Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics

From MaRDI portal

Publication:1240771

Jump to:navigation, search

DOI10.1016/0024-3795(77)90069-6zbMath0364.15021OpenAlexW2057503509MaRDI QIDQ1240771

Joseph B. Kruskal

Publication date: 1977

Published in: Linear Algebra and its Applications (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/0024-3795(77)90069-6

Mathematics Subject Classification ID

Linear inference, regression (62J99) Multilinear algebra, tensor calculus (15A69)

Related Items

Estimating latent trends in multivariate longitudinal data via Parafac2 with functional and structural constraints, Efficient Identification of Butterfly Sparse Matrix Factorizations, Low Permutation-rank Matrices: Structural Properties and Noisy Completion, Identifiability for the k-secant variety of the Segre-Veronese varieties, Separability of Hermitian tensors and PSD decompositions, Unnamed Item, Identifiability of Bifactor Models, $(L_r,L_r,1)$-Decompositions, Sparse Component Analysis, and the Blind Separation of Sums of Exponentials, Effective Criteria for Specific Identifiability of Tensors and Forms, Linear convergence of an alternating polar decomposition method for low rank orthogonal tensor approximations, Decomposition Algorithms for Tensors and Polynomials, A generalization of Kruskal’s theorem on tensor decomposition, On the description of identifiable quartics, Optimal estimation of high-dimensional Gaussian location mixtures, Complete decomposition of symmetric tensors in linear time and polylogarithmic precision, Tensor Codes and Their Invariants, Random effect models for multivariate mixed data: A Parafac-based finite mixture approach, Nonlinear algebra and applications, CPclus: Candecomp/Parafac clustering model for three-way data, Embedded varieties, \(X\)-ranks and uniqueness or finiteness of the solutions, On the multigraded Hilbert function of lines and rational curves in multiprojective spaces, From non-defectivity to identifiability, Generic identifiability of the DINA model and blessing of latent dependence, A note on weaker conditions for identifying restricted latent class models for binary responses, On Best Low Rank Approximation of Positive Definite Tensors, Further results on tensor nuclear norms, Estimation under group actions: recovering orbits from invariants, Identifiability of hidden Markov models for learning trajectories in cognitive diagnosis, An algorithm for the non-identifiability of rank-3 tensors, Constructions of perfect bases for classes of 3-tensors, Hankel Tensor Decompositions and Ranks, Hermitian Tensor Decompositions, The Epsilon-Alternating Least Squares for Orthogonal Low-Rank Tensor Approximation and Its Global Convergence, Unnamed Item, Convex Relaxation Approaches for Strictly Correlated Density Functional Theory, On the hyperdeterminant for 2×2×3 arrays, Subtracting a best rank-1 approximation may increase tensor rank, Exact line and plane search for tensor optimization, Comment: The Challenges of Multiple Causes, Tensor Representation of Rank-Metric Codes, Pencil-Based Algorithms for Tensor Rank Decomposition are not Stable, Fiber Sampling Approach to Canonical Polyadic Decomposition and Application to Tensor Completion, Unnamed Item, Multilinear Control Systems Theory, Coupled Canonical Polyadic Decompositions and (Coupled) Decompositions in Multilinear Rank-$(L_r,n,L_r,n,1)$ Terms---Part I: Uniqueness, Brain Regions Identified as Being Associated With Verbal Reasoning Through the Use of Imaging Regression via Internal Variation, Global estimation of finite mixture and misclassification models with an application to multiple equilibria, A Recursive Eigenspace Computation for the Canonical Polyadic Decomposition, Guarantees for Existence of a Best Canonical Polyadic Approximation of a Noisy Low-Rank Tensor, New Riemannian Preconditioned Algorithms for Tensor Completion via Polyadic Decomposition, Decoupling Multivariate Polynomials Using First-Order Information and Tensor Decompositions, Two-dimensional direction finding of coherent signals with a linear array of vector hydrophones, Maximal rank of an element of a tensor product, The Optimization Landscape for Fitting a Rank-2 Tensor with a Rank-1 Tensor, About the maximal rank of 3-tensors over the real and the complex number field, Generic