Study on potential functions for tensors of rank four in differential geometry.
problem Applying potential functions to tensors of rank four.
method Analyzing the inverse problem and intrinsic perspective.
result Negative result: Potential functions cannot be applied to tensors of rank four.
Four algorithms improve sparse tensor BR1Approx with theoretical guarantees.
problem Sparse tensor best rank-1 approximation.
method Four approximation algorithms exploiting multilinearity and sparsity.
result Theoretical worst-case approximation lower bounds for all algorithms.
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.
Paper identifies tensor ranks via prior predictive matching, solving system of equations.
problem Determining the latent dimensions (ranks) in tensor factorization models.
method Prior predictive moment matching to transform moment matching conditions into a log-linear system of equations.
result Identifies which tensor models have identifiable ranks and derives rank estimators.
Researchers describe the metric structure of compact ECS manifolds.
problem Understanding the metric structure of compact rank-one ECS manifolds.
method Analyzing pseudo-Riemannian manifolds with nonzero parallel Weyl tensor.
result Compact rank-one ECS manifolds are either translational or noncompact.
Tensors play a central role in many modern machine learning and signal processing applications. In such applications, the target tensor is usually of low rank, i.e., can be expressed as a sum of a small number of rank one tensors. This motivates us to consider the problem of low rank tensor recovery from a class of lin…
FasTR efficiently solves sparse and unit-rank tensor regression problems.
problem Sparse and unit-rank tensor regression problems in tensor data analysis.
method FasTR decomposes tensor coefficients into component vectors and estimates each with ℓ1 regularized regression, solving in parallel. result FasTR computes better solutions faster than baseline models.
The paper studies Stein-Weiss operators on symmetric tensors, extending previous work.
problem Understanding Stein-Weiss operators on symmetric tensors of arbitrary rank.
method Analyzing the decomposition of tensor spaces into irreducible components and computing Weitzenbock formulas.
result Unified framework for second-order Stein-Weiss operators and tools for geometric analysis.
GLSKF improves tensor completion by capturing both global and local variations.
problem Tensor completion with missing entries, especially in data with spatial or temporal side information.
method Integrates smoothness-constrained low-rank factorization with a locally correlated residual process.
result GLSKF achieves superior performance and scalability on real-world datasets.
Extends multivariate regression for tensor-variate data, identifying brain regions and facial characteristics.
problem Challenges in fitting regression models with multivariate responses and covariates.
method Low-rank tensor formats on regression coefficients and tensor-variate normal distribution for errors.
result Maximum likelihood estimators for tensor-on-tensor regression via block-relaxation algorithms.
Small initialization improves tensor recovery from noisy data.
problem Recovering low-tubal-rank tensors from noisy measurements.
method Factorized gradient descent with small initialization.
result Achieves nearly minimax optimal recovery error.
Paper proposes a new model for imputing missing spatiotemporal traffic data.
problem Missing data and sparsity in spatiotemporal traffic data.
method Low-rank tensor completion (LRTC) framework with truncated nuclear norm (TNN).
result The proposed model outperforms state-of-the-art imputation models in various scenarios.
HOTCAKE compresses CNNs by decomposing kernels into smaller parts.
problem Compressing deep CNNs without significant accuracy loss.
method Input channel decomposition, guided Tucker rank selection, higher order Tucker decomposition, fine-tuning.
result HOTCAKE produces highly compressed CNN models with good accuracy.
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.
We find necessary and sufficient conditions for a Riemannian four-dimensional manifold (M,g) with anti-self-dual Weyl tensor to be locally conformal to a Ricci--flat manifold. These conditions are expressed as the vanishing of scalar and tensor conformal invariants. The invariants obstruct the existence of parallel …
Paper proposes a new tensor imputation method for spatiotemporal traffic data with missing patterns.
problem Imputation of corrupted or incomplete traffic data.
method Truncated tensor Schatten p-norm (TSpN) for spatiotemporal traffic data imputation.
result The proposed method outperforms other state-of-the-art tensor-based imputation models in various missing cases.
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.
Non-asymptotic tail bounds for Kostlan-Shub-Smale field on sphere
problem Estimating rank-R symmetric signal tensor from Gaussian observation
method Profile maximum likelihood estimator
result Finite-(k,d) error bound recovers asymptotically optimal rate
Derives Bochner formulas for the Weyl tensor on 4D Einstein manifolds.
problem Understanding the geometry of Einstein manifolds through the Weyl tensor.
method Derives higher-order Bochner type formulas for the Weyl tensor on 4D Einstein manifolds.
result Proves a second Bochner type formula for the Weyl tensor.
Geometrically, tensors of fixed rank form a minimal submanifold.
problem Understanding the geometric properties of tensors of fixed rank.
method Geometric analysis of tensors in Euclidean space.
result Real tensors of fixed multilinear rank form a minimal submanifold.
Develops methods to estimate high rank tensors from noisy data.
problem Estimating high rank tensors from noisy observations.
method Generative latent variable tensor model, polynomial-time spectral algorithm.
result Achieves computationally optimal rate for signal tensor estimation.
A simple self-supervised model for tensor RPCA using deep unfolding.
problem Tensor robust principal component analysis (RPCA) challenges in practical applications.
method Deep unfolding with only four hyperparameters.
result Competitive or superior performance compared to supervised methods, even in data-starved scenarios.
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.
Efficiently reduces tensor ranks using mean-field approximation.
problem Low-rank approximation of non-negative tensors.
method Mean-field approximation of tensor rank reduction.
result Our algorithm achieves faster and competitive tensor rank reduction.
