STARK learns structured dictionaries for tensor data.
problem Representing multidimensional data with structured dictionaries.
method Solves a convex relaxation of a nonconvex rank-1 tensor recovery problem.
result Empirical results show promising performance for tensors of any order.
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.
A new algorithm completes rank-1 tensors with minimal samples and time.
problem Completing rank-1 tensors with minimal samples and time.
method Gauss-Jordan on random linear systems.
result Gauss-Jordan algorithm uses O(d2logd) samples and runs in O(md2) time. A tensor-based model reduces weight parameters and spatial structure for high-order data classification.
problem High-order data classification with limited training samples and preserved spatial structure.
method Rank-1 FNN model based on modified feedforward neural network with rank-1 canonical decomposition and new learning algorithm.
result The proposed model outperforms state-of-the-art methods, especially in cases with small training samples.
Study explores warped geometries of tensor manifolds, finding non-geodesic connections for some parameters.
problem Investigate non-geodesic connections in warped Segre-Veronese manifolds.
method Investigate a one-parameter family of warped geometries, presenting closed expressions for maps and distance.
result Segre-Veronese manifolds are not geodesically connected in Euclidean geometry but can be for some warping parameters.
New compact ECS manifolds with rank 2 discovered, differing from previous rank 1 examples.
problem Finding new compact ECS manifolds with rank 2.
method Constructing new examples of compact pseudo-Riemannian manifolds with parallel Weyl tensor, rank 1 or 2.
result New compact ECS manifolds of rank 2, locally homogeneous, and geodesically incomplete.
New method reduces tensor completion sample complexity to nearly optimal levels.
problem Low rank tensor completion with noisy measurements.
method Using atomic-norm and max-quasi-norm for tensor completion.
result Optimal sample complexity of O(dN) achieved for tensor completion. New method solves rank-1 L1-norm TUCKER2 decomposition efficiently.
problem Exact solution to rank-1 L1-norm TUCKER2 decomposition of tensors.
method Proved equivalent to combinatorial optimization, derived two algorithms.
result L1-TUCKER2 outperforms other methods in tensor approximation for outlier-corrupted data.
In this paper, we provide local and global convergence guarantees for recovering CP (Candecomp/Parafac) tensor decomposition. The main step of the proposed algorithm is a simple alternating rank-1 update which is the alternating version of the tensor power iteration adapted for asymmetric tensors. Local convergence g…
GETF efficiently decomposes large-scale Boolean tensors.
problem Efficiently factorizing large-scale Boolean tensors.
method Geometric Expansion for all-order Tensor Factorization (GETF).
result GETF significantly improves reconstruction accuracy and efficiency.
This work provides a guaranteed tensor recovery method by combining low-rankness and smoothness priors.
problem Guaranteed tensor recovery with theoretical guarantees for low-rank and smoothness priors.
method Developed a new regularization term that combines low-rankness and smoothness priors, proving exact recovery guarantees.
result Rigorously proved exact recovery guarantees for tensor completion and tensor robust principal component analysis.
We study stability and local minimizing properties of Lp- norms of Riemannian curvature tensor denoted by Rp by variational methods. We compute the Hessian of Rp at compact rank 1 symmetric spaces and prove that they are stable for Rp for certain values of p > 2. A similar resu…
This paper tackles tensor recovery from noisy and multi-level quantized measurements.
problem Tensors from multi-level quantized measurements.
method Nonconvex optimization problem with alternating proximal gradient descent.
result The recovery error diminishes to zero with increasing tensor dimensions.
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. 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.
DeepTD learns CNN weights from non-overlapping patches using tensor decomposition.
problem Learning weights of a deep convolutional neural network (CNN).
method Deep Tensor Decomposition (DeepTD) based on rank-1 tensor decomposition.
result DeepTD is data-efficient and works as soon as sample size exceeds total number of weights.
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.
New method proves exact recovery for tensor decomposition under reshuffling.
problem Numerical defects limit practical applications of tensor decomposition.
method Proves exact-recovery property for latent convex tensor decomposition using reshuffling.
result Generalized LCTD achieves exact recovery under reshuffling.
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.
Paper proves conditions for nonconvex matrix recovery to avoid spurious local minima.
problem Ensuring no spurious local minima in nonconvex matrix recovery.
method Sharp restricted isometry bounds proof technique.
result RIP constant of δ < 1/2 is necessary and sufficient for exact recovery.
Two methods improve tensor recovery in Ising models, revealing gene interactions.
problem Improving tensor recovery in Ising models for complex data structures.
method Pseudolikelihood and interaction screening approaches for tensor learning.
result Both methods achieve tensor recovery with sample size logarithmic in nodes, exponential in strength and degree.
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.
Study reveals efficient recovery of multi-modal signals via Bayesian methods and sequential learning.
problem Recovering multiple high-dimensional signals from correlated modalities.
method Bayesian Approximate Message Passing and Sequential Curriculum Learning.
result Sequential learning strategy optimally recovers weak signals in multi-modal settings.
This work proves exact low tubal rank tensor recovery from Gaussian measurements.
problem Low rank tensor recovery from Gaussian measurements.
method Careful choice of atomic set and computation of Gaussian width for atomic norm.
result Exact recovery of tensors with tubal rank r from O(r(n1+n2−r)n3) Gaussian measurements. New tensor recovery method uses Riemannian optimization on Segre manifold.
problem Recovering low-rank tensors from noisy measurements.
method Riemannian Gradient Descent (RGD) and Riemannian Gauss-Newton (RGN) algorithms over the Segre manifold.
result Proven convergence rates for RGD and RGN under mild noise assumptions.
