Related papers: Approximation of Points on Low-Dimensional Manifol…
The problem of finding suitable point embedding or geometric configurations given only Euclidean distance information of point pairs arises both as a core task and as a sub-problem in a variety of machine learning applications. In this…
Manifold hypothesis states that data points in high-dimensional space actually lie in close vicinity of a manifold of much lower dimension. In many cases this hypothesis was empirically verified and used to enhance unsupervised and…
We give a dimensionality reduction procedure to approximate the sum of distances of a given set of $n$ points in $R^d$ to any "shape" that lies in a $k$-dimensional subspace. Here, by "shape" we mean any set of points in $R^d$. Our…
We characterize collections of orthogonal projections for which it is possible to reconstruct a vector from the magnitudes of the corresponding projections. As a result we are able to show that in an $M$-dimensional real vector space a…
Recent medical image reconstruction techniques focus on generating high-quality medical images suitable for clinical use at the lowest possible cost and with the fewest possible adverse effects on patients. Recent works have shown…
The motion measurement of point targets constitutes a fundamental problem in photogrammetry, with extensive applications across various engineering domains. Reconstructing a point's 3D motion just from the images captured by only a…
Let $X$ be a $d$-dimensional random vector and $X_\theta$ its projection onto the span of a set of orthonormal vectors $\{\theta_1,...,\theta_k\}$. Conditions on the distribution of $X$ are given such that if $\theta$ is chosen according to…
Reconstructing a 3D shape based on a single sketch image is challenging due to the large domain gap between a sparse, irregular sketch and a regular, dense 3D shape. Existing works try to employ the global feature extracted from sketch to…
For the space of functions that can be approximated by linear chirps, we prove a reconstruction theorem by random sampling at arbitrary rates.
High-dimensional data analysis has been an active area, and the main focuses have been variable selection and dimension reduction. In practice, it occurs often that the variables are located on an unknown, lower-dimensional nonlinear…
This paper deals with the reconstruction of a discrete measure $\gamma_Z$ on $\mathbb{R}^d$ from the knowledge of its pushforward measures $P_i\#\gamma_Z$ by linear applications $P_i: \mathbb{R}^d \rightarrow \mathbb{R}^{d_i}$ (for instance…
This paper proposes a new method which builds a simplex based approximation of a $d-1$-dimensional manifold $M$ separating a $d$-dimensional compact set into two parts, and an efficient algorithm classifying points according to this…
Exact reconstruction of an image from measurements of its Discrete Fourier Transform (DFT) typically requires all DFT coefficients to be available. However, incorporating the prior assumption that the image contains only integer values…
Microstructure reconstruction is an important cornerstone to the inverse materials design concept. In this work, a general algorithm is developed to reconstruct a three-dimensional microstructure from given descriptors. Based on…
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 paper, we propose a novel lower dimensional representation of a shape sequence. The proposed dimension reduction is invertible and computationally more efficient in comparison to other related works. Theoretically, the differential…
We present an efficient and automatic approach for accurate reconstruction of instances of big 3D objects from multiple, unorganized and unstructured point clouds, in presence of dynamic clutter and occlusions. In contrast to conventional…
This paper studies the statistical query (SQ) complexity of estimating $d$-dimensional submanifolds in $\mathbb{R}^n$. We propose a purely geometric algorithm called Manifold Propagation, that reduces the problem to three natural geometric…
Infinite-dimensional, holomorphic functions have been studied in detail over the last several decades, due to their relevance to parametric differential equations and computational uncertainty quantification. The approximation of such…
Phase retrieval is a nonlinear inverse problem that arises in a wide range of imaging modalities, from electron microscopy to Fourier ptychography. In particular, the reconstruction is facilitated when the sensing matrix is i.i.d. random,…