Related papers: Manifold Approximation by Moving Least-Squares Pro…
Manifold learning offers nonlinear dimensionality reduction of high-dimensional datasets. In this paper, we bring geometry processing to bear on manifold learning by introducing a new approach based on metric connection for generating a…
Manifold fitting aims to reconstruct a low-dimensional manifold from high-dimensional data, whose framework is established by Fefferman et al. \cite{fefferman2020reconstruction,fefferman2021reconstruction}. This paper studies the recovery…
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…
We present an approach to analyze $C^1(\mathbb{R}^m)$ functions that addresses limitations present in the Active Subspaces (AS) method of Constantine et al.(2015; 2014). Under appropriate hypotheses, our Active Manifolds (AM) method…
Manifold learning using deep neural networks been shown to be an effective tool for building sophisticated prior image models that can be applied to noise reduction in low-dose CT. We propose a new iterative CT reconstruction algorithm,…
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…
We propose a manifold matching approach to generative models which includes a distribution generator (or data generator) and a metric generator. In our framework, we view the real data set as some manifold embedded in a high-dimensional…
We are interested in derivative-free optimization of high-dimensional functions. The sample complexity of existing methods is high and depends on problem dimensionality, unlike the dimensionality-independent rates of first-order methods.…
When performing classification tasks, raw high dimensional features often contain redundant information, and lead to increased computational complexity and overfitting. In this paper, we assume the data samples lie on a single underlying…
Multidimensional scaling (MDS) is a dimensionality reduction tool used for information analysis, data visualization and manifold learning. Most MDS procedures embed data points in low-dimensional Euclidean (flat) domains, such that…
Optimization over the Stiefel manifold is a fundamental computational problem in many scientific and engineering applications. Despite considerable research effort, high-dimensional optimization problems over the Stiefel manifold remain…
This paper introduces a new method for semi-supervised learning on high dimensional nonlinear manifolds, which includes a phase of unsupervised basis learning and a phase of supervised function learning. The learned bases provide a set of…
Many algorithms in machine learning and computational geometry require, as input, the intrinsic dimension of the manifold that supports the probability distribution of the data. This parameter is rarely known and therefore has to be…
System identification has greatly benefited from deep learning techniques, particularly for modeling complex, nonlinear dynamical systems with partially unknown physics where traditional approaches may not be feasible. However, deep…
High-dimensional datasets are well-approximated by low-dimensional structures. Over the past decade, this empirical observation motivated the investigation of detection, measurement, and modeling techniques to exploit these low-dimensional…
We introduce Joint Multidimensional Scaling, a novel approach for unsupervised manifold alignment, which maps datasets from two different domains, without any known correspondences between data instances across the datasets, to a common…
The general aim of manifold estimation is reconstructing, by statistical methods, an $m$-dimensional compact manifold $S$ on ${\mathbb R}^d$ (with $m\leq d$) or estimating some relevant quantities related to the geometric properties of $S$.…
Modern machine learning increasingly leverages the insight that high-dimensional data often lie near low-dimensional, non-linear manifolds, an idea known as the manifold hypothesis. By explicitly modeling the geometric structure of data…
This article introduces a new data-driven approach that leverages a manifold embedding generated by the invertible neural network to improve the robustness, efficiency, and accuracy of the constitutive-law-free simulations with limited…
Modern information processing relies on the axiom that high-dimensional data lie near low-dimensional geometric structures. This paper revisits the problem of data-driven learning of these geometric structures and puts forth two new…