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.
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48 results for “tensor transformations”
Two new methods for injective ray transforms of tensor fields on surfaces are introduced.
problem Injective ray transforms for tensor fields on surfaces.
method Two approaches: weights varying along geodesic flow and transverse ray transforms.
result Constructive solutions for tensor tomography on simple surfaces.
Local invertibility of higher order tensor transforms on compact manifolds.
problem Invertibility of higher order tensor transforms on compact manifolds.
method Local invertibility of transverse and mixed ray transforms of tensors on compact Riemannian manifolds.
result Local invertibility of transverse and mixed ray transforms of tensors for specific dimensions.
This work solves TRPCA under linear transforms, recovering low-rank and sparse components.
problem Exact recovery of tensor low-rank and sparse components from their sum.
method Convex optimization with weighted tensor nuclear norm and ℓ1-norm.
result The convex program exactly recovers the components under certain incoherence conditions.
Paper improves tensor completion by reducing sample entries needed.
problem Reducing the number of required sample entries for tensor completion.
method Utilizes multi-rank and unitary transformation in tensor singular value decomposition.
result Provides a bound on the number of required sample entries for tensor completion.
Injectivity of X-ray transform on solenoidal tensors proven for certain manifolds.
problem Injectivity of X-ray transform on solenoidal tensors on manifolds with hyperbolic trapped set.
method Proof of equivalence principle concerning injectivity and surjectivity of operators on symmetric solenoidal tensors.
result Injectivity of X-ray transform on solenoidal tensors of any order for surfaces with hyperbolic trapped set.
The article recovers tensor fields from partial data using weighted divergent ray transforms.
problem Recovering tensor fields from partial data.
method Weighted divergent ray transforms, unique continuation property of fractional Laplacian, explicit reconstruction formulas.
result Recovery of symmetric -tensor fields and unique continuation for vector fields and symmetric 2-tensor fields.
TEAFormers preserve multi-dimensional time series structures for better forecasting.
problem Traditional Transformers flatten multi-dimensional time series data, losing critical multi-dimensional relationships.
method Tensor-Augmented Transformer (TEAFormer) with Tensor-Augmentation (TEA) module.
result Significant performance enhancements in time series forecasting across benchmarks.
Local invertibility of ray transforms on convex manifolds.
problem Invertibility of ray transforms on compact Riemannian manifolds with strictly convex boundary.
method Local invertibility results for transverse and mixed ray transforms of 1 and 1+1 tensors.
result Local invertibility of ray transforms near boundary points, leading to global results.
Study natural operators transforming tensor fields, proving all bilinear ones are of order one.
problem Understanding natural differential operators on tensor fields.
method Proved all bilinear operators are of order one, then classified operators in specific cases.
result Full classification of natural differential operators on tensor fields.
Support theorem for tensor fields on Riemannian manifolds.
problem Determining the support of tensor fields from ray transforms.
method Support theorem for transverse ray transform of rank 2 tensor fields.
result Support of tensor fields vanishes on geodesics where ray transform vanishes.
Study on non-Berwaldian Landsberg spaces using conformal transformation.
problem Existence and properties of non-Berwaldian Landsberg spaces.
method Using conformal transformation to analyze Berwald and Landsberg tensors.
result Necessary condition for a Landsberg space to be Berwaldian is given.
Study extends geodesic ray transform results to orientable surfaces.
problem Characterize and stabilize mixed and transverse ray transforms on surfaces.
method Algebraic arguments applied to various geometries and ray transforms.
result Characterization of kernel and stability for mixed and transverse ray transforms on orientable surfaces.
The paper explores the injectivity of the light ray transform on Lorentzian manifolds.
problem Injectivity of the light ray transform on functions and tensors.
method Analyzes injectivity conditions for scalar and tensor fields on stationary and static Lorentzian manifolds.
result Injectivity of the light ray transform on functions and tensors is proven under specific conditions.
Researchers prove injectivity and stability for mixed ray transform on simple manifolds.
problem Injectivity and stability of mixed ray transform for tensor fields.
method Analyzing tensor fields on 3D compact simple Riemannian manifolds with boundary.
result Injectivity and stability estimates for normal operator on generic 3D simple manifolds.
