相关论文: Near-Optimal Column-Based Matrix Reconstruction
We consider the problem of approximately reconstructing a partially-observed, approximately low-rank matrix. This problem has received much attention lately, mostly using the trace-norm as a surrogate to the rank. Here we study low-rank…
Projected gradient descent and its Riemannian variant belong to a typical class of methods for low-rank matrix estimation. This paper proposes a new Nesterov's Accelerated Riemannian Gradient algorithm by efficient orthographic retraction…
We investigate the problem of recovering a partially observed high-rank matrix whose columns obey a nonlinear structure such as a union of subspaces, an algebraic variety or grouped in clusters. The recovery problem is formulated as the…
We consider a reconstruction problem of a reduced stable positive network system with the preservation of the original interconnection structure based on an $H^2$ optimal model reduction problem with constraints. To this end, we define an…
Consider the approximate sparse recovery problem: given Ax, where A is a known m-by-n dimensional matrix and x is an unknown (approximately) sparse n-dimensional vector, recover an approximation to x. The goal is to design the matrix A such…
An arbitrary $m\times n$ Boolean matrix $M$ can be decomposed {\em exactly} as $M =U\circ V$, where $U$ (resp. $V$) is an $m\times k$ (resp. $k\times n$) Boolean matrix and $\circ$ denotes the Boolean matrix multiplication operator. We…
We address the problem of estimating a high-dimensional matrix from linear measurements, with a focus on designing optimal rank-adaptive algorithms. These algorithms infer the matrix by estimating its singular values and the corresponding…
We describe an algorithm for sampling a low-rank random matrix $Q$ that best approximates a fixed target matrix $P\in\mathbb{C}^{n\times m}$ in the following sense: $Q$ is unbiased, i.e., $\mathbb{E}[Q] = P$; $\mathsf{rank}(Q)\leq r$; and…
We present a new algorithm for finding a near optimal low-rank approximation of a matrix $A$ in $O(nnz(A))$ time. Our method is based on a recursive sampling scheme for computing a representative subset of $A$'s columns, which is then used…
This dissertation is devoted to provide advanced nonconvex nonsmooth variational models of (Magnetic Resonance Image) MRI reconstruction, efficient learnable image reconstruction algorithms and parameter training algorithms that improve the…
The topic of recovery of a structured model given a small number of linear observations has been well-studied in recent years. Examples include recovering sparse or group-sparse vectors, low-rank matrices, and the sum of sparse and low-rank…
The Nystr\"om method is a convenient heuristic method to obtain low-rank approximations to kernel matrices in nearly linear complexity. Existing studies typically use the method to approximate positive semidefinite matrices with low or…
We study the problem of recovering an incomplete $m\times n$ matrix of rank $r$ with columns arriving online over time. This is known as the problem of life-long matrix completion, and is widely applied to recommendation system, computer…
In many areas of machine learning, it becomes necessary to find the eigenvector decompositions of large matrices. We discuss two methods for reducing the computational burden of spectral decompositions: the more venerable Nystom extension…
We give some new methods for perfect reconstruction from frame and sampling erasures in finitely many steps. By bridging an erasure set we mean replacing the erased Fourier coefficients of a function with respect to a frame by appropriate…
We study the problem of residual error estimation for matrix and vector norms using a linear sketch. Such estimates can be used, for example, to quickly assess how useful a more expensive low-rank approximation computation will be. The…
CUR matrix decomposition is a randomized algorithm that can efficiently compute the low rank approximation for a given rectangle matrix. One limitation with the existing CUR algorithms is that they require an access to the full matrix A for…
We present a matrix-factorization algorithm that scales to input matrices with both huge number of rows and columns. Learned factors may be sparse or dense and/or non-negative, which makes our algorithm suitable for dictionary learning,…
Alternating Minimization is a widely used and empirically successful heuristic for matrix completion and related low-rank optimization problems. Theoretical guarantees for Alternating Minimization have been hard to come by and are still…
Suppose we are given a matrix that is formed by adding an unknown sparse matrix to an unknown low-rank matrix. Our goal is to decompose the given matrix into its sparse and low-rank components. Such a problem arises in a number of…