Paper studies tensor models using random matrix theory.
problem Analyzing asymmetric order-d spiked tensor models with Gaussian noise.
method Uses variational definition of singular vectors and values, constructs equivalent spiked symmetric block-wise random matrix from tensor contractions.
result Characterizes asymptotic singular values and alignments of singular vectors with true spike components.
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 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.
New concentration inequalities for tensors with heavy-tailed coefficients.
problem Developing bounds for Euclidean functions of tensors with sub-Weibull distributions.
method Extending concentration inequalities to sub-Weibull random tensors, using new inequalities for heavy-tailed random variables and martingale analysis.
result Established a phase transition between sub-gaussian and heavy-tailed regimes for Euclidean functions of tensors.
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.
Estimates the probability of a random symmetric tensor being close to rank-one.
problem Estimating the probability of a random symmetric tensor being close to rank-one.
method Using Weyl's tube formula and techniques from Random Matrix theory, we study metric invariants of the real Veronese variety.
result Explicit formula for the reach and curvature coefficients of the real Veronese variety with respect to the Bombieri-Weyl metric.
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 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.
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.
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.
The paper analyzes tensor recovery from symmetric rank-one measurements using information theory.
problem Recovering tensors with low symmetric rank from symmetric rank-one measurements.
method Covering numbers argument, Carbery-Wright inequality, orthogonal polynomials, Fano's inequality.
result Near-optimal sample complexity bounds for log-concave distributions.
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…
Tensorized random projections reduce high-dimensional tensor size efficiently.
problem Efficiently reducing the dimension of very high-dimensional tensors.
method Proposes two tensorized random projection maps using TT and CP decompositions.
result TT format offers superior performance in terms of required random projection size.
The aim of this paper is twofold. On the one hand, it provides a review of the links between random tensor models, seen as quantum gravity theories, and the PL-manifolds representation by means of edge-colored graphs (crystallization theory). On the other hand, the core of the paper is to establish results about the to…
Paper proposes robust tensor regression method for tensor data analysis.
problem Outliers in tensor data analysis can make existing methods sensitive.
method Nonconvex relaxation of tensor tubal rank in optimization framework.
result Global convergence of proposed estimation algorithm under mild assumptions.
Study Gaussian-process limits of neural networks using tensor programs.
problem Understanding the behavior of neural networks as they approach infinite width.
method Quantitative analysis through tensor programs and Wasserstein distance.
result Explicit finite-width error bounds, showing convergence to Gaussian-process limits.
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.
New algorithms solve tensor problems with random components using SDP.
problem Exact tensor nuclear norm, decomposition, and completion for random tensors.
method Degree-4 Sum of Squares (SOS) semidefinite programs.
result Exact solutions for tensor nuclear norm, decomposition, and completion with random asymmetric components.
Tensor train (TT) decomposition provides a space-efficient representation for higher-order tensors. Despite its advantage, we face two crucial limitations when we apply the TT decomposition to machine learning problems: the lack of statistical theory and of scalable algorithms. In this paper, we address the limitations…
Sparse sampling method for tensor factorization and completion of high rank tensors.
problem Completion of high rank tensors with missing data in recommendation systems.
method Sparse measurements and message-passing algorithms in a high-dimensional limit.
result Theoretical insights and performance analysis of tensor factorization in dense limit.
Gradient descent in tensor factorization favors low-rank solutions.
problem Tackling implicit regularization in tensor factorization problems.
method Gradient descent with small random initialization for overparametrized tensor factorization.
result Gradient descent leads to implicit regularization towards low tubal rank solutions.
Stochastic trace estimation with tensor train random vectors
problem Stochastic trace estimation for large-scale matrices
method Gaussian random tensor train vectors
result Median-of-means variant achieves dimension-independent guarantees
Develops a new feature theory for robust machine learning.
problem Creating robust machine learning features from training data.
method Stochastic tensor space feature theory with Karhunen-Loeve expansion and hierarchical subspaces.
result Dramatic increases in accuracy for predicting Alzheimer's disease stages.
Develops path integral for spiked tensor model dynamics.
problem Dynamics of spiked tensor model with random initial conditions.
method Path integral approach applied to partial differential equations.
result Large-N saddle point equations dominated by melonic diagrams. 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.
Paper analyzes infinite-width attention layers using Tensor Programs.
problem Capturing the infinite-width limit of attention layers.
method Tensor Programs framework to rigorously identify the limit distribution.
result Derives exact form of infinite-width limit distribution without Gaussian approximations.
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.
This work develops efficient methods for computing moments of Gaussian mixtures.
problem Efficient computation of moments for Gaussian mixtures with large dimensions.
method Theory and numerical methods for implicit computations with moment tensors of Gaussian mixtures.
result Reduced computational and storage costs for moment tensors of Gaussian mixtures.
