Related papers: Dimension reduction, exact recovery, and error est…
Dimension reduction is widely regarded as an effective way for decreasing the computation, storage and communication loads of data-driven intelligent systems, leading to a growing demand for statistical methods that allow analysis (e.g.,…
Consider the problem of reconstructing a multidimensional signal from an underdetermined set of measurements, as in the setting of compressed sensing. Without any additional assumptions, this problem is ill-posed. However, for signals such…
Many statistical $M$-estimators are based on convex optimization problems formed by the combination of a data-dependent loss function with a norm-based regularizer. We analyze the convergence rates of projected gradient and composite…
Recovering continuous-time dynamics from discrete observations is difficult because local supervision (e.g., pointwise regression targets, derivative approximations, or equation residuals) loses fidelity as the observation interval grows.…
This article presents two novel adaptive-sparse polynomial dimensional decomposition (PDD) methods for solving high-dimensional uncertainty quantification problems in computational science and engineering. The methods entail global…
In recent years, large high-dimensional data sets have become commonplace in a wide range of applications in science and commerce. Techniques for dimension reduction are of primary concern in statistical analysis. Projection methods play an…
X-ray ptychography is one of the versatile techniques for nanometer resolution imaging. The magnitude of the diffraction patterns is recorded on a detector and the phase of the diffraction patterns is estimated using phase retrieval…
Identifying a low-dimensional informed parameter subspace offers a viable path to alleviating the dimensionality challenge in the sampled-based solution to large-scale Bayesian inverse problems. This paper introduces a novel gradient-based…
Given a set of points \F in a high dimensional space, the problem of finding a union of subspaces \cup_i V_i\subset \R^N that best explains the data \F increases dramatically with the dimension of \R^N. In this article, we study a class of…
We provide here a framework to analyze the phase transition phenomenon of slice inverse regression (SIR), a supervised dimension reduction technique introduced by \cite{Li:1991}. Under mild conditions, the asymptotic ratio $\rho= \lim p/n$…
Complex, oscillatory data arises from a large variety of biological, physical, and social systems. However, the inherent oscillation and ubiquitous noise pose great challenges to current methodology such as linear and nonlinear time series…
This paper considers the recovery of continuous signals in infinite dimensional spaces from the magnitude of their frequency samples. It proposes a sampling scheme which involves a combination of oversampling and modulations with complex…
The Fourier inversion of phased coherent diffraction patterns offers images without the resolution and depth-of-focus limitations of lens-based tomographic systems. We report on our recent experimental images inverted using recent…
Multidimensional scaling is an important dimension reduction tool in statistics and machine learning. Yet few theoretical results characterizing its statistical performance exist, not to mention any in high dimensions. By considering a…
This paper introduces a new unsupervised method for dimensionality reduction via regression (DRR). The algorithm belongs to the family of invertible transforms that generalize Principal Component Analysis (PCA) by using curvilinear instead…
In this work, we revisit fast dimension reduction approaches, as with random projections and random sampling. Our goal is to summarize the data to decrease computational costs and memory footprint of subsequent analysis. Such dimension…
We present a collection of algorithms which utilize dimensional reduction to perform mesh refinement and study possibly singular solutions of time-dependent partial differential equations. The algorithms are inspired by constructions used…
We propose a multifidelity dimension reduction method to identify a low-dimensional structure present in many engineering models. The structure of interest arises when functions vary primarily on a low-dimensional subspace of the…
In this paper, we address the problem of predicting a response variable in the context of both, spatially correlated and high-dimensional data. To reduce the dimensionality of the predictor variables, we apply the sufficient dimension…
Deep learning has been widely used for solving image reconstruction tasks but its deployability has been held back due to the shortage of high-quality training data. Unsupervised learning methods, such as the deep image prior (DIP),…