Robust tensor recovery plays an instrumental role in robustifying tensor decompositions for multilinear data analysis against outliers, gross corruptions and missing values and has a diverse array of applications. In this paper, we study the problem of robust low-rank tensor recovery in a convex optimization framework,…
Estimates spatio-temporal Hawkes processes using tensor recovery.
problem Estimating influence functions for spatio-temporal Hawkes processes.
method Formulates influence function as a tensor kernel, assumes low-rank structure, solves as convex optimization problem.
result Provides theoretical guarantees and demonstrates efficiency with simulations.
Robust tensor ring completion improves tensor recovery accuracy and efficiency.
problem Tensor completion sensitivity to sparse components.
method Robust Tensor Ring Completion (RTRC) with weighted nuclear norms and l1 regularization.
result Exact recovery guarantees and superior performance in various tasks.
Paper provides conditions for local recovery of tensor data's Kronecker-structured dictionaries.
problem Local recovery of Kronecker-structured dictionaries for tensor data.
method Derives sufficient conditions for local recovery of coordinate dictionaries.
result Sufficient conditions guarantee recovery of individual coordinate dictionaries up to specified error.
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.
Tensor PCA problem analyzed with statistical query lower bounds.
problem Estimating the expected value of a rank-1 tensor from Gaussian samples.
method Sharp analysis of optimal sample complexity in the Statistical Query model.
result SQ algorithms with polynomial query complexity fail in the conjectured hard phase and have sub-optimal sample complexity.
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.
Paper proves tensor ring completion with high probability using convex optimization.
problem Recovering a multi-dimensional array from limited measurements.
method Tensor ring decomposition and convex optimization.
result High probability exact recovery with n^{d/2} r^2 ln^7(n^{d/2}) samples.
New method improves traffic data recovery for streaming data.
problem Improve data quality in traffic data for ITS.
method Online robust tensor recovery algorithm leveraging spatio-temporal correlations and local consistency.
result Significantly improved computational efficiency and high recovery accuracy.
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 proves sufficient conditions for tensor recovery using t-RIP with random measurements.
problem Establish robust recovery guarantees for low-tubal-rank tensors.
method Probabilistic arguments and random sub-Gaussian distributions to ensure t-RIP conditions.
result Minimal number of linear measurements nearly optimal for tensor recovery.
Paper shows moderate RIP is insufficient for avoiding spurious local minima in matrix recovery.
problem The need for moderate RIP to avoid spurious local minima in matrix recovery.
method Analyzes the necessity of RIP constants and provides counterexamples.
result Counterexamples show spurious local minima exist even with moderate RIP.
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…
Tensor decompositions have rich applications in statistics and machine learning, and developing efficient, accurate algorithms for the problem has received much attention recently. Here, we present a new method built on Kruskal's uniqueness theorem to decompose symmetric, nearly orthogonally decomposable tensors. Unlik…
New findings on tensor decomposition complexity, showing polynomial functions can estimate the largest component under certain conditions.
problem The complexity of tensor decomposition, especially for low-degree polynomials.
method Modeling a slightly larger component in a random tensor decomposition and using polynomial functions to estimate it.
result Polynomial functions can accurately estimate the largest component when r≪n3/2 but fail when r≫n3/2. Nonnegative low-rank matrix recovery can have spurious local minima.
problem Nonnegative low-rank matrix recovery problems can have spurious local minima.
method Investigated projected gradient methods for nonnegative low-rank recovery problems.
result Benign nonconvexity holds in the fully-observed case with RIP constant δ=0 but fails in the partially-observed case and higher-rank ground truths.
Study recovers spike order in noisy tensor estimation without SNR assumptions.
problem Estimating multiple signal vectors from noisy tensor observations.
method Gradient flow optimization of a nonconvex function.
result Determines sample complexity for efficient permutation recovery.
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.
This work shows that a simple local search can recover true principal components in non-negative rank-1 RPCA.
problem Recovering true principal components in non-negative rank-1 robust principal component analysis with noisy measurements.
method Using the Burer-Monteiro approach to cast RPCA as a non-convex and non-smooth ℓ1 optimization problem. result The low-dimensional formulation of symmetric and asymmetric positive rank-1 RPCA has a unique global solution and no spurious local solutions.
The paper analyzes deflation for estimating a low-rank spike in large tensors with noise.
problem Estimating a low-rank symmetric spike in large tensors with additive Gaussian noise.
method Characterization of deflation performance in terms of vector alignments and weights.
result Understanding deflation mechanism in noisy conditions and designing more efficient methods.
The subdifferential of convex functions of the singular spectrum of real matrices has been widely studied in matrix analysis, optimization and automatic control theory. Convex analysis and optimization over spaces of tensors is now gaining much interest due to its potential applications to signal processing, statistics…
Paper explores limits of high-order clustering with planted structures.
problem Statistical and computational limits of high-order clustering with planted structures.
method Developed methods for detection and recovery of clusters, identified signal-to-noise ratio boundaries.
result Sharp boundaries of signal-to-noise ratio for statistical and computational feasibility.
RTC-GTNLN model recovers traffic data from missing values and noise.
problem Simultaneous missing data and noise in traffic data.
method Gradient tensor nuclear L1-L2 norm for robust tensor completion.
result RTC-GTNLN model outperforms existing methods in complex recovery scenarios.