Paper reviews multi-way graph signal processing for tensor data.
problem Maximizing use of multi-way structure in irregular tensor data.
method Generalizes GSP to multi-way data, focusing on graph signals across tensor modes.
result Synthesizes common themes in combining GSP with tensor analysis.
The paper studies phase transitions in random matrices and tensor unfolding for detecting signals.
problem Phase transitions in singular values and vectors of large random matrices.
method Analysis of singular values and vectors of long rectangular random matrices, and tensor unfolding algorithm for asymmetric rank-one spiked tensor models.
result An exact threshold for tensor unfolding to detect signals, independent of unfolding procedure.
Detects missing tensor signals in a KS subspace with high probability.
problem Detecting tensor signals with many missing entities in a KS subspace.
method Projecting the signal onto the KS subspace and bounding residual energy.
result Reliable detection is possible if the missing signal cardinality exceeds KS subspace dimensions.
Study analyzes Hotelling-type tensor deflation for spiked tensors, providing insights into signal and noise.
problem Characterizing singular values and alignments in Hotelling-type tensor deflation.
method Asymptotic study of Hotelling-type tensor deflation in large dimensional regime using random tensor theory.
result Characterization of singular values and alignments at each step of the deflation procedure.
Efficient biclustering of tensor data for identifying similar signal patterns over time.
problem Identifying similar signal patterns over time in multi-dimensional data.
method Spectral decomposition to build biclusters.
result Quality of biclusters evaluated using synthetic and real data.
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.
Unified approach to tensor PCA and related problems using tensor cumulants.
problem Statistical inference on invariant distributions, particularly tensor PCA.
method Definition and analysis of tensor cumulants to unify and extend previous results.
result Unified explanation of hardness and subexponential-time algorithms for tensor PCA.
New model encodes multivariate signals more efficiently with sparsity and low-rank constraints.
problem Efficiently encoding multivariate signals with sparsity and low-rank constraints.
method Multivariate convolutional sparse coding with tensor algebra, CP decomposition, and alternating optimization.
result Proves model closely related to Kruskal tensor regression problem with theoretical guarantees.
Study analyzes accuracy of tensor deflation in noisy conditions.
problem Analyzing accuracy of tensor deflation in noisy conditions.
method Asymptotic study of Hotelling-type tensor deflation in large tensor dimensions.
result Characterization of estimated singular values and singular vector alignments.
Sharp analysis of power iteration for tensor PCA, improving convergence and stopping criteria.
problem Analyzing the power iteration algorithm for tensor PCA to improve convergence and stopping criteria.
method Sharp bounds on the number of iterations, revealing a smaller algorithmic threshold, proposing a stopping criterion.
result Sharp bounds on the number of iterations required for power method to converge, revealing a smaller algorithmic threshold than previously conjectured.
Study uses random matrix theory to improve tensor approximation accuracy.
problem Improving tensor approximation accuracy in the presence of noise.
method Random matrix theory applied to tensor unfoldings.
result Characterizes spectral behavior of tensor unfoldings and predicts reconstruction performance.
Signals are generally modeled as a superposition of exponential functions in spectroscopy of chemistry, biology and medical imaging. For fast data acquisition or other inevitable reasons, however, only a small amount of samples may be acquired and thus how to recover the full signal becomes an active research topic. Bu…
Study on tensor signal estimation from incomplete data.
problem Estimating a rank-one tensor signal from noisy, incomplete data.
method Reduction to random matrix model for spectral analysis.
result Loss of performance due to incomplete data.
Enhances tensor regression for interpretability and performance.
problem Interpreting and modeling multidimensional tensor data with structural heterogeneity.
method Generalized Nonnegative Structured Kruskal Tensor Regression (NS-KTR) with hybrid regularization and nonnegativity constraints.
result NS-KTR outperforms conventional methods in synthetic and real hyperspectral datasets.
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.
Efficient algorithm for tensor PCA with improved time complexity.
problem Tensor PCA problem with signal-to-noise ratio constraint.
method Counting specific weighted hypergraphs to solve tensor PCA.
result Algorithm runs in time nC+o(1) for λn−4p signal-to-noise ratio. Graphical notation simplifies tensor operations and decompositions.
problem Complex tensor operations are difficult to understand and represent.
method Introduces graphical notation to represent tensor operations.
result Simplified representation of tensor operations and decompositions.
New estimator reduces bias and variance in tensor and matrix denoising.
problem Optimal bias-variance tradeoff in matrix and tensor estimation.
method One-step variant of higher-order SVD (HOSVD) estimator.
result Achieves optimal bias-variance tradeoff in both matrix and tensor settings.
New method estimates tensors from noisy data with missing entries.
problem Tensor estimation from noisy observations with missing entries.
method Sign series representation for tensor completion, addressing low- and high-rank signals.
result Excess risk bounds, estimation error rates, and sample complexities established.
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.
Tensors or {\em multi-way arrays} are functions of three or more indices (i,j,k,⋯) -- similar to matrices (two-way arrays), which are functions of two indices (r,c) for (row,column). Tensors have a rich history, stretching over almost a century, and touching upon numerous disciplines; but they have only recent…
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.
New algorithms recover sparse tensor principal components efficiently.
problem Recovering sparse tensor principal components from noisy data.
method Family of algorithms interpolating between polynomial-time and exhaustive search, tailored for sparse and highly sparse regimes.
result Our algorithms recover sparse vectors for signal-to-noise ratios beyond previous limits, with time complexity ildeO(np+t). 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.