Uniqueness Conditions for the Canonical Polyadic Decomposition and INDSCAL, Successive Rank-One Approximations for Nearly Orthogonally Decomposable Symmetric Tensors, Explicit Candecomp/Parafac solutions for a contrived 2\(\times 2\times 2\) array of rank three, Overcomplete Order-3 Tensor Decomposition, Blind Deconvolution, and Gaussian Mixture Models, Segre Functions in Multiprojective Spaces and Tensor Analysis, Non-minimum tensor rank Gabidulin codes, Sufficient and Necessary Conditions for the Identifiability of the $Q$-matrix, Identifying Latent Structures in Restricted Latent Class Models, On approximation algorithm for orthogonal low-rank tensor approximation, New Uniqueness Conditions for the Canonical Polyadic Decomposition of Third-Order Tensors, Robust Joint Design of Linear Relay Precoder and Destination Equalizer for Dual-Hop Amplify-and-Forward MIMO Relay Systems, Topology of tensor ranks, An approximation method of CP rank for third-order tensor completion, Rank of a Hadamard product, Linearly dependent subsets of Segre varieties, On the identifiability of ternary forms, Reverse-order law for core inverse of tensors, Condition numbers for the tensor rank decomposition, An exploratory diagnostic model for ordinal responses with binary attributes: identifiability and estimation, An exponential lower bound for the degrees of invariants of cubic forms and tensor actions, Algebraic geometry and representation theory in the study of matrix multiplication complexity and other problems in theoretical computer science, Fast nonnegative tensor factorizations with tensor train model, Latent class model with conditional dependency per modes to cluster categorical data, The symmetric rank and decomposition of \(m\)-order \(n\)-dimensional \((n = 2,3,4)\) symmetric tensors over the binary field, Criteria for SLOCC and LU equivalence of generic multi-qudit states, Unnamed Item, Rank properties and computational methods for orthogonal tensor decompositions, An Algebraic Approach to Nonorthogonal General Joint Block Diagonalization, Identifiability for a class of symmetric tensors, A sparse latent class model for cognitive diagnosis, Symmetric Hermitian decomposability criterion, decomposition, and its applications, Erasure Coding for Fault-Oblivious Linear System Solvers, The Condition Number of Join Decompositions, Mixed membership Gaussians, The smoothed complexity of Frank-Wolfe methods via conditioning of random matrices and polytopes, Algorithms for Sparse Nonnegative Tucker Decompositions, Unnamed Item, Unnamed Item, Uni-mode uniqueness conditions for CANDECOMP/PARAFAC decomposition of \(n\)-way arrays with linearly dependent loadings, On generic identifiability of symmetric tensors of subgeneric rank, On global convergence of alternating least squares for tensor approximation, The average condition number of most tensor rank decomposition problems is infinite, Finding Planted Subgraphs with Few Eigenvalues using the Schur--Horn Relaxation, Fast matrix multiplication and its algebraic neighbourhood, Factor uniqueness of the structural Parafac model, Latent theme dictionary model for finding co-occurrent patterns in process data, Constrained Candecomp/Parafac via the Lasso, The special sign indeterminacy of the direct-fitting Parafac2 model: some implications, cautions, and recommendations for simultaneous component analysis, Learning Paths from Signature Tensors, On the Problem of Decoupling Multivariate Polynomials, A robust Parafac model for compositional data, Diagonalizable higher degree forms and symmetric tensors, Low rank Tucker-type tensor approximation to classical potentials, Tensor Rank is Hard to Approximate, On the identifiability of binary Segre products, Real identifiability vs. complex identifiability, Nonlinear system modeling and identification using Volterra‐PARAFAC models, On the best rank-1 approximation to higher-order symmetric tensors, The typical rank of tall three-way arrays, INDCLAS: a three-way hierarchical classes model, Numerical multilinear algebra and its applications, Semi-parametric estimation for conditional independence multivariate finite mixture models, Degeneracy in Candecomp/Parafac explained for \(p\times p\times 2\) arrays of rank \(p+1\) or higher, Independent component analysis for~three-way data with