Related papers: A Nonlinear Approach to Dimension Reduction
Deep neural networks have proved very successful on archetypal tasks for which large training sets are available, but when the training data are scarce, their performance suffers from overfitting. Many existing methods of reducing…
We give two different and simple constructions for dimensionality reduction in $\ell_2$ via linear mappings that are sparse: only an $O(\varepsilon)$-fraction of entries in each column of our embedding matrices are non-zero to achieve…
Johnson--Lindenstrauss Transforms are powerful tools for reducing the dimensionality of data while preserving key characteristics of that data, and they have found use in many fields from machine learning to differential privacy and more.…
We consider the problem of nonlinear dimensionality reduction: given a training set of high-dimensional data whose ``intrinsic'' low dimension is assumed known, find a feature extraction map to low-dimensional space, a reconstruction map…
The Johnson-Lindenstrauss lemma allows dimension reduction on real vectors with low distortion on their pairwise Euclidean distances. This result is often used in algorithms such as $k$-means or $k$ nearest neighbours since they only use…
Nitsche's method is a standard device for weakly imposing Dirichlet boundary conditions, but for the stabilized nonsymmetric formulation the available $L^2$-error analysis for Poisson's equation still predicts a half-order loss, whereas…
In 1984, Johnson and Lindenstrauss proved that any finite set of data in a high-dimensional space can be projected to a lower-dimensional space while preserving the pairwise Euclidean distance between points up to a bounded relative error.…
We devise a new embedding technique, which we call measured descent, based on decomposing a metric space locally, at varying speeds, according to the density of some probability measure. This provides a refined and unified framework for the…
This paper investigates the double descent phenomenon in two-layer neural networks, focusing on the role of L1 regularization and representation dimensions. It explores an alternative double descent phenomenon, named sparse double descent.…
We discuss mathematical methods to derive Nonlinear Schr\"odinger equations (NLS) in "low dimensional" settings, i.e. the 3-dimensional physical space e.g. to 2 or 1 space dimensions. Beside from the case the system exhibits an internal…
For a set $X$ of $N$ points in $\mathbb{R}^D$, the Johnson-Lindenstrauss lemma provides random linear maps that approximately preserve all pairwise distances in $X$ -- up to multiplicative error $(1\pm \epsilon)$ with high probability --…
Inspired by the numerical immersed boundary method, we introduce regularized Stokes immersed boundary problems in two dimensions to describe regularized motion of a 1-D closed elastic string in a 2-D Stokes flow, in which a regularized…
A novel perturbative method, proposed by Panda {\it et al.} [1] to solve the Helmholtz equation in two dimensions, is extended to three dimensions for general boundary surfaces. Although a few numerical works are available in the literature…
In recent years, manifold learning has become increasingly popular as a tool for performing non-linear dimensionality reduction. This has led to the development of numerous algorithms of varying degrees of complexity that aim to recover man…
For a given metric measure space $(X,d,\mu)$ we consider finite samples of points, calculate the matrix of distances between them and then reconstruct the points in some finite-dimensional space using the multidimensional scaling (MDS)…
The Johnson-Lindenstrauss Lemma states that there exist linear maps that project a set of points of a vector space into a space of much lower dimension such that the Euclidean distance between these points is approximately preserved. This…
We developed a Nonlinear Level-set Learning (NLL) method for dimensionality reduction in high-dimensional function approximation with small data. This work is motivated by a variety of design tasks in real-world engineering applications,…
In this paper, we consider robust optimization problems in high dimensions. Because a real-world dataset may contain significant noise or even specially crafted samples from some attacker, we are particularly interested in the optimization…
Recent advances show that two-dimensional linear discriminant analysis (2DLDA) is a successful matrix based dimensionality reduction method. However, 2DLDA may encounter the singularity issue theoretically and the sensitivity to outliers.…
We develop theory for nonlinear dimensionality reduction (NLDR). A number of NLDR methods have been developed, but there is limited understanding of how these methods work and the relationships between them. There is limited basis for using…