Related papers: A Super-Resolution Framework for Tensor Decomposit…
We study the problem of recovering an atomic measure on the unit 2-sphere $\mathbb{S}^2$ given finitely many moments with respect to spherical harmonics. The analysis relies on the formulation of this problem as an optimization problem on…
The development of energy selective, photon counting X-ray detectors allows for a wide range of new possibilities in the area of computed tomographic image formation. Under the assumption of perfect energy resolution, here we propose a…
Atomic norm methods have recently been proposed for spectral super-resolution with flexibility in dealing with missing data and miscellaneous noises. A notorious drawback of these convex optimization methods however is their lower…
This paper develops a mathematical theory of super-resolution. Broadly speaking, super-resolution is the problem of recovering the fine details of an object---the high end of its spectrum---from coarse scale information only---from samples…
Convex optimization recently emerges as a compelling framework for performing super resolution, garnering significant attention from multiple communities spanning signal processing, applied mathematics, and optimization. This article offers…
We prove a generalization to Jennrich's uniqueness theorem for tensor decompositions in the undercomplete setting. Our uniqueness theorem is based on an alternative definition of the standard tensor decomposition, which we call…
Tensor completion can estimate missing values of a high-order data from its partially observed entries. Recent works show that low rank tensor ring approximation is one of the most powerful tools to solve tensor completion problem. However,…
The matrix spectral and nuclear norms appear in enormous applications. The generalizations of these norms to higher-order tensors is becoming increasingly important but unfortunately they are NP-hard to compute or even approximate. Although…
This work considers the problem of super-resolution. The goal is to resolve a Dirac distribution from knowledge of its discrete, low-pass, Fourier measurements. Classically, such problems have been dealt with parameter estimation methods.…
This chapter studies the problem of decomposing a tensor into a sum of constituent rank one tensors. While tensor decompositions are very useful in designing learning algorithms and data analysis, they are NP-hard in the worst-case. We will…
This work studies the combinatorial optimization problem of finding an optimal core tensor shape, also called multilinear rank, for a size-constrained Tucker decomposition. We give an algorithm with provable approximation guarantees for its…
Symmetric tensor decomposition is an important problem with applications in several areas for example signal processing, statistics, data analysis and computational neuroscience. It is equivalent to Waring's problem for homogeneous…
We consider imaging of two partially coherent sources and derive the ultimate quantum limits for estimating the separation, location, relative intensity, and coherence factor. We show that super-resolution in the separation is achievable…
We consider the problem of solving mixed random linear equations with $k$ components. This is the noiseless setting of mixed linear regression. The goal is to estimate multiple linear models from mixed samples in the case where the labels…
Hyperspectral super-resolution refers to the problem of fusing a hyperspectral image (HSI) and a multispectral image (MSI) to produce a super-resolution image (SRI) that has fine spatial and spectral resolution. State-of-the-art methods…
We develop fast spectral algorithms for tensor decomposition that match the robustness guarantees of the best known polynomial-time algorithms for this problem based on the sum-of-squares (SOS) semidefinite programming hierarchy. Our…
Higher-order tensors have received increased attention across science and engineering. While most tensor decomposition methods are developed for a single tensor observation, scientific studies often collect side information, in the form of…
In this work, we study the tensor ring decomposition and its associated numerical algorithms. We establish a sharp transition of algorithmic difficulty of the optimization problem as the bond dimension increases: On one hand, we show the…
The popular Alternating Least Squares (ALS) algorithm for tensor decomposition is efficient and easy to implement, but often converges to poor local optima---particularly when the weights of the factors are non-uniform. We propose a…
We consider the problem of automatically decomposing operations over tensors or arrays so that they can be executed in parallel on multiple devices. We address two, closely-linked questions. First, what programming abstraction should…