Related papers: Perturbation Analysis of Singular Values in Concat…
Modern data analysis increasingly requires identifying shared latent structure across multiple high-dimensional datasets. A commonly used model assumes that the data matrices are noisy observations of low-rank matrices with a shared…
Graph matrices are a type of matrix which has played a crucial role in analyzing the sum of squares hierarchy on average case problems. However, except for rough norm bounds, little is known about graph matrices. In this paper, we take a…
Matrix perturbation bounds (such as Weyl and Davis-Kahan) are used abundantly in many areas of mathematics and data science. Many bounds (such as the above two) involve the spectral norm of the noise matrix and are sharp in worst case…
We propose new algorithms for singular value decomposition (SVD) of very large-scale matrices based on a low-rank tensor approximation technique called the tensor train (TT) format. The proposed algorithms can compute several dominant…
Estimating the size of the modeling error is crucial for robust control. Over the years, numerous metrics have been developed to quantify the model error in a control relevant manner. One of the most important such metrics is the structured…
In this paper, we show that the SVD of a matrix can be constructed efficiently in a hierarchical approach. Our algorithm is proven to recover the singular values and left singular vectors if the rank of the input matrix $A$ is known.…
In signal processing and identification, generalized singular value decomposition (GSVD), related to a sequence of matrices in product/quotient form are essential numerical linear algebra tools. On behalf of the growing demand for efficient…
A problem of paramount importance in both pure (Restricted Invertibility problem) and applied mathematics (Feature extraction) is the one of selecting a submatrix of a given matrix, such that this submatrix has its smallest singular value…
Singular value decomposition is central to many problems in engineering and scientific fields. Several quantum algorithms have been proposed to determine the singular values and their associated singular vectors of a given matrix. Although…
Pseudospectral analysis serves as a powerful tool in matrix computation and the study of both linear and nonlinear dynamical systems. Among various numerical strategies, random sampling, especially in the form of rank-$1$ perturbations,…
Consider the problem of estimating the entries of a large matrix, when the observed entries are noisy versions of a small random fraction of the original entries. This problem has received widespread attention in recent times, especially…
Over the past decade, various matrix completion algorithms have been developed. Thresholded singular value decomposition (SVD) is a popular technique in implementing many of them. A sizable number of studies have shown its theoretical and…
Deep learning methods achieve great success recently on many computer vision problems, with image classification and object detection as the prominent examples. In spite of these practical successes, optimization of deep networks remains an…
This paper considers the problem of updating the rank-k truncated Singular Value Decomposition (SVD) of matrices subject to the addition of new rows and/or columns over time. Such matrix problems represent an important computational kernel…
Recently, the singular value decomposition (SVD) was applied to standard Gaussian ensembles of Random Matrix Theory (RMT) to determine the scale invariance in the spectral fluctuations without performing any unfolding procedure. Here, SVD…
Estimating singular subspaces from noisy matrices is a fundamental problem with wide-ranging applications across various fields. Driven by the challenges of data integration and multi-view analysis, this study focuses on estimating shared…
This paper establishes a variant of Stewart's theorem (Theorem~6.4 of Stewart, {\em SIAM Rev.}, 15:727--764, 1973) for the singular subspaces associated with the SVD of a matrix subject to perturbations. Stewart's original version uses both…
In this paper, we study the relative perturbation bounds for joint eigenvalues of commuting tuples of normal $n \times n$ matrices. Some Hoffman-Wielandt type relative perturbation bounds are proved using the Clifford algebra technique. A…
We evaluate performance of associative memory in a neural network by based on the singular value decomposition (SVD) of image data stored in the network. We consider the situation in which the original image and its highly coarse-grained…
Combined perturbation bounds are presented for eigenvalues and eigenspaces of Hermitian matrices or singular values and singular subspaces of general matrices. The bounds are derived based on the smooth decompositions and elementary…