The rational homotopy type of certain manifolds is determined by their cohomology and Bianchi-Massey tensor.
problem Determining the rational homotopy type of (n-1)-connected manifolds.
method Defining the Bianchi-Massey tensor and using it to prove the rational homotopy type is determined by cohomology and tensor.
result The rational homotopy type of (n-1)-connected manifolds of dimension up to 5n-3 is determined by their cohomology and Bianchi-Massey tensor.
New invariant P helps classify simply-connected 8-manifolds.
problem Classifying simply-connected 8-manifolds.
method Introducing a rational homotopy invariant P and using it to classify manifolds.
result Simply-connected 8-manifolds are classified by the value of P.
New method encodes manifold homotopy types into algebra structures, extending previous bounds.
problem Encoding the real homotopy type of compact manifolds into algebraic structures.
method Homotopy transfer of unital DGCA structure from de Rham algebra to cohomology.
result Multiplication vanishes for all k ≥ ℓ-1 in minimal unital C∞-algebra for certain dimensions.
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.
A new tensor decomposition method that minimizes KL divergence.
problem Tensor reconstruction accuracy.
method Legendre decomposition, based on information geometry.
result Minimizes KL divergence and improves tensor reconstruction accuracy.
A Matlab toolbox for tensor operations based on t-product.
problem Extending matrix operations to tensors.
method Developed a Matlab toolbox implementing tensor operations based on t-product.
result Implemented several tensor operations including SVD, spectral norm, and nuclear norm.
Paper improves tensor completion using unitary transforms.
problem Robust tensor completion for various datasets.
method Transformed tensor SVD with unitary matrices.
result Recovered images have better PSNR than traditional methods.
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.
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.
Compatible tensors form a special Jordan algebra.
problem Understanding the algebraic structure of compatible tensors.
method Proving tensors form a Jordan algebra through symmetrized product properties.
result Riemann, Weyl, and curvature compatible tensors form a special Jordan algebra.
Efficiently decomposes large tensors using stochastic gradients.
problem Efficiently decomposing large tensors for multiway data analysis.
method Stochastic gradients computed via MTTKRP kernel for efficient computation.
result Advantages and scalability demonstrated for large-scale problems.
New spectral tensor network algorithms solve continuous tensor problems.
problem Continuous tensor decomposition and orbit recovery problems over infinite groups.
method Leverage tensor networks to design spectral algorithms.
result Solve continuous multi-reference alignment over infinite SO(2) group.
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…
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.
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 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. Paper solves TRPCA problem for tensor data with new tensor nuclear norm.
problem Exact recovery of tensor low-rank and sparse components.
method Introduces tensor-tensor product and new tensor nuclear norm to solve TRPCA.
result The new tensor nuclear norm guarantees exact recovery of tensor data.
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.
Proposes tensor Q-rank for better tensor rank recovery in complex data.
problem Improving tensor rank recovery for complex data with low sampling rate.
method Introduces tensor Q-rank and two selection methods for Q, proposing VMTQN and MOTQN models. result Demonstrates superior performance in tensor completion problems compared to TNN-based methods.
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.
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 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.
Graphical models and tensor networks are shown to be dual.
problem No specific problem stated; focuses on the duality between models.
method Study of tensor hypernetworks on hypergraphs and their correspondence to graphical models.
result Tensor hypernetworks on hypergraphs correspond to graphical models of the dual hypergraph.
New algorithm decomposes 3rd order tensors efficiently.
problem Handling high-rank 3rd order tensors efficiently.
method Sum-of-Squares hierarchy for quasi-polynomial time decomposition.
result First efficient algorithm for decomposing super-linear rank 3rd order tensors.
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.
Optimal low rank tensor recovery requires a minimum number of entries for accurate reconstruction.
problem Exact recovery of high order tensors of low rank from a subset of their entries.
method Riemannian optimization algorithm with initial value from a spectral method, leveraging tensor restricted isometry property and curvature of the manifold.
result Tensor of size nimesnimes⋯imesn of ranks (r,⋯,r) can be reconstructed with high probability from O((rd+dnr)log(d)) entries. 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.
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.
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.
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.
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.
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.
Fast tensor decomposition for latent variable models.
problem Efficiently decompose tensors for latent variable models.
method Randomized tensor CP decomposition using sketching techniques.
result Fastest algorithm for both sparse and dense tensors.
Study on curvature tensors, discovering new Osserman tensors.
problem Investigate properties of curvature tensors and their relations.
method Introduce quasi-Clifford curvature tensors and analyze their properties.
result Discovered an Osserman curvature tensor not satisfying the duality principle.
Derdzinski and Shen's theorem on the restrictions posed by a Codazzi tensor on the Riemann tensor holds more generally when a Riemann-compatible tensor exists. Several properties are shown to remain valid in this broader setting. Riemann compatibility is equivalent to the Bianchi identity of the new "Codazzi deviation …
This paper studies how key tensor properties are inherited in subtensors of tensor train decompositions.
problem Theoretical development of property inheritance for subtensors in tensor train decompositions.
method Theoretical analysis of incoherence and condition number preservation, and tensor train rank preservation through fiber-wise sampling.
result Key tensor properties (incoherence and condition number) can be well preserved to subtensors formed via fiber-wise sampling.
Proposes a method to recover sparse tensors with covariate info.
problem Sparse tensor with high missing entries and many zeros.
method Covariate-assisted Sparse Tensor Completion (COSTCO) using latent components.
result 23% accuracy improvement over baseline in advertisement dataset.
Proposes a scalable algorithm for large-scale probabilistic tensor analysis.
problem Leveraging time constraints to capture evolving tensor data.
method Introduces a new tensor data split strategy and an efficient algorithm for stochastic Alternating Direction Method of Multipliers.
result Demonstrates that P2T2F is a highly effective and efficiently scalable algorithm. Introduces STTM to improve STM's performance.
problem Overfitting and curse of dimensionality in SVM.
method Replaces rank-one tensor in STM with tensor train.
result STTM outperforms SVM and STM.
Many problems can be formulated as recovering a low-rank tensor. Although an increasingly common task, tensor recovery remains a challenging problem because of the delicacy associated with the decomposition of higher order tensors. To overcome these difficulties, existing approaches often proceed by unfolding tensors i…
This paper studies geometric structures in tensor representations using tree-based formats.
problem Understanding geometric structures in tensor representations.
method Using minimal subspaces and tree structures, the paper introduces a new format for tensors and analyzes their manifold geometry.
result The set of tensors in the tree-based format with bounded or fixed tree-based rank forms analytic Banach manifolds.