Related papers: From low to high-dimensional moments without magic
A set of orthonormal polynomials is proposed for image reconstruction from projection data. The relationship between the projection moments and image moments is discussed in detail, and some interesting properties are demonstrated.…
An algorithm to efficiently compute the moments of volumetric images is disclosed. The approach demonstrates a reduction in processing time by reducing the computational complexity significantly. Specifically, the algorithm reduces…
Random projection has been widely used in data classification. It maps high-dimensional data into a low-dimensional subspace in order to reduce the computational cost in solving the related optimization problem. While previous studies are…
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…
Random Projection is a foundational research topic that connects a bunch of machine learning algorithms under a similar mathematical basis. It is used to reduce the dimensionality of the dataset by projecting the data points efficiently to…
This paper considers the approximate reconstruction of points, x \in R^D, which are close to a given compact d-dimensional submanifold, M, of R^D using a small number of linear measurements of x. In particular, it is shown that a number of…
We introduce visual deprojection: the task of recovering an image or video that has been collapsed along a dimension. Projections arise in various contexts, such as long-exposure photography, where a dynamic scene is collapsed in time to…
High resolution reconstruction of complicated objects from incomplete and noisy data can be achieved by solving modulation equations iteratively under physical constraints. This direct demodulation method is a powerful technique for dealing…
In many linear inverse problems, we want to estimate an unknown vector belonging to a high-dimensional (or infinite-dimensional) space from few linear measurements. To overcome the ill-posed nature of such problems, we use a low-dimension…
We consider the problem of recovering elements of a low-dimensional model from linear measurements. From signal and image processing to inverse problems in data science, this question has been at the center of many applications. Lately,…
This paper, broadly speaking, covers the use of randomness in two main areas: low-rank approximation and kernel methods. Low-rank approximation is very important in numerical linear algebra. Many applications depend on matrix decomposition…
We describe an algorithm that, given any full-rank matrix A having fewer rows than columns, can rapidly compute the orthogonal projection of any vector onto the null space of A, as well as the orthogonal projection onto the row space of A,…
Fitting linear regression models can be computationally very expensive in large-scale data analysis tasks if the sample size and the number of variables are very large. Random projections are extensively used as a dimension reduction tool…
The random sampling task performed by Google's Sycamore processor gave us a glimpse of the "Quantum Supremacy era". This has definitely shed some spotlight on the power of random quantum circuits in this abstract task of sampling outputs…
We introduce sparse random projection, an important dimension-reduction tool from machine learning, for the estimation of discrete-choice models with high-dimensional choice sets. Initially, high-dimensional data are compressed into a…
In this work, we study distance metric learning (DML) for high dimensional data. A typical approach for DML with high dimensional data is to perform the dimensionality reduction first before learning the distance metric. The main…
We consider the problem of extracting a low-dimensional, linear latent variable structure from high-dimensional random variables. Specifically, we show that under mild conditions and when this structure manifests itself as a linear space…
We discuss the application of random projections to conic programming: notably linear, second-order and semidefinite programs. We prove general approximation results on feasibility and optimality using the framework of formally real Jordan…
Random projections offer an appealing and flexible approach to a wide range of large-scale statistical problems. They are particularly useful in high-dimensional settings, where we have many covariates recorded for each observation. In…
We consider a phase retrieval problem, where we want to reconstruct a $n$-dimensional vector from its phaseless scalar products with $m$ sensing vectors, independently sampled from complex normal distributions. We show that, with a suitable…