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…
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48 results for “tensor transformation”
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.
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.
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.
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.
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.
We first give a constructive answer to the attenuated tensor tomography problem on simple surfaces. We then use this result to propose two approaches to produce vector-valued integral transforms which are fully injective over tensor fields. The first approach is by construction of appropriate weights which vary along t…
This work studies the Tensor Robust Principal Component Analysis (TRPCA) problem, which aims to exactly recover the low-rank and sparse components from their sum. Our model is motivated by the recently proposed linear transforms based tensor-tensor product and tensor SVD. We define a new transforms depended tensor rank…
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.
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.
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.
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.
Given a Lorentzian manifold, the light ray transform of a function is its integrals along null geodesics. This paper is concerned with the injectivity of the light ray transform on functions and tensors, up to the natural gauge for the problem. First, we study the injectivity of the light ray transform of a scalar func…
For smooth compact connected manifolds with strictly convex boundary, no conjugate points and a hyperbolic trapped set, we prove an equivalence principle concerning the injectivity of the X-ray transform on symmetric solenoidal tensors and the surjectivity of an operator on the set of solenoidal tensors…
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…
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.
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.
In this paper, we study the long existence problem of non Berwaldian Landsberg spaces using the conformal transformation point of view. Under conformal transformation, the Berwald and Landesberg tensors are calculated in terms of the T-tensor. By giving examples, we show that under conformal transformation, there are L…
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.
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.
We consider the mixed ray transform of tensor fields on a three-dimensional compact simple Riemannian manifold with boundary. We prove the injectivity of the transform, up to natural obstructions, and establish stability estimates for the normal operator on generic three dimensional simple manifold in the case of 1+1 a…
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.
Let (M, g) be a simple, real analytic, Riemannian manifold with boundary and of dimension n>=3. In this work, we prove a support theorem for the transverse ray transform of tensor fields of rank 2 defined over such manifolds. More specifically, given a symmetric tensor field f of rank 2, we show that if the transverse …
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…
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.
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.
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.
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.
We study natural differential operators transforming two tensor fields into a tensor field. First, it is proved that all bilinear operators are of order one, and then we give the full classification of such operators in several concrete situations.
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…
We propose a novel multilinear dynamical system (MLDS) in a transform domain, named -MLDS, to model tensor time series. With transformations applied to a tensor data, the latent multidimensional correlations among the frontal slices are built, and thus resulting in the computational independence in the tra…
Enhances stock movement prediction using Higher Order Transformers for multimodal time-series data.
problem Predicting stock movements in financial markets with complex dynamics.
method Introduced Higher Order Transformers, extending self-attention and transformer architecture to capture complex market dynamics. Employed low-rank tensor decomposition and kernel attention to manage computational complexity. Integrated technical and fundamental analysis from historical prices and tweets.
result Demonstrated effectiveness of the method on the Stocknet dataset, improving stock movement prediction.
Study inverse problems for twisted geodesic flows on manifolds.
problem Understanding inverse problems for twisted geodesic flows.
method Generalized ray transforms and tensor tomography.
result New insights into rigidity problems for twisted geodesic flows.
Consider a Riemannian manifold in dimension with strictly convex boundary. We prove the local invertibility, up to potential fields, of the geodesic ray transform on tensor fields of rank four near a boundary point. This problem is closely related with elastic \textit{qP}-wave tomography. Under the condition …
New model fills in missing traffic data efficiently.
problem Missing data in large-scale spatiotemporal traffic data.
method Developed scalable tensor learning model LSTC-Tubal for imputation.
result LSTC-Tubal achieves high accuracy with lower computational cost.
A new method computes Greeks for multi-asset options using tensor trains and Fourier transforms.
problem Efficient computation of Greeks for multi-asset options with high accuracy and low sample complexity.
method Tensor train (TT) representations of Fourier-based pricing functions, combined with numerical differentiation or analytical approaches.
result Significant speed-ups of up to over Monte Carlo simulations while maintaining comparable accuracy.
Tensor trains speed up option pricing for multi-asset options.
problem Speeding up option pricing for multi-asset options.
method Tensor train learning algorithms to compress functions with parameter dependence.
result The proposed method outperforms Monte Carlo-based pricing in computational complexity.