Related papers: Approximation of Points on Low-Dimensional Manifol…

We study the approximate nearest neighbour method for cost-sensitive classification on low-dimensional manifolds embedded within a high-dimensional feature space. We determine the minimax learning rates for distributions on a smooth…

We consider the problem of efficiently approximating and encoding high-dimensional data sampled from a probability distribution $\rho$ in $\mathbb{R}^D$, that is nearly supported on a $d$-dimensional set $\mathcal{M}$ - for example…

From a sufficiently large point sample lying on a compact Riemannian submanifold of Euclidean space, one can construct a simplicial complex which is homotopy-equivalent to that manifold with high confidence. We describe a corresponding…

In this paper we study the reconstruction of moving object densities from undersampled dynamic X-ray tomography in two dimensions. A particular motivation of this study is to use realistic measurement protocols for practical applications,…

The choice of data representation is a key factor in the success of deep learning in geometric tasks. For instance, DUSt3R recently introduced the concept of viewpoint-invariant point maps, generalizing depth prediction and showing that all…

Let $\mathbf{P}=\{ p_1, p_2, \ldots p_n \}$ and $\mathbf{Q} = \{ q_1, q_2 \ldots q_m \}$ be two point sets in an arbitrary metric space. Let $\mathbf{A}$ represent the $m\times n$ pairwise distance matrix with $\mathbf{A}_{i,j} = d(p_i,…

A common observation in data-driven applications is that high dimensional data has a low intrinsic dimension, at least locally. In this work, we consider the problem of estimating a $d$ dimensional sub-manifold of $\mathbb{R}^D$ from a…

Curve reconstruction from unstructured points in a plane is a fundamental problem with many applications that has generated research interest for decades. Involved aspects like handling open, sharp, multiple and non-manifold outlines,…

We start by considering the problem of estimating intrinsic distances on a smooth submanifold. We show that minimax optimality can be obtained via a reconstruction of the surface, and discuss the use of a particular mesh construction -- the…

We give a polynomial time algorithm to decode multivariate polynomial codes of degree $d$ up to half their minimum distance, when the evaluation points are an arbitrary product set $S^m$, for every $d < |S|$. Previously known algorithms can…

Real world data often exhibit low-dimensional geometric structures, and can be viewed as samples near a low-dimensional manifold. This paper studies nonparametric regression of H\"{o}lder functions on low-dimensional manifolds using deep…

We establish the fundamental limits of lossless analog compression by considering the recovery of arbitrary m-dimensional real random vectors x from the noiseless linear measurements y=Ax with n x m measurement matrix A. Our theory is…

We investigate the classes of functions whose minimization diagrams can be approximated efficiently in \Re^d. We present a general framework and a data-structure that can be used to approximate the minimization diagram of such functions.…

This paper introduces an efficient multi-linear nonparametric (kernel-based) approximation framework for data regression and imputation, and its application to dynamic magnetic-resonance imaging (dMRI). Data features are assumed to reside…

Reduced model spaces, such as reduced basis and polynomial chaos, are linear spaces $V_n$ of finite dimension $n$ which are designed for the efficient approximation of families parametrized PDEs in a Hilbert space $V$. The manifold…

Supervised 3D reconstruction has witnessed a significant progress through the use of deep neural networks. However, this increase in performance requires large scale annotations of 2D/3D data. In this paper, we explore inexpensive 2D…

This paper is concerned with the question of reconstructing a vector in a finite-dimensional complex Hilbert space when only the magnitudes of the coefficients of the vector under a redundant linear map are known. We present new…

In the usual trace reconstruction problem, the goal is to exactly reconstruct an unknown string of length $n$ after it passes through a deletion channel many times independently, producing a set of traces (i.e., random subsequences of the…

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…

Traditional sampling schemes often assume that the sampling locations are known. Motivated by the recent bioimaging technique known as cryogenic electron microscopy (cryoEM), we consider the problem of reconstructing an unknown 3D structure…