Related papers: The Mathematical Foundations of Manifold Learning
Deep neural networks excel at learning the training data, but often provide incorrect and confident predictions when evaluated on slightly different test examples. This includes distribution shifts, outliers, and adversarial examples. To…
Real world data often lie on low-dimensional Riemannian manifolds embedded in high-dimensional spaces. This motivates learning degenerate normalizing flows that map between the ambient space and a low-dimensional latent space. However, if…
One of the prevailing trends in the machine- and deep-learning community is to gravitate towards the use of increasingly larger models in order to keep pushing the state-of-the-art performance envelope. This tendency makes access to the…
A neural population responding to multiple appearances of a single object defines a manifold in the neural response space. The ability to classify such manifolds is of interest, as object recognition and other computational tasks require a…
Based on the manifold hypothesis, real-world data often lie on a low-dimensional manifold, while normalizing flows as a likelihood-based generative model are incapable of finding this manifold due to their structural constraints. So, one…
Data living on manifolds commonly appear in many applications. Often this results from an inherently latent low-dimensional system being observed through higher dimensional measurements. We show that under certain conditions, it is possible…
Smoothness and low dimensional structures play central roles in improving generalization and stability in learning and statistics. This work combines techniques from semi-infinite constrained learning and manifold regularization to learn…
The hyperbolic manifold is a smooth manifold of negative constant curvature. While the hyperbolic manifold is well-studied in the literature, it has gained interest in the machine learning and natural language processing communities lately…
The last decade has witnessed an experimental revolution in data science and machine learning, epitomised by deep learning methods. Indeed, many high-dimensional learning tasks previously thought to be beyond reach -- such as computer…
We study the multiple manifold problem, a binary classification task modeled on applications in machine vision, in which a deep fully-connected neural network is trained to separate two low-dimensional submanifolds of the unit sphere. We…
Deep learning has had tremendous success at learning low-dimensional representations of high-dimensional data. This success would be impossible if there was no hidden low-dimensional structure in data of interest; this existence is posited…
Manifold learning is a hot research topic in the field of computer science and has many applications in the real world. A main drawback of manifold learning methods is, however, that there is no explicit mappings from the input data…
Few-shot learning algorithms aim to learn model parameters capable of adapting to unseen classes with the help of only a few labeled examples. A recent regularization technique - Manifold Mixup focuses on learning a general-purpose…
Manifold learning is a fundamental task at the core of data analysis and visualisation. It aims to capture the simple underlying structure of complex high-dimensional data by preserving pairwise dissimilarities in low-dimensional…
Manifold Learning is a class of algorithms seeking a low-dimensional non-linear representation of high-dimensional data. Thus manifold learning algorithms are, at least in theory, most applicable to high-dimensional data and sample sizes to…
Many approaches in the field of machine learning and data analysis rely on the assumption that the observed data lies on lower-dimensional manifolds. This assumption has been verified empirically for many real data sets. To make use of this…
Research on manifold learning within a density ridge estimation framework has shown great potential in recent work for both estimation and de-noising of manifolds, building on the intuitive and well-defined notion of principal curves and…
In order to avoid the curse of dimensionality, frequently encountered in Big Data analysis, there was a vast development in the field of linear and nonlinear dimension reduction techniques in recent years. These techniques (sometimes…
A theory of graded manifolds can be viewed as a generalization of differential geometry of smooth manifolds. It allows one to work with functions which locally depend not only on ordinary real variables, but also on $\mathbb{Z}$-graded…
Matrix factorization is a popular framework for modeling low-rank data matrices. Motivated by manifold learning problems, this paper proposes a quadratic matrix factorization (QMF) framework to learn the curved manifold on which the dataset…