Novel deep learning model for multivariate time series prediction.
problem Challenges in multivariate time series prediction with correlations and complex temporal patterns.
method Temporal Tensor Transformation Network (TTNT) that transforms multivariate time series into tensors for improved feature extraction.
result TTNT outperforms state-of-the-art methods in window-based predictions across various tasks.
Local invertibility of higher rank tensor fields on curved manifolds proven.
problem Local invertibility of geodesic ray transform on tensor fields of rank four.
method Proved local invertibility up to potential fields on Riemannian manifolds with strictly convex boundary.
result Local invertibility of tensor fields of rank four on curved manifolds proven.
The following result is proved: Consider a 4-dimensional Kaehler manifold M with nonvanishing Bochner tensor B. Then any holomorphic transformation of M, which preserves B is a homothety.
A new MLDS model captures nonlinear tensor time series with improved accuracy and efficiency.
problem Modeling and analyzing nonlinear tensor time series data.
method Transform-based multilinear dynamical system (MLDS) with EM algorithm for parameter estimation.
result Significantly higher prediction accuracy and exponential improvement in training time compared to state-of-the-art models.
Paper shows invertibility of tensor X-ray transform on certain manifolds.
problem Invertibility of tensor X-ray transform on asymptotically conic manifolds.
method Used 1-cusp pseudodifferential operator algebra and modified solenoidal gauge condition.
result Invertibility of tensor X-ray transform up to natural obstruction.
Historically tensor calculus emerged in an attempt to formalize Rie- mann's ideas. We show that tensor calculus can be based also on Lie's idea of a transformation group and this approach leads quite naturally to the concept of deformation of a transformation group and the Kodaira- Spencer map.
Tensor network architecture for classification and regression using wavelet transformations.
problem Efficiently performing classification and regression tasks on complex data.
method Tensor network layers based on MERA and MPS, with adaptive fine-graining.
result Adaptive fine-graining improves model performance without loss in accuracy.
On Radon transforms on torimath.DG
We show injectivity of the X-ray transform and the -plane Radon transform for distributions on the -torus, lowering the regularity assumption in the recent work by Abouelaz and Rouvière. We also show solenoidal injectivity of the X-ray transform on the -torus for tensor fields of any order, allowing the tensor…
In this paper we study the local magnetic ray transform of symmetric tensor fields up to rank two on a Riemannian manifold of dimension with boundary. In particular, we consider the magnetic ray transform of the combinations of tensors of different orders due to the nature of magnetic flows. We show that such …
Inverts rank m symmetric tensor fields using line integrals.
problem Recovering symmetric tensor fields from line integrals.
method Computes normal operator and presents inversion formula.
result Recovering rank m tensor fields from data .
DTCCA learns nonlinear transformations of multi-view data for high-order correlation.
problem Learning complex nonlinear transformations of multiple data views.
method Maximizes high-order canonical correlation by jointly learning transformations of each view using a reformulated tensor decomposition.
result DTCCA efficiently handles high-dimensional and large number of views, overcoming scalability issues.
New method forecasts multilinear data using tensor autoregression.
problem Forecasting 2D data in big data.
method L-Transform Tensor autoregressive (L-TAR) method.
result Statistical independence achieved through invertible discrete linear transforms.
The X-ray transform on the periodic slab , , has a non-trivial kernel due to the symmetry of the manifold and presence of trapped geodesics. For tensor fields gauge freedom increases the kernel further, and the X-ray transform is not solenoidally injective unless . We characterize t…
Characterizes the range of a tensor field transform in Schwartz space.
problem Range characterization of a tensor field transform in Schwartz space.
method Differential and integral equations for characterizing the range in different dimensions.
result Range characterization for the operator on Schwartz space of rank m tensor fields.
In this paper we consider the geodesic X-ray transform with attenuation coefficient as a combination of smooth complex function and 1-form. We show that attenuated X-ray transform applied to the pair of tensors is injective modulo the natural obstruction.
Enhanced Transformer solves math problems better with explicit relation encoding.
problem Improving Transformer models for solving math word problems.
method Integrates Tensor-Product Representations and TP-Attention mechanism.
result Sets new state of the art on the Mathematics Dataset.
