Tensor decomposition is a relevant topic, both for theoretical and applied mathematics, due to its interdisciplinary nature, which ranges from multilinear algebra and algebraic geometry to numerical ...analysis, algebraic statistics, quantum physics, signal processing, artificial intelligence, etc. The starting point behind the study of a decomposition relies on the idea that knowledge of elementary components of a tensor is fundamental to implement procedures that are able to understand and efficiently handle the information that a tensor encodes. Recent advances were obtained with a systematic application of geometric methods: secant varieties, symmetries of special decompositions, and an analysis of the geometry of finite sets. Thanks to new applications of theoretic results, criteria for understanding when a given decomposition is minimal or unique have been introduced or significantly improved. New types of decompositions, whose elementary blocks can be chosen in a range of different possible models (e.g., Chow decompositions or mixed decompositions), are now systematically studied and produce deeper insights into this topic. The aim of this Special Issue is to collect papers that illustrate some directions in which recent researches move, as well as to provide a wide overview of several new approaches to the problem of tensor decomposition.
Variations of Andrews-Beck type congruences Chan, Song Heng; Mao, Renrong; Osburn, Robert
Journal of mathematical analysis and applications,
03/2021, Volume:
495, Issue:
2
Journal Article
Peer reviewed
Open access
We prove three variations of recent results due to Andrews on congruences for NT(m,k,n), the total number of parts in the partitions of n with rank congruent to m modulo k. We also conjecture new ...congruences and relations for NT(m,k,n) and for a related crank-type function.
Receptor activator of nuclear factor-kappa B (RANK) ligand (RANKL) binds RANK on the surface of osteoclast precursors to trigger osteoclastogenesis. Recent studies have indicated that osteocytic ...RANKL has an important role in osteoclastogenesis during bone remodelling; however, the role of osteoblastic RANKL remains unclear. Here we show that vesicular RANK, which is secreted from the maturing osteoclasts, binds osteoblastic RANKL and promotes bone formation by triggering RANKL reverse signalling, which activates Runt-related transcription factor 2 (Runx2). The proline-rich motif in the RANKL cytoplasmic tail is required for reverse signalling, and a RANKL(Pro29Ala) point mutation reduces activation of the reverse signalling pathway. The coupling of bone resorption and formation is disrupted in RANKL(Pro29Ala) mutant mice, indicating that osteoblastic RANKL functions as a coupling signal acceptor that recognizes vesicular RANK. RANKL reverse signalling is therefore a potential pharmacological target for avoiding the reduced bone formation associated with inhibition of osteoclastogenesis.
RANKL is a TNF family member that mediates osteoclast formation, activation, and survival by activating RANK. The proresorptive effects of RANKL are prevented by binding to its soluble inhibitor ...osteoprotegerin (OPG). Recombinant human OPG‐Fc recognizes RANKL from multiple species and reduced bone resorption and increased bone volume, density, and strength in a number of rodent models of bone disease. The clinical development of OPG‐Fc was discontinued in favor of denosumab, a fully human monoclonal antibody that specifically inhibits primate RANKL. Direct binding assays showed that denosumab bound to human RANKL but not to murine RANKL, human TRAIL, or other human TNF family members. Denosumab did not suppress bone resorption in normal mice or rats but did prevent the resorptive response in mice challenged with a human RANKL fragment encoded primarily by the fifth exon of the RANKL gene. To create mice that were responsive to denosumab, knock‐in technology was used to replace exon 5 from murine RANKL with its human ortholog. The resulting “huRANKL” mice exclusively express chimeric (human/murine) RANKL that was measurable with a human RANKL assay and that maintained bone resorption at slightly reduced levels versus wildtype controls. In young huRANKL mice, denosumab and OPG‐Fc each reduced trabecular osteoclast surfaces by 95% and increased bone density and volume. In adult huRANKL mice, denosumab reduced bone resorption, increased cortical and cancellous bone mass, and improved trabecular microarchitecture. These huRANKL mice have potential utility for characterizing the activity of denosumab in a variety of murine bone disease models.
Bone‐related diseases, such as osteoporosis and rheumatoid arthritis, affect hundreds of millions of people worldwide and pose a tremendous burden to health care. By deepening our understanding of ...the molecular mechanisms of bone metabolism and bone turnover, it became possible over the past years to devise new and promising strategies for treating such diseases. In particular, three tumor necrosis factor (TNF) family molecules, the receptor activator of NF‐κB (RANK), its ligand RANKL, and the decoy receptor of RANKL, osteoprotegerin (OPG), have attracted the attention of scientists and pharmaceutical companies alike. Genetic experiments revolving around these molecules established their pivotal role as central regulators of osteoclast development and osteoclast function. RANK‐RANKL signaling not only activates a variety of downstream signaling pathways required for osteoclast development, but crosstalk with other signaling pathways also fine‐tunes bone homeostasis both in normal physiology and disease. In addition, RANKL and RANK have essential roles in lymph node formation, establishment of the thymic microenvironment, and development of a lactating mammary gland during pregnancy. Consequently, novel drugs specifically targeting RANK, RANKL, and their signaling pathways in osteoclasts are expected to revolutionize the treatment of various ailments associated with bone loss, such as arthritis, periodontal disease, cancer metastases, and osteoporosis.
