Geometric Structures in Tensor Representations (Final Release)

The main goal of this paper is to study the geometric structures associated with the representation of tensors in subspace based formats. To do this we use a property of the so-called minimal subspaces which allows us to describe the tensor representation by means of a rooted tree. By using the tree structure and the dimensions of the associated minimal subspaces, we introduce, in the underlying algebraic tensor space, the set of tensors in a tree-based format with either bounded or fixed tree-based rank. This class contains the Tucker format and the Hierarchical Tucker format (including the Tensor Train format). In particular, we show that the set of tensors in the tree-based format with bounded (respectively, fixed) tree-based rank of an algebraic tensor product of normed vector spaces is an analytic Banach manifold. Indeed, the manifold geometry for the set of tensors with fixed tree-based rank is induced by a fibre bundle structure and the manifold geometry for the set of tensors with bounded tree-based rank is given by a finite union of connected components. In order to describe the relationship between these manifolds and the natural ambient space, we introduce the definition of topological tensor spaces in the tree-based format. We prove under natural conditions that any tensor of the topological tensor space under consideration admits best approximations in the manifold of tensors in the tree-based format with bounded tree-based rank. In this framework, we also show that the tangent (Banach) space at a given tensor is a complemented subspace in the natural ambient tensor Banach space and hence the set of tensors in the tree-based format with bounded (respectively, fixed) tree-based rank is an immersed submanifold. This fact allows us to extend the Dirac-Frenkel variational principle in the framework of topological tensor spaces.Comment: Some errors are corrected and Lemma 3.22 is improve

Principal bundle structure of matrix manifolds

In this paper, we introduce a new geometric description of the manifolds of matrices of fixed rank. The starting point is a geometric description of the Grassmann manifold $\mathbb{G}_r(\mathbb{R}^k)$ of linear subspaces of dimension $r<k$ in $\mathbb{R}^k$ which avoids the use of equivalence classes. The set $\mathbb{G}_r(\mathbb{R}^k)$ is equipped with an atlas which provides it with the structure of an analytic manifold modelled on $\mathbb{R}^{(k-r)\times r}$. Then we define an atlas for the set $\mathcal{M}_r(\mathbb{R}^{k \times r})$ of full rank matrices and prove that the resulting manifold is an analytic principal bundle with base $\mathbb{G}_r(\mathbb{R}^k)$ and typical fibre $\mathrm{GL}_r$, the general linear group of invertible matrices in $\mathbb{R}^{k\times k}$. Finally, we define an atlas for the set $\mathcal{M}_r(\mathbb{R}^{n \times m})$ of non-full rank matrices and prove that the resulting manifold is an analytic principal bundle with base $\mathbb{G}_r(\mathbb{R}^n) \times \mathbb{G}_r(\mathbb{R}^m)$ and typical fibre $\mathrm{GL}_r$. The atlas of $\mathcal{M}_r(\mathbb{R}^{n \times m})$ is indexed on the manifold itself, which allows a natural definition of a neighbourhood for a given matrix, this neighbourhood being proved to possess the structure of a Lie group. Moreover, the set $\mathcal{M}_r(\mathbb{R}^{n \times m})$ equipped with the topology induced by the atlas is proven to be an embedded submanifold of the matrix space $\mathbb{R}^{n \times m}$ equipped with the subspace topology. The proposed geometric description then results in a description of the matrix space $\mathbb{R}^{n \times m}$, seen as the union of manifolds $\mathcal{M}_r(\mathbb{R}^{n \times m})$, as an analytic manifold equipped with a topology for which the matrix rank is a continuous map

Proper Generalized Decomposition for Nonlinear Convex Problems in Tensor Banach Spaces

Tensor-based methods are receiving a growing interest in scientific computing for the numerical solution of problems defined in high dimensional tensor product spaces. A family of methods called Proper Generalized Decompositions methods have been recently introduced for the a priori construction of tensor approximations of the solution of such problems. In this paper, we give a mathematical analysis of a family of progressive and updated Proper Generalized Decompositions for a particular class of problems associated with the minimization of a convex functional over a reflexive tensor Banach space.Comment: Accepted in Numerische Mathemati

