Related papers: Matrix estimation by Universal Singular Value Thre…
Matrix factorization (MF) has been widely used to discover the low-rank structure and to predict the missing entries of data matrix. In many real-world learning systems, the data matrix can be very high-dimensional but sparse. This poses an…
The Randomized Singular Value Decomposition (RSVD) is a widely used algorithm for efficiently computing low-rank approximations of large matrices, without the need to construct a full-blown SVD. Of interest, of course, is the approximation…
Singular value decomposition (SVD) and matrix inversion are ubiquitous in scientific computing. Both tasks are computationally demanding for large scale matrices. Existing algorithms can approximatively solve these problems with a given…
The tensor-train (TT) decomposition is widely used to compress large tensors into a more compact form by exploiting their inherent data structures. A fundamental approach for constructing the TT format is the well-known TT-SVD method, which…
We address the collective matrix completion problem of jointly recovering a collection of matrices with shared structure from partial (and potentially noisy) observations. To ensure well--posedness of the problem, we impose a joint low rank…
We consider the problem of noisy 1-bit matrix completion under an exact rank constraint on the true underlying matrix $M^*$. Instead of observing a subset of the noisy continuous-valued entries of a matrix $M^*$, we observe a subset of…
Bayesian methods for low-rank matrix completion with noise have been shown to be very efficient computationally. While the behaviour of penalized minimization methods is well understood both from the theoretical and computational points of…
Efficiently computing a subset of a correlation matrix consisting of values above a specified threshold is important to many practical applications. Real-world problems in genomics, machine learning, finance other applications can produce…
Given (orthonormal) approximations $\tilde{U}$ and $\tilde{V}$ to the left and right subspaces spanned by the leading singular vectors of a matrix $A$, we discuss methods to approximate the leading singular values of $A$ and study their…
The task of estimating a matrix given a sample of observed entries is known as the \emph{matrix completion problem}. Most works on matrix completion have focused on recovering an unknown real-valued low-rank matrix from a random sample of…
Matrix completion aims to reconstruct a data matrix based on observations of a small number of its entries. Usually in matrix completion a single matrix is considered, which can be, for example, a rating matrix in recommendation system.…
It has been observed that the performances of many high-dimensional estimation problems are universal with respect to underlying sensing (or design) matrices. Specifically, matrices with markedly different constructions seem to achieve…
Random sampling has become a critical tool in solving massive matrix problems. For linear regression, a small, manageable set of data rows can be randomly selected to approximate a tall, skinny data matrix, improving processing time…
In computer vision, the estimation of the fundamental matrix is a basic problem that has been extensively studied. The accuracy of the estimation imposes a significant influence on subsequent tasks such as the camera trajectory…
Low-rank matrix approximations are often used to help scale standard machine learning algorithms to large-scale problems. Recently, matrix coherence has been used to characterize the ability to extract global information from a subset of…
We study the problem of robust matrix completion (RMC), where the partially observed entries of an underlying low-rank matrix is corrupted by sparse noise. Existing analysis of the non-convex methods for this problem either requires the…
This work studies the Generalized Singular Value Thresholding (GSVT) operator ${\text{Prox}}_{g}^{{\sigma}}(\cdot)$, \begin{equation*} {\text{Prox}}_{g}^{{\sigma}}(B)=\arg\min\limits_{X}\sum_{i=1}^{m}g(\sigma_{i}(X)) +…
Concatenating matrices is a common technique for uncovering shared structures in data through singular value decomposition (SVD) and low-rank approximations. The fundamental question arises: How does the singular value spectrum of the…
Matrix completion is the problem of recovering a low rank matrix by observing a small fraction of its entries. A series of recent works [KOM12,JNS13,HW14] have proposed fast non-convex optimization based iterative algorithms to solve this…
We develop fixed-point algorithms for the approximation of structured matrices with rank penalties. In particular we use these fixed-point algorithms for making approximations by sums of exponentials, or frequency estimation. For the basic…