Related papers: Local Averaging Accurately Distills Manifold Struc…
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 neural networks (DNNs) experience significant performance degradation when processing noisy labels, primarily due to overfitting on mislabeled data. Current mainstream approaches attempt to mitigate this issue by passively filtering…
The Gaussian kernel and its traditional normalizations (e.g., row-stochastic) are popular approaches for assessing similarities between data points. Yet, they can be inaccurate under high-dimensional noise, especially if the noise magnitude…
In the study of high-dimensional data, it is often assumed that the data set possesses an underlying lower-dimensional structure. A practical model for this structure is an embedded compact manifold with boundary. Since the underlying…
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…
3D point clouds are often perturbed by noise due to the inherent limitation of acquisition equipments, which obstructs downstream tasks such as surface reconstruction, rendering and so on. Previous works mostly infer the displacement of…
We assume that $M_0$ is a $d$-dimensional $C^{2,1}$-smooth submanifold of $R^n$. Let $K_0$ be the convex hull of $M_0,$ and $B^n_1(0)$ be the unit ball. We assume that $ M_0 \subseteq \partial K_0 \subseteq B^n_1(0).$ We also suppose that…
The manifold hypothesis suggests that high-dimensional data often lie on or near a low-dimensional manifold. Estimating the dimension of this manifold is essential for leveraging its structure, yet existing work on dimension estimation is…
Modern signal and image acquisition systems are able to capture data that is no longer real-valued, but may take values on a manifold. However, whenever measurements are taken, no matter whether manifold-valued or not, there occur tiny…
We propose a kernel-spectral embedding algorithm for learning low-dimensional nonlinear structures from high-dimensional and noisy observations, where the datasets are assumed to be sampled from an intrinsically low-dimensional manifold and…
In this paper, we consider the fundamental problem of approximation of functions on a low-dimensional manifold embedded in a high-dimensional space, with noise affecting both in the data and values of the functions. Due to the curse of…
Existing dimensionality reduction methods are adept at revealing hidden underlying manifolds arising from high-dimensional data and thereby producing a low-dimensional representation. However, the smoothness of the manifolds produced by…
Manifold learning is a central task in modern statistics and data science. Many datasets (cells, documents, images, molecules) can be represented as point clouds embedded in a high dimensional ambient space, however the degrees of freedom…
It has been shown that perturbing the input during training implicitly regularises the gradient of the learnt function, leading to smoother models and enhancing generalisation. However, previous research mostly considered the addition of…
Analyzing large volumes of high-dimensional data is an issue of fundamental importance in data science, molecular simulations and beyond. Several approaches work on the assumption that the important content of a dataset belongs to a…
Nonlinear manifold learning from unorganized data points is a very challenging unsupervised learning and data visualization problem with a great variety of applications. In this paper we present a new algorithm for manifold learning and…
In this manuscript, we propose an efficient manifold denoiser based on landmark diffusion and optimal shrinkage under the complicated high dimensional noise and compact manifold setup. It is flexible to handle several setups, including the…
Deep neural networks have proven to be highly effective when large amounts of data with clean labels are available. However, their performance degrades when training data contains noisy labels, leading to poor generalization on the test…
The problem of source localization with ad hoc microphone networks in noisy and reverberant enclosures, given a training set of prerecorded measurements, is addressed in this paper. The training set is assumed to consist of a limited number…
Inspired by edge detection based on the decay behavior of wavelet coefficients, we introduce a (near) linear-time algorithm for detecting the local regularity in non-uniformly sampled multivariate signals. Our approach quantifies regularity…