Study geodesic ray transform on 2D manifolds with conjugate points.
problem Understanding geodesic ray transform on manifolds with conjugate points.
method Decomposition into pseudodifferential operator and Fourier integral operators using method of stationary phase.
result Explicit computation of principal symbol and cancellation of singularities.
Study geodesic X-ray transforms on hyperbolic surfaces, proposing new reconstruction methods.
problem Inverting geodesic X-ray transforms for symmetric tensor fields on asymptotically hyperbolic surfaces.
method Developed a decomposition theorem for m-tensor fields, used Guillemin-Kazhdan operators and 0-calculus, and provided explicit reconstruction methods.
result Explicit reconstruction methods for even tensor fields from their X-ray transform or normal operator.
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.
OGRe simplifies tensor calculations in general relativity.
problem Complex tensor calculations in general relativity.
method Object-oriented design for tensor calculus, automatic transformations, and optimized algorithms.
result Eliminates user errors and simplifies tensor calculations.
We study the geodesic X-ray transform on Cartan-Hadamard manifolds, and prove solenoidal injectivity of this transform acting on functions and tensor fields of any order. The functions are assumed to be exponentially decaying if the sectional curvature is bounded, and polynomially decaying if the sectional curvature de…
The paper develops methods to generate invariant quantities in Metric-Affine Geometry.
problem Developing methods to generate invariant quantities in Metric-Affine Geometry.
method The paper introduces a theorem to generate invariant quantities under transformations of the affine connection, proving invariance conditions.
result Theorem establishing conditions for invariance of functionals under transformations of the affine connection.
New formula extracts full local information from ray transform data.
problem Determining symmetric tensor fields from ray transform data.
method Deriving explicit formula for Saint Venant operator.
result Explicit formula for extracting full local information.
Wavelet transform clusters climate data into biomes.
problem Understanding climate biomes through complex data.
method Discrete wavelet transform for coarse-graining, ensemble classification, information theory.
result Efficient consensus clustering identifies climate biomes.
New proof confirms noncompact locally conformally flat manifolds are compact.
problem Rigidity of Schouten tensor under conformal transformations.
method Proof of Cheng's theorem using modified Schouten tensor.
result Noncompact locally conformally flat manifolds are compact.
We present a simple way of generating the infinite set of Jacobi tensors, compatible with a given one, via the "gauge transformations" of the functions on Jacobi manifold. We consider also some applications of this result to the construction of bi-Hamiltonian systems on Jacobi manifolds.
Sharp stability estimate for geodesic ray transform on simple manifolds.
problem Stability estimate for geodesic ray transform of tensor fields.
method Microlocal analysis and structured range of geodesic ray transform.
result Sharp stability estimate for tensor fields of order 0, 1, and 2.
Algorithm learns polynomial transformations of Gaussian distributions.
problem Learning high-dimensional polynomial transformations of Gaussian distributions.
method Polynomial-time algorithms for smoothed settings, tensor ring decomposition.
result First end-to-end guarantees for learning pushforwards under neural networks.
Assume (M,g,Ω) is a closed, oriented Riemannian surface equipped with an Anosov magnetic flow. We establish certain results on the surjectivity of the adjoint of the magnetic ray transform, and use these to prove the injectivity of the magnetic ray transform on sums of tensors of degree at most two. In the final sectio…
Generative model improves tabular data density estimation.
problem Challenges in estimating tabular data distribution.
method Tensor contraction layers and transformers in VAEs.
result Embedding representations improve density estimation metrics.
A new method estimates rare events using tensor trains.
problem Estimating rare event probabilities in high-dimensional problems.
method Approximating optimal importance distribution via tensor-train decompositions and compositions.
result Better variance reduction and efficient computation of rare event probabilities.
Study of a 3D manifold with a circulant structure whose cube is the identity.
problem Characterizing a specific type of Riemannian manifold.
method Analyzing a tensor structure on a 3D manifold with a circulant property and its properties.
result An important characteristic identity for the fundamental tensor is derived.
We establish an equivalence principle between the solenoidal injectivity of the geodesic ray transform acting on symmetric -tensors and the existence of invariant distributions or smooth first integrals with prescribed projection over the set of solenoidal -tensors. We work with compact simple manifolds, but seve…