A New Ensemble Probabilistic Method for Short-Term Photovoltaic Power Forecasting

The high penetration of photovoltaic (PV) systems led to their growing impact on the planning and operation of actual distribution systems. However, the uncertainties due to the intermittent nature of solar energy complicate these tasks. Therefore, high-quality methods for forecasting the PV power are now essential, and many tools have been developed in order to provide useful and consistent forecasts. This chapter deals with probabilistic forecasting methods of PV system power, since they have recently drawn the attention of researchers as appropriate tools to cope with the unavoidable uncertainties of solar source. A new multi-model probabilistic ensemble is proposed; it properly combines a Bayesian-based and a quantile regression-based probabilistic method as individual predictors. Numerical applications based on actual irradiance data give evidence of the probabilistic performances of the proposed method in terms of both sharpness and calibration

Geomorphological Evolution of Volcanic Cliffs in Coastal Areas: The Case of Maronti Bay (Ischia Island)

The morphoevolution of coastal areas is due to the interactions of multiple continental and marine processes that define a highly dynamic environment. These processes can occur as rapid catastrophic events (e.g., landslides, storms, and coastal land use) or as slower continuous processes (i.e., wave, tidal, and current actions), creating a multi-hazard scenario. Maronti Bay (Ischia Island, Southern Italy) can be classified as a pocket beach that represents an important tourist and environmental area for the island, although it has been historically affected by slope instability, sea cliff recession, and coastal erosion. In this study, the historical morphoevolution of the shoreline was analysed by means of a dataset of aerial photographs and cartographic information available in the literature over a 25-year period. Furthermore, the role of cliff recession and its impact on the beach was also explored, as in recent years, the stability condition of the area was worsened by the occurrence of a remarkable landslide in 2019. The latter was reactivated following a cloudburst on the 26th of November 2022 that affected the whole Island and was analysed with the Dem of Difference technique. It provided an estimate of the mobilised volumes and showed how the erosion and deposition areas were distributed and modified by wave action. The insights from this research can be valuable in developing mitigation strategies and protective measures to safeguard the surrounding environment and ensure the safety of residents and tourists in this multi-hazard environment

BEHAVIOR ANALYSIS OF TAEKWONDO ATHLETES ACCORDING TO THE ROUNDS OF THE CHAMPIONSHIP

The purpose of this study was to analyse the technical and tactical behaviour of taekwondo athletes in a university level taekwondo championship according to the round. A total of 169 matches, consisting of 1088 performances were videotaped. The results showed that athletes when compete in a final and semifinal round performed more direct actions, simultaneous counterattacks, linear kicks, actions to the chest, and with left leg than when competing in prior rounds, whose technical-tactical behavior is characterized by perform more anticipatory actions than competitors in the final and semifinals. This information serves to focus taekwondo training in direct attacks to the chest and perform simultaneous counterattacks, imitating the final round competitors’ behavior

How does the human RUNX3 gene induce apoptosis in gastric cancer? Latest data, reflections and reactions.

RUNX3 is the oldest known gene in the RUNX family. Data have demonstrated its function to be thoroughly involved the neurogenesis of the dorsal root ganglia, T-cell differentiation and tumorigenesis of gastric epithelium. As a TGF-beta target, RUNX3 protein is believed to be involved in TGF-beta-mediated tumor suppressor pathway; however, little is known about its role in apoptosis. According to recent data reported by Yamamura et al., (J Biol Chem 2006; 281:5267-76), RUNX3 interacts with FoxO3a/FKHRL1 expressed in gastric cancer cells to activate Bim and induce apoptosis. The cooperation between RUNX3 and the PI3K/Akt signaling pathway component FoxO3a/FKHRL1 suggests the putative role of RUNX3 in the homoeostasis of gastric cells and in stomach cancer control. Here we discuss recent breakthroughs in our understanding of the mechanisms of RUNX3 in gastric malignancy and comment on possible future trends and perspectives