Tachibana operator applied to invariant submanifolds of Lorentzian trans-Sasakian manifolds.
problem Analyzing geometric properties of invariant submanifolds in Lorentzian trans-Sasakian manifolds.
method Application of Tachibana operator to invariant submanifolds using various tensors.
result Results discussed in terms of geometry, including a non-trivial example.
The study characterizes symmetries in Kaehler manifolds.
problem Understanding symmetries in Kaehler manifolds.
method Analyzing specific types of Kaehler manifolds: constant holomorphic sectional curvature, semisymmetric, and holomorphically pseudosymmetric.
result Characterization results and geometric interpretation of the complex Tachibana tensor.
Survey on manifolds satisfying generalized Einstein conditions.
problem Characterizing semi-Riemannian manifolds under specific curvature conditions.
method Analyzing the difference tensor R.C-C.R expressed as linear combinations of Tachibana tensors.
result Recent results on manifolds and submanifolds satisfying generalized Einstein conditions.
The paper proves rigidity results for manifolds with special holonomy.
problem Proving rigidity results for compact Riemannian manifolds with special holonomy.
method Using divergence free Weyl tensors and curvature operators, the paper proves similar results for manifolds with special holonomy.
result The paper proves that manifolds with special holonomy are locally symmetric or conformally equivalent to a quotient of the sphere.
The paper proves manifold rigidity under curvature conditions.
problem Rigidity of manifolds with harmonic curvature and curvature operator positivity.
method Analyzes conditions on complete manifolds to prove constant sectional curvature.
result Rigidity holds for manifolds with harmonic curvature and curvature operator positivity.
The paper characterizes metallic pseudo-Riemannian manifolds using conjugate connections and tensor structures.
problem Characterizing metallic pseudo-Riemannian manifolds.
method Using conjugate connections and tensor structures, the paper derives new characterizations and conditions for these manifolds.
result A necessary and sufficient condition for a non-integrable metallic pseudo-Riemannian manifold to be a quasi metallic pseudo-Riemannian manifold is derived.
The paper explores symmetries in Kähler manifolds using Ricci tensor properties.
problem Investigating symmetries in Kähler manifolds involving Ricci tensor.
method Analyzing properties of Kähler-Einstein spaces and their generalizations.
result Clarified the geometric role of holomorphic Ricci pseudosymmetry and established new criteria for Kähler manifolds to be Einstein.
We derive point-wise and integral rigidity/gap results for a closed manifold with harmonic Weyl curvature in any dimension. In particular, there is a generalization of Tachibana's theorem for non-negative curvature operator. The key ingredients are new Bochner-Weitzenböck-Lichnerowicz type formulas for the Weyl tensor,…
We present a rough classification of differential forms on a Riemannian manifold, we consider definitions and properties of conformal Killing forms on a compact Riemannian manifold and define Tachibana numbers as an analog of the well known Betti numbers. We state the conditions that characterize these numbers. In the …
We present definitions and properties of conformal Killing, Killing and planarity forms on a Riemannian manifold and determine Tachibana, Killing and planarity numbers as an analog of the well known Betti numbers. We state some set of conditions to characterize these numbers. Moreover, we formulate the main results on …
The purpose of this paper is to define the r-th Tachibana number t(r) of an n-dimentional closed and oriented Riemannian manifold (M,g) as the dimension of the space of all conformal Killing r-forms for r=1,2,...,n-1 and to formulate some properties of these numbers as an analogue to properties of the r-th Betty number…
Study of hypersurfaces in curved spaces with specific curvature properties.
problem Characterizing hypersurfaces in spaces of constant curvature with particular curvature properties.
method Investigates hypersurfaces isometrically immersed in semi-Riemannian spaces of constant curvature, focusing on the curvature tensor and its properties.
result Hypersurfaces in the specified spaces satisfy a Roter type equation, linking their curvature tensor to specific tensor products.
Throughout the history of Einstein manifolds, differential geometers have shown great interest in finding the relationships between curvature and the topology of Einstein manifolds. In the paper, first, we prove that a compact Einstein manifold (M,g) with Einstein constant α>0 is a homo-logical sphere when the mini…
In the present paper we show spectral properties of a littleknown natural Riemannian second-order differential operator acting on differential forms.
The question of whether a Sasakian metric can admit an additional compatible (K-)contact structure is addressed. In the complete case if the second structure is also assumed Sasakian, works of Tachibana-Yu and Tanno show that the manifold must be 3-Sasakian or an odd dimensional sphere with constant curvature. Some ext…
The study shows conditions for Kähler manifolds to have rational cohomology.
problem Conditions for Kähler manifolds to have rational cohomology.
method Analyzes the Kähler curvature operator and its eigenvalues.
result Compact Kähler manifolds have rational cohomology under certain conditions.
