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.
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.
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.
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.
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.
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.
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…
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.
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.
Proposes a new tensor grid method for image completion.
problem Image completion from missing data.
method Low-rank tensor grid with two-stage density matrix renormalization group initialization and alternating least squares factorization.
result The proposed tensor grid method outperforms existing methods in image recovery accuracy.
New result on tensor recovery without strong assumptions.
problem Recoverability of randomly compressed tensors with low CP rank.
method Deriving restricted isometry property (R.I.P.) via set covering techniques.
result The tensor is recoverable if the number of measurements is proportional to the model parameters.
Study of Langevin dynamics for tensor PCA recovery in high dimensions.
problem Recovering hidden signal vectors (spikes) from noisy Gaussian tensor observations.
method Langevin dynamics approach for nonconvex optimization.
result Sample complexity matches the single-spike case but degrades for all spikes.
QAOA matches classical tensor power iteration in spiked tensor model recovery.
problem Statistical estimation in spiked tensor model with computational gap.
method Analysis of QAOA performance on spiked tensor model.
result QAOA weak recovery threshold matches tensor power iteration.
New tensor recovery method improves efficiency under strict complementarity.
problem Efficiently recovering low-rank tensors using tensor nuclear norm.
method Developed strict complementarity condition for tensor nuclear norm ball and applied to gradient methods.
result Standard gradient methods achieve linear convergence and nearly linear runtime under strict complementarity.
SGD recovers multiple signal vectors in noisy tensor PCA.
problem Estimating multiple signal vectors from noisy tensor observations.
method Online stochastic gradient descent (SGD) in high dimensions with detailed analysis of correlations.
result Sequential elimination of correlations allows recovery of all spikes from Np−2 samples. We tackle tensor denoising with unknown permutations, achieving optimal recovery with polynomial estimators.
problem Structured tensor denoising with unknown permutations in recommendation systems, neuroimaging, etc.
method Developed a constrained least-squares estimator in a block-wise polynomial family.
result Achieved the minimax error bound with polynomial estimators of degree up to (m−2)(m+1)/2. Paper proposes a method for estimating sparse and low-rank tensors from sketchings.
problem Estimating sparse and low-rank tensors from limited data.
method Two-stage non-convex implementation using sparse tensor decomposition and thresholded gradient descent.
result Exact and stable recovery of tensors in noisy and noiseless cases with high probability.
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.
Efficiently recovers low-tubal-rank tensors from few measurements.
problem Recovering tensors with low tubal-rank from limited measurements.
method Factorization and factorized gradient descent.
result Factorized gradient descent reduces computational costs and storage requirements.
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.
The completion of tensors, or high-order arrays, attracts significant attention in recent research. Current literature on tensor completion primarily focuses on recovery from a set of uniformly randomly measured entries, and the required number of measurements to achieve recovery is not guaranteed to be optimal. In add…
New method for tensor completion from specific mode observations.
problem Recovering multiway data tensors from partial observations.
method Tensor train decomposition for fiber-wise observations.
result Deterministic recovery guarantees for specific observation patterns.
Recovering a low-rank tensor from incomplete information is a recurring problem in signal processing and machine learning. The most popular convex relaxation of this problem minimizes the sum of the nuclear norms of the unfoldings of the tensor. We show that this approach can be substantially suboptimal: reliably recov…
Develops a two-stage approach for robust tensor completion of visual data.
problem Estimating missing values in high-order data with outliers.
method Coarse-to-fine framework and M-estimator-based robust tensor ring recovery.
result Superior performance compared to state-of-the-art robust algorithms.
AMP algorithm for matrix tensor product model provides recovery conditions.
problem Generalization of standard spiked matrix models with multiple pairwise observations.
method Approximate message passing with optimal weighing and combining of estimates.
result Asymptotically exact performance description and necessary/sufficient recovery conditions.
The paper provides recovery guarantees for CNNs with multiple kernels under polynomial sample and computational complexities.
problem Parameter recovery for non-overlapping CNNs with multiple kernels.
method Showed local strong convexity of squared loss for most popular activations, used tensor methods for initialization, and proved convergence of gradient descent.
result Gradient descent following tensor initialization converges to the global optimal with polynomial time complexity.
Unified framework for coupled tensor completion improves recovery accuracy.
problem Improving recovery accuracy in coupled tensor completion.
method Unified framework using tensor ring (TR) decomposition with shared latent factors and novel optimization model.
result The proposed method achieves superior recovery accuracy on real-world data compared to state-of-the-art methods.
New framework extends ICA for non-independent variables, identifying pairwise mean independence.
problem Non-independent variables complicating ICA recovery.
method Algebraic recovery algorithm based on least-squares optimization over the orthogonal group.
result Pairwise mean independence is identifiable, robust to independence constraints.