New tools in nonlinear random matrices improve understanding of the Sum of Squares hierarchy.
problem Improving the Sum of Squares (SoS) hierarchy's performance on average-case problems.
method Developed new tools in nonlinear random matrices and applied them to analyze the SoS hierarchy.
result Subexponential-time SoS lower bounds for various problems, offering evidence for the low-degree likelihood ratio hypothesis.
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.
Estimates metric tensor on neuromanifolds using Fisher information and random methods.
problem Computing the metric tensor on high-dimensional neuromanifolds efficiently and accurately.
method Deterministic bounds and unbiased random estimators based on Hutchinson's trace method.
result An efficient random estimator with bounded standard deviation.
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…
This paper conducts a rigorous analysis for provable estimation of multidimensional arrays, in particular third-order tensors, from a random subset of its corrupted entries. Our study rests heavily on a recently proposed tensor algebraic framework in which we can obtain tensor singular value decomposition (t-SVD) that …
A new algorithm speeds up CP decomposition for large tensors.
problem Efficiently processing large-scale tensors in real-time.
method Randomized online CP decomposition (ROCP) algorithm.
result ROCP reduces computing time and memory usage significantly.
A new tensor completion method handles missing data with missing not at random entries.
problem Handling missing data in tensors where the probability of observation depends on other entries.
method Estimate propensities using convex relaxation, then use higher-order SVD with inverse propensities weights.
result Finite-sample error bounds on the completed tensor are provided.
Tensor network decomposition, originated from quantum physics to model entangled many-particle quantum systems, turns out to be a promising mathematical technique to efficiently represent and process big data in parsimonious manner. In this study, we show that tensor networks can systematically partition structured dat…
New method approximates high-dimensional probability densities efficiently.
problem Approximating high-dimensional probability densities accurately and efficiently.
method Hierarchical tensor-network approach using randomized SVD and linear equations.
result The method effectively approximates high-dimensional densities with linear complexity.
Surveying random sections on Kähler manifolds, leading to metrics.
problem Understanding statistics of random sections on Kähler manifolds.
method Analyzing tensor powers of line bundles.
result Induced metrics from random sections.
Corrected whitening restores orthogonality in high-dimensional spherical Gaussian mixtures.
problem In high-dimensional data, standard whitening fails to preserve orthogonality of mixture means.
method Derived exact limits for whitened means dot products using random matrix theory, constructed a corrected whitening matrix.
result Corrected whitening allows for improved estimation of spherical Gaussian mixtures in the large-dimensional regime.
Paper generalizes tensor-train approximation for complex random variables.
problem Characterizing intractable high-dimensional random variables.
method Extends inverse Rosenblatt transform to general reference measures and integrates into deep variable transformation framework.
result Deep inverse Rosenblatt transport significantly expands tensor approximations for complex random variables.
Estimates joint probability distribution from 1-way marginals using low-rank tensors and random projections.
problem Nonparametric estimation of joint probability mass function (PMF) from limited data.
method Low-rank tensor decomposition and random projections to link data to PMF estimation.
result Estimates joint density from 1-way marginals using transformed space and novel algorithm.
Research proves the semi-classical limit of Liouville conformal field theory, describing deterministic geometry from random fluctuations.
problem Proving the semi-classical limit of Liouville conformal field theory.
method Probabilistic definition of Liouville theory, proving existence of semi-classical limit, defining classical stress-energy tensor.
result Existence and description of the semi-classical limit in terms of a massive Gaussian free field with Robin boundary conditions.
Paper refutes conjecture on tensor power iteration convergence in overcomplete models.
problem Understanding convergence of tensor power iteration in overcomplete random tensors.
method Analysis of tensor power iteration dynamics from random initialization.
result Polynomially many steps are necessary for convergence, refutes logarithmic conjecture.
TEC combines multiple RPSTMs to classify big tensors efficiently.
problem Tensor classification for big data applications.
method Tensor Ensemble Classifier (TEC) using Random Projection-based Support Tensor Machine (RPSTM).
result TEC provides statistically consistent predictions with reduced computational cost.
A new method reduces the complexity of tensor products from cubic to quadratic, improving both speed and accuracy.
problem Efficiently computing high-dimensional tensor products for polynomial kernels.
method Complex-to-Real (CtR) modification of sketches using complex random projections.
result Achieves state-of-the-art performance in accuracy and speed.
Tensor factorization arises in many machine learning applications, such knowledge base modeling and parameter estimation in latent variable models. However, numerical methods for tensor factorization have not reached the level of maturity of matrix factorization methods. In this paper, we propose a new method for CP te…
TTRP method preserves distances in high-dimensional data with reduced storage and speed.
problem Preserving distances in high-dimensional datasets efficiently and accurately.
method Tensor train random projection (TTRP) using TT-ranks of one.
result TTRP is an expected isometric projection with bounded variance.
Sketching is a randomized dimensionality-reduction method that aims to preserve relevant information in large-scale datasets. Count sketch is a simple popular sketch which uses a randomized hash function to achieve compression. In this paper, we propose a novel extension known as Higher-order Count Sketch (HCS). While …