Sasakian manifolds are odd-dimensional counterpart to Kahler manifolds. They can be defined as contact manifolds equipped with an invariant Kahler structure on their symplectic cone. The quotient of this cone by the homothety action is a complex manifold called Vaisman. We study harmonic forms and Hodge decomposition o…
Study on real hypersurfaces in products of complex space forms, proving rigidity and nonexistence results.
problem Existence and properties of totally umbilical real hypersurfaces in complex space forms.
method Analyzing shape operators and local product structures in products of complex space forms.
result Nonexistence and rigidity results for totally umbilical real hypersurfaces in products of complex space forms.
In the present paper, we introduce and investigate the notion of a semi concurrent vector field on a Finsler manifold. We show that some special Finsler manifolds admitting such vector fields turn out to be Riemannian. We prove that Tachibana's characterization of Finsler manifolds admitting a concurrent vector field l…
The paper defines minimal norm tensors for curvature and divergence tensors, explaining Weyl and Cotten tensors.
problem Understanding curvature tensors and their minimal norm.
method Analyzing minimal norm tensors for third and fourth covariant tensors, including Riemannian curvature and divergence.
result Weyl tensor and Cotten tensor are identified as minimal norm tensors of Riemannian curvature and divergence tensors, respectively.
A new tree method for tensor data improves regression accuracy.
problem Efficiently modeling tensor data for regression problems.
method Scalar-output regression tree models for scalar-on-tensor problems, and tensor-on-tensor problems using additive tree ensemble approaches.
result The tensor-input tree (TT) method outperforms tensor-input GP models in efficiency and accuracy.
Curvature tensors can always be matched to a metric tensor under certain conditions.
problem Sectionally positive curvature tensors and their relationship to metric tensors.
method Existence and uniqueness of a metric tensor gab such that Rabcdgbd=gacλ. result A metric tensor gab can be found for sectionally positive curvature tensors, and it is unique up to a constant factor. Extends geometrical description of tensor manifolds in tree-based formats.
problem Geometrical description of tensor manifolds in tree-based formats.
method Provided a new geometrical description of manifolds of tensors in tree-based format.
result Geometrical description compatible with Tucker format.
The tensor-tensor product (t-product) [M. E. Kilmer and C. D. Martin, 2011] is a natural generalization of matrix multiplication. Based on t-product, many operations on matrix can be extended to tensor cases, including tensor SVD, tensor spectral norm, tensor nuclear norm [C. Lu, et al., 2018] and many others. The line…
Paper optimizes tensor deflation for non-orthogonal signals.
problem Recovering low-rank signals from noisy tensors with correlated components.
method Developed an asymptotic analysis and optimized deflation procedure using random tensor theory.
result Proposed an efficient tensor deflation algorithm that optimizes a parameter introduced in the deflation mechanism.
Paper proposes a new method for exact recovery in robust tensor principal component analysis.
problem Exact recovery of low-rank and sparse components in tensors.
method Proposes a new method based on tensor-tensor product and t-SVD to solve a convex optimization problem.
result Exact recovery achieved in a deterministic fashion without randomness assumptions.
We introduce Bayesian multi-tensor factorization, a model that is the first Bayesian formulation for joint factorization of multiple matrices and tensors. The research problem generalizes the joint matrix-tensor factorization problem to arbitrary sets of tensors of any depth, including matrices, can be interpreted as u…
Tensor fields depending on other tensor fields are considered. The concept of extended tensor fields is introduced and the theory of differentiation for such fields is developed.
The paper tackles tensor factorization and completion from noisy data.
problem Sparse nonnegative tensor factorization and completion from partial and noisy observations.
method Minimizes the sum of maximum likelihood estimation and tensor ℓ0 norm with nonnegativity constraints. result Error bounds and minimax lower bounds are established for the proposed model.
The paper examines properties of W-curvature tensor in relativistic space-times.
problem Investigating the properties and implications of the W-curvature tensor in relativistic space-times. method Analyzing the semi-symmetry and divergence properties of the energy-momentum tensor in relation to the W-curvature tensor. result Space-times with specific properties of the W-curvature tensor are classified as Einstein or Codazzi type. The integrability conditions for the existence of a conformal Killing-Yano tensor of arbitrary order are worked out in all dimensions and expressed in terms of the Weyl tensor. As a consequence, the integrability conditions for the existence of a Killing-Yano tensor are also obtained. By means of such conditions, it is…
New tensors reveal full curvature structure from Riemann tensor.
problem Limited information from Ricci contraction of Riemann tensor.
method Contracting double dual of Riemann tensor to reveal full curvature.
result New tensors provide canonical parents of Einstein tensor.