Symmetric Tensors and Symmetric Tensor Rank Comon, Pierre; Golub, Gene; Lim, Lek-Heng ...
SIAM journal on matrix analysis and applications,
01/2008, Volume:
30, Issue:
3
Journal Article
Peer reviewed
Open access
A symmetric tensor is a higher order generalization of a symmetric matrix. In this paper, we study various properties of symmetric tensors in relation to a decomposition into a symmetric sum of outer ...product of vectors. A rank-1 order-$k$ tensor is the outer product of $k$ nonzero vectors. Any symmetric tensor can be decomposed into a linear combination of rank-1 tensors, each of which is symmetric or not. The rank of a symmetric tensor is the minimal number of rank-1 tensors that is necessary to reconstruct it. The symmetric rank is obtained when the constituting rank-1 tensors are imposed to be themselves symmetric. It is shown that rank and symmetric rank are equal in a number of cases and that they always exist in an algebraically closed field. We will discuss the notion of the generic symmetric rank, which, due to the work of Alexander and Hirschowitz J. Algebraic Geom., 4 (1995), pp. 201-222, is now known for any values of dimension and order. We will also show that the set of symmetric tensors of symmetric rank at most $r$ is not closed unless $r=1$.
Hyperspectral image (HSI) enjoys great advantages over more traditional image types for various applications due to the extra knowledge available. For the nonideal optical and electronic devices, HSI ...is always corrupted by various noises, such as Gaussian noise, deadlines, and stripings. The global correlation across spectrum (GCS) and nonlocal self-similarity (NSS) over space are two important characteristics for HSI. In this paper, a nonlocal low-rank regularized CANDECOMP/PARAFAC (CP) tensor decomposition (NLR-CPTD) is proposed to fully utilize these two intrinsic priors. To make the rank estimation more accurate, a new manner of rank determination for the NLR-CPTD model is proposed. The intrinsic GCS and NSS priors can be efficiently explored under the low-rank regularized CPTD to avoid tensor rank estimation bias for denoising performance. Then, the proposed HSI denoising model is performed on tensors formed by nonlocal similar patches within an HSI. The alternating direction method of multipliers-based optimization technique is designed to solve the minimum problem. Compared with state-of-the-art methods, the proposed algorithm can greatly promote the denoising performance of an HSI in various quality assessments.
The estimation of large covariance and precision matrices is fundamental in modern multivariate analysis. However, problems arise from the statistical analysis of large panel economic and financial ...data. The covariance matrix reveals marginal correlations between variables, while the precision matrix encodes conditional correlations between pairs of variables given the remaining variables. In this paper, we provide a selective review of several recent developments on the estimation of large covariance and precision matrices. We focus on two general approaches: a rank-based method and a factor-model-based method. Theories and applications of both approaches are presented. These methods are expected to be widely applicable to the analysis of economic and financial data.
Let X⊂Pr be an integral and non-degenerate variety. A “cost-function” (for the Zariski topology, the semialgebraic one, or the Euclidean one) is a semicontinuous function w:=1,+∞)∪+∞ such that w(a)=1 ...for a non-empty open subset of X. For any q∈Pr, the rank rX,w(q) of q with respect to (X,w) is the minimum of all ∑a∈Sw(a), where S is a finite subset of X spanning q. We have rX,w(q)<+∞ for all q. We discuss this definition and classify extremal cases of pairs (X,q). We give upper bounds for all rX,w(q) (twice the generic rank) not depending on w. This notion is the generalization of the case in which the cost-function w is the constant function 1. In this case, the rank is a well-studied notion that covers the tensor rank of tensors of arbitrary formats (PARAFAC or CP decomposition) and the additive decomposition of forms. We also adapt to cost-functions the rank 1 decomposition of real tensors in which we allow pairs of complex conjugate rank 1 tensors.
Abstract To treat systemic bone loss as in osteoporosis and/or focal osteolysis as in rheumatoid arthritis or periodontal disease, most approaches target the osteoclasts, the cells that resorb bone. ...Bisphosphonates are currently the most widely used antiresorptive therapies. They act by binding the mineral component of bone and interfere with the action of osteoclasts. The nitrogen-containing bisphosphonates, such as alendronate, act as inhibitors of farnesyl-pyrophosphate synthase, which leads to inhibition of the prenylation of many intracellular signaling proteins. The discovery of RANKL and the essential role of RANK signaling in osteoclast differentiation, activity and survival have led to the development of denosumab, a fully human monoclonal antibody. Denosumab acts by binding to and inhibiting RANKL, leading to the loss of osteoclasts from bone surfaces. In phase 3 clinical studies, denosumab was shown to significantly reduce vertebral, nonvertebral and hip fractures compared with placebo and increase areal BMD compared with alendronate. In this review, we suggest that the key pharmacological differences between denosumab and the bisphosphonates reside in the distribution of the drugs within bone and their effects on precursors and mature osteoclasts. This may explain differences in the degree and rapidity of reduction of bone resorption, their potential differential effects on trabecular and cortical bone, and the reversibility of their actions.
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