DeepTensor uses deep networks to efficiently decompose tensors with improved performance and robustness.
problem Efficiently decomposing tensors with deep learning to capture nonlinear structures.
method Low-rank tensor decomposition using deep generative networks trained to minimize approximation error.
result DeepTensor outperforms classical methods like SVD and PCA in various applications, including image denoising and 3D MRI.
ScaledGD algorithm estimates low-rank tensors efficiently from corrupted data.
problem Estimating meaningful information from corrupted tensor data.
method Scaled gradient descent (ScaledGD) algorithm with tailored spectral initializations.
result ScaledGD achieves linear convergence at a constant rate independent of condition number.
New estimator for tensor weights with improved bias.
problem Estimating tensor weights from noisy data.
method Random matrix theory and KKT conditions.
result Asymptotically unbiased estimator for tensor rank.
New method tackles heterogeneous breast cancer imaging data.
problem Challenges in multimodality breast cancer imaging data.
method Developed a multilayer tensor learning method to incorporate heterogeneity.
result Outperforms other existing methods in predicting disease status.
We propose a new framework for the analysis of low-rank tensors which lies at the intersection of spectral graph theory and signal processing. As a first step, we present a new graph based low-rank decomposition which approximates the classical low-rank SVD for matrices and multi-linear SVD for tensors. Then, building …
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. This paper tackles efficient cooperative control for large-scale traffic signals using tensor-based deep learning.
problem Efficient training and control for large-scale multi-intersection traffic signals.
method Tensor representation, multi-task learning, imitation learning, proximal policy optimization.
result The proposed model achieves better performance compared to existing methods.
We prove the correspondence between the information geometry of a signal filter and a Kähler manifold. The information geometry of a minimum-phase linear system with a finite complex cepstrum norm is a Kähler manifold. The square of the complex cepstrum norm of the signal filter corresponds to the Kähler potential. The…
Tensor analysis tackles complex multidimensional data across fields.
problem Efficiently extracting information from high-dimensional data.
method Interdisciplinary approach combining statistics, optimization, and numerical linear algebra.
result Significant progress in tensor analysis over the last decade.
New CSC model extracts EEG signals with low noise sensitivity.
problem Analyzing noisy EEG signals during anesthesia.
method Kruskal CSC model using Kruskal decomposition for low-rank tensor activations.
result TC-FISTA efficiently extracts robust, sparse, and interpretable EEG encodings.
We consider the Principal Component Analysis problem for large tensors of arbitrary order k under a single-spike (or rank-one plus noise) model. On the one hand, we use information theory, and recent results in probability theory, to establish necessary and sufficient conditions under which the principal component ca…
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.
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.
New method tackles tensor regression with robust Kaczmarz approach.
problem Reconstructing tensor signals from corrupted measurements.
method Quantile-based randomized Kaczmarz method for tensor linear systems.
result Improved convergence and robustness to adversarial corruptions.
Efficient method for tensor linear form inference with noisy incomplete data.
problem Statistical inference of tensor linear forms with incomplete and noisy observations.
method Initial estimate + debiasing + one-step power iteration.
result Optimal uncertainty quantification and statistical-to-computational gaps examined.
Novel tensor perturbation bounds for orthogonal iteration methods.
problem Developing robust bounds for tensor reconstruction and subspace estimation.
method Blockwise tensor perturbation bounds for high-order orthogonal iteration (HOOI).
result Upper bounds for singular subspace estimation converge linearly and tensor reconstruction error bound is characterized by a simple quantity.
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…
New approach uses random matrix theory to understand tensor estimation performance.
problem Understanding the performance of estimators for low-rank signals in noisy tensors.
method Developed a new approach using random matrix theory to study random tensors.
result Discovered a fixed-point equation that matches the performance of the maximum likelihood estimator.
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.
SMPI recovers tensor spikes from noisy data with improved performance.
problem Recovering tensor spikes corrupted by Gaussian noise.
method Selective Multiple Power Iterations (SMPI) with polynomial random initializations and symmetrized tensor power iterations.
result SMPI outperforms existing algorithms and approaches theoretical optimal recovery.
Paper proposes a new tensor model for mixed memberships and provides error bounds.
problem Estimating mixed memberships in higher-order multiway data.
method Tensor mixed-membership blockmodel, higher-order orthogonal iteration algorithm (HOOI), simplex corner-finding algorithm.
result Consistency of estimation procedure with error bounds under specific conditions.
Develops a new tensor model for clustering with degree correction.
problem Clustering with unknown degree heterogeneity in multiway data.
method Degree-corrected tensor block model with estimation guarantees.
result Demonstrates an intrinsic statistical-to-computational gap for tensors of order three or greater.
Neuroimaging modalities such as functional magnetic resonance imaging (fMRI) and electroencephalography (EEG) provide information about neurological functions in complementary spatiotemporal resolutions; therefore, fusion of these modalities is expected to provide better understanding of brain activity. In this paper, …
Improved tensor rank learning for CPD models using a generalized hyperbolic prior.
problem Inaccurate tensor rank determination leads to overfitting or underfitting in CPD models.
method Introduced a generalized hyperbolic prior for automatic tensor rank learning in probabilistic CPD models.
result Significantly improved performance in learning both low and high tensor ranks, even for low SNR cases.