Adaptive algorithm learns tensor network structures from data.
problem Identifying optimal tensor network structure from data.
method Greedy approach starting from rank one tensor, small rank increments.
result Adaptive algorithm identifies efficient tensor network structures.
Tensorized Rademacher projections outperform Gaussian projections in reducing tensor dimensions.
problem Reducing the dimension of high-dimensional tensors for machine learning.
method Tensorized Rademacher random projections using Tensor Train decomposition.
result Tensorized Rademacher projections can replace Gaussian projections in tensor compression.
The paper shows how gradient flow on over-parametrized tensor decomposition behaves like deflation.
problem Understanding the training dynamics of gradient flow on tensor decomposition.
method Empirical observation and mathematical proof of gradient flow dynamics for orthogonally decomposable tensors.
result Gradient flow dynamics for orthogonally decomposable tensors follows a tensor deflation process, recovering all tensor components.
New method tackles non-smooth tensor data for better recovery.
problem Non-smooth changes in tensor data degrade traditional t-SVD methods.
method Learnable tensor nuclear norm, Alternating Proximal Multiplier Method (APMM), multi-objective tensor recovery framework.
result The proposed method effectively recovers tensor data with non-smooth changes.
We solve linear equations with tensors of any rank.
problem Solving linear equations involving tensors of arbitrary rank.
method Developed a systematic approach for tensors of rank 3 and generalized to arbitrary rank.
result Derived a solution for tensors of arbitrary rank.
Proposes FATTNN for tensor-on-tensor regression with improved prediction and reduced computation.
problem Tensor-on-tensor regression with complex tensor structures and nonlinear relationships.
method Integrates tensor factor models into deep neural networks to handle nonlinearity and reduce data dimensionality.
result Significant improvements in prediction accuracy and computational efficiency over traditional methods.
We present a novel nonnegative tensor decomposition method, called Legendre decomposition, which factorizes an input tensor into a multiplicative combination of parameters. Thanks to the well-developed theory of information geometry, the reconstructed tensor is unique and always minimizes the KL divergence from an inpu…
In this paper, we study robust tensor completion by using transformed tensor singular value decomposition (SVD), which employs unitary transform matrices instead of discrete Fourier transform matrix that is used in the traditional tensor SVD. The main motivation is that a lower tubal rank tensor can be obtained by usin…
ScaledGD algorithm estimates low-rank tensors efficiently from corrupted data.
problem Estimating meaningful information from corrupted tensor data.
method Scaled gradient descent (ScaledGD) algorithm with tailored spectral initializations.
result ScaledGD achieves linear convergence at a constant rate independent of condition number.
TRNN combines tensor geometry with neural network nonlinearity for HD data.
problem Modeling high-dimensional data with preserved tensor geometry and nonlinear interactions.
method Introduces TRNN that integrates tensor geometry and neural network nonlinearity.
result TRNN preserves tensor geometry while offering nonlinearity.
Graphical notation simplifies tensor operations and decompositions.
problem Complex tensor operations are difficult to understand and represent.
method Introduces graphical notation to represent tensor operations.
result Simplified representation of tensor operations and decompositions.
Tensor decomposition is a well-known tool for multiway data analysis. This work proposes using stochastic gradients for efficient generalized canonical polyadic (GCP) tensor decomposition of large-scale tensors. GCP tensor decomposition is a recently proposed version of tensor decomposition that allows for a variety of…
New method for tensor classification with missing data.
problem Handling incomplete tensor data in high-dimensional classification.
method High-dimensional tensor linear discriminant analysis with TGMM and Tensor LDA-MD.
result Established convergence rates and minimax optimal bounds for misclassification rate.
The paper characterizes integrability of tensors on manifolds.
problem Analyzing integrability conditions for various tensor types on manifolds.
method Analytical and geometric characterizations of integrability for different tensor types, using Nijenhuis tensors.
result Integrability of tensors is equivalent to algebraic constancy coupled with vanishing of Nijenhuis-type tensors.
New methods solve tensor-on-tensor regression with unknown rank, revealing benefits of over-parameterization.
problem Connecting tensor responses to tensor covariates with unknown intrinsic rank.
method Riemannian gradient descent and Riemannian Gauss-Newton methods for tensor-on-tensor regression.
result Riemannian optimization methods converge linearly and quadratically to a statistically optimal estimate in rank over-parameterized settings.
Introduces tensor bandits for multi-dimensional online decision making.
problem Optimal decision making in multi-dimensional online scenarios.
method Stochastic low-rank tensor bandits, tensor elimination, tensor epoch-greedy, tensor ensemble sampling.
result Tensor elimination and tensor epoch-greedy algorithms outperform existing methods.