representation learning offers a solution by embedding data onto matching manifolds, it typically relies on an encoder to estimate densities on…
Embedding is a common technique for analyzing multi-dimensional data. However, the embedding projection cannot always form significant and interpretable visual structures that foreshadow underlying data patterns. We propose an approach that…
Manifold learning flows are a class of generative modelling techniques that assume a low-dimensional manifold description of the data. The embedding of such a manifold into the high-dimensional space of the data is achieved via learnable…
The manifold hypothesis presumes that high-dimensional data lies on or near a low-dimensional manifold. While the utility of encoding geometric structure has been demonstrated empirically, rigorous analysis of its impact on the learnability…
We consider the problem of learning a manifold from a teacher's demonstration. Extending existing approaches of learning from randomly sampled data points, we consider contexts where data may be chosen by a teacher. We analyze learning from…
Geometric representation learning in preserving the intrinsic geometric and topological properties for discrete non-Euclidean data is crucial in scientific applications. Previous research generally mapped non-Euclidean discrete data into…
This work proposes an algorithm for explicitly constructing a pair of neural networks that linearize and reconstruct an embedded submanifold, from finite samples of this manifold. Our such-generated neural networks, called Flattening…
Embedding high-dimensional data into a low-dimensional space is an indispensable component of data analysis. In numerous applications, it is necessary to align and jointly embed multiple datasets from different studies or experimental…
Consider a set of $n$ data points in the Euclidean space $\mathbb{R}^d$. This set is called dataset in machine learning and data science. Manifold hypothesis states that the dataset lies on a low-dimensional submanifold with high…
A common belief in high-dimensional data analysis is that data are concentrated on a low-dimensional manifold. This motivates simultaneous dimension reduction and regression on manifolds. We provide an algorithm for learning gradients on…
Deep learning has achieved a remarkable performance breakthrough in several fields, most notably in speech recognition, natural language processing, and computer vision. In particular, convolutional neural network (CNN) architectures…
Deep learning can extract rich data representations if provided sufficient quantities of labeled training data. For many tasks however, annotating data has significant costs in terms of time and money, owing to the high standards of subject…