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. Introduces nondecreasing rank for matrices and tensors, developing methods and applications.
problem Finding low-rank approximations for matrices and tensors with monotonic constraints.
method Developed a variant of hierarchical alternating least squares algorithm for finding low ND rank approximations.
result Low ND rank factorizations can be found and interpreted for real-world datasets.
Paper improves MVSC using tensor low-rank modeling.
problem Improving multi-view spectral clustering.
method Structured tensor low-rank norm for MVSC optimization.
result Proposed method outperforms state-of-the-art methods.
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.
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.
New tensor completion method reduces impact of outliers.
problem Recover tensors from incomplete data with outliers.
method Proposes a new correntropy-based objective function and half-quadratic minimization.
result Demonstrates robust performance with real and synthetic data.
New method for tensor recovery with fewer samples.
problem Recovering low-TT-rank tensors from few samples.
method Minimizing a weighted sum of nuclear norms of unfoldings.
result Significantly fewer samples required for recovery.
Bayesian model identifies outliers and determines tensor rank in streaming data.
problem Outliers and over-fitting in streaming tensor factorization.
method Variational Bayesian Inference for robust tensor rank determination and outlier identification.
result Model accurately identifies sparse outliers and determines tensor rank.
Paper extends tensor recovery method for low CP-rank tensors.
problem Recovery of low-rank tensors from few measurements.
method Iterative Hard Thresholding with tensor version of RIP.
result Exact recovery of tensors with low CP-rank is guaranteed.
New framework for low-rank tensor analysis on graphs.
problem Low-rank tensor analysis on non-Euclidean domains.
method Graph-based low-rank decomposition and convex optimization.
result Significant speed-up and performance enhancement at low SNR.
Tensor rank and low-rank tensor decompositions have many applications in learning and complexity theory. Most known algorithms use unfoldings of tensors and can only handle rank up to n⌊p/2⌋ for a p-th order tensor in Rnp. Previously no efficient algorithm can decompose 3rd order ten…
New method uses low-rank tensor factor analysis for better image restoration.
problem Restoring images from limited data.
method Low-rank tensor factor analysis combined with ADMM.
result The method outperforms traditional approaches, especially at low sampling rates.
Paper develops RGN method for estimating low-rank tensors from noisy measurements.
problem Estimating low-rank tensors from noisy linear measurements.
method Riemannian Gauss-Newton (RGN) method for efficient low-rank tensor estimation.
result First local quadratic convergence guarantee of RGN for low-rank tensor estimation in noisy settings.
Derives smooth homogeneous structures for low-rank tensors.
problem Understanding the geometry of low-rank tensors.
method Analyzes sets of fixed CP, multilinear, and TT rank tensors to derive smooth homogeneous manifolds.
result Derives Riemannian metrics with complete geodesics.
Efficient tensor completion method using rank minimization on TR latent space.
problem High model sensitivity and exponential model possibilities in TR decomposition.
method Nuclear norm regularization on latent TR factors, ADMM scheme.
result Superior performance and efficiency compared to state-of-the-art algorithms.
BRTR improves robust tensor completion with automatic rank detection.
problem Robust tensor completion from incomplete data with outliers.
method Bayesian robust tensor ring decomposition (BRTR) with variational Bayesian (VB) algorithm.
result Automatic detection of TR rank and improved performance over state-of-the-art methods.
Study classifies Lie groups with specific metric properties.
problem Investigating metrics on Lie groups with zero Schouten-Weyl tensor.
method Complete classification through Lie algebra structure constants.
result Complete classification of metric Lie groups.
Proposes a model to relate a tensor feature to a univariate outcome using sparse and low-rank components.
problem Relating a univariate outcome to a feature tensor with sparse and low-rank components.
method Divide-and-conquer strategy, stagewise estimation procedure for unit-rank tensor regression.
result The stagewise solution paths converge to those of regularized regression as step size goes to zero.
Unified framework for statistical inference of low-rank tensors.
problem Statistical inference for tensors in high-dimensional data.
method Unified framework using debiasing and tangent space projection.
result Achieves asymptotic normality and minimax-optimal confidence intervals.
A low-rank tensor model simplifies multi-dimensional Markov chains.
problem Simplifying the dynamics of multi-dimensional Markov chains.
method Low-rank tensor decomposition for multi-dimensional state spaces.
result Our tensor model requires fewer parameters and samples than conventional methods.
Improved machine learning with reduced tensor rank constraints and dropout.
problem Efficiently approximating large tensors in machine learning.
method Tree tensor networks with CP rank constraints and tensor dropout.
result Low-rank TTN classifier achieves 90.3% accuracy in Fashion-MNIST.
Tensor regression networks improve neural network compression and regularization.
problem Improving neural network compression and regularization with low-rank tensor approximations.
method Investigating various low-rank tensor approximations in tensor regression networks.
result Tensor regression networks with Global Average Pooling layer outperformed in deep CNNs, while shallow CNNs with tensor regression and dropout achieved lower test error.
Paper develops inference methods for low-rank tensors without debiasing.
problem Statistical inference for low-rank tensor models.
method Two-iteration alternating minimization for asymptotic distribution.
result Asymptotic distributions and confidence regions for singular subspaces.
Paper determines rank of unknown low-rank data.
problem Determining the rank of unknown low-rank data.
method Analyzes various data models and provides upper bounds on rank based on sampled entries.
result Upper bounds on rank are equal to the true rank in many cases.