We define the determinantal divisor rank of an integral matrix to be the number of invariant factors which equal 1. Some properties of the determinantal divisor rank are proved, which are analogous to known properties of the usual rank. These include the Frobenious inequality for the rank of a product and a relation between the rank of a submatrix of a matrix and that of its complementary subma...

We show that determining Kapranov rank of tropical matrices is not only NP-hard over any field but if Diophantine equations over the rational numbers is undecidable, determining Kapranov rank over the rational numbers is also undecidable. We prove that Kapranov rank of tropical matrices is not bounded in terms of tropical rank, answering a question of Develin, Santos, and Sturmfels [4].

Recently, fundamental conditions on the sampling patterns have been obtained for finite completability of low-rank matrices or tensors given the corresponding ranks. In this paper, we consider the scenario where the rank is not given and we aim to approximate the unknown rank based on the location of sampled entries and some given completion. We consider a number of data models, including singl...

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].. The Econometric Society is collaborating with JSTOR to di...

Andreas-Stephan ELSENHANS a and Jörg JAHNEL a a Universität Göttingen, Mathematisches Institut, Bunsenstraße 3–5, D-37073 Göttingen, Germany 1 Abstract. We construct explicit examples of K3 surfaces over Q which are of degree 2 and the geometric Picard rank of which is equal to 1. We construct, particularly, examples in the form w2 = det M where M is a symmetric (3× 3)-matrix of ternary quadrat...

Abstract We introduce a novel approach to exploit mixed precision arithmetic for low-rank approximations. Our is based on the observation that singular vectors associated with small values can be stored in lower precisions while preserving high accuracy overall. provide an explicit criterion determine which level of needed each vector. apply this block (BLR) matrices, most whose off-diagonal bl...

Abstract The denoising of 2D images through low-rank methods is a relevant topic in digital image processing. This paper proposes novel method that trains learning network to predict the optimal thresholds singular value decomposition involved images. To improve results, we apply block-matching algorithm and classify each 3D block according four parameters, which increase specificity for predic...