an application from~atmospheric science, Sufficient conditions for uniqueness in Candecomp/Parafac and Indscal with random component matrices, A concise proof of Kruskal's theorem on tensor decomposition, Unnamed Item, Perturbation analysis for matrix joint block diagonalization, Numerical tensor calculus, Imposing Independence Constraints in the CP Model, Remarks on the Symmetric Rank of Symmetric Tensors, Nonlinear predictive controller based on S-PARAFAC Volterra models applied to a communicating two-tank system, Degeneracy in Candecomp/Parafac and Indscal explained for several three-sliced arrays with a two-valued typical rank, Model-based clustering of time-evolving networks through temporal exponential-family random graph models, Resonator Networks, 2: Factorization Performance and Capacity Compared to Optimization-Based Methods, An inexact augmented Lagrangian method for computing strongly orthogonal decompositions of tensors, Identification of Wiener channels using a tensor approach, Theoretical grounding for estimation in conditional independence multivariate finite mixture models, Efficient Evaluation of Noncommutative Polynomials Using Tensor and Noncommutative Waring Decompositions, Homotopy techniques for tensor decomposition and perfect identifiability, Semialgebraic Geometry of Nonnegative Tensor Rank, Three-way clustering of multi-tissue multi-individual gene expression data using semi-nonnegative tensor decomposition, On Koopman mode decomposition and tensor component analysis, An operator theoretic approach to nonparametric mixture models, Incremental CP Tensor Decomposition by Alternating Minimization Method, On the identifiability of diagnostic classification models, Stand-alone results and applications in multilinear algebra, The Hitchhiker guide to: secant varieties and tensor decomposition, Three-mode factor analysis by means of Candecomp/Parafac, Comparison of Accuracy and Scalability of Gauss--Newton and Alternating Least Squares for CANDECOMC/PARAFAC Decomposition, Unnamed Item, Time-Varying Autoregression with Low-Rank Tensors, A spectral algorithm for latent Dirichlet allocation, A note on the ranks of 2 × 2 × 2 and 2 × 2 × 2 × 2 tensors, Minimality and uniqueness for decompositions of specific ternary forms, Tensor factorisation for narrowband single channel source decomposition, Nonexistence of \(n\)-qubit unextendible product bases of size \(2^n-5\), Estimating multivariate latent-structure models, Matrices with special reference to applications in psychometrics, Typical rank and indscal dimensionality for symmetric three-way arrays of order \(I\times 2\times 2\) or \(I\times 3\times 3\), Blind PARAFAC signal detection for polarization sensitive array, 4D near-field source localization using cumulant, On the spectral problem for trivariate functions, A seminorm regularized alternating least squares algorithm for canonical tensor decomposition, A new method of moments for latent variable models, Dimensionality reduction in higher-order signal processing and rank-\((R_1,R_2,\ldots,R_N)\) reduction in multilinear algebra, On the nuclear norm and the singular value decomposition of tensors, Set evincing the ranks with respect to an embedded variety (symmetric tensor rank and tensor rank), A very brief introduction to nonnegative tensors from the geometric viewpoint, Smoothed analysis for tensor methods in unsupervised learning, Three-way analysis of imprecise data, A user-oriented overview of multiway methods and software, Local unitary equivalence of generic multi-qubits based on the CP decomposition, Joint frequency, 2-D DOA, and polarization estimation using parallel factor analysis, Functional parallel factor analysis for functions of one- and two-dimensional arguments, Symmetry transformations for square sliced three-way arrays, with applications to their typical rank, On Kruskal's uniqueness condition for the Candecomp/Parafac decomposition, A randomized algorithm for a tensor-based generalization of the singular value decomposition, Canonical polyadic decomposition of third-order tensors: relaxed uniqueness conditions and algebraic algorithm, Nonparametric mixture models with conditionally independent multivariate component densities, A variational approach of the rank function, Uniqueness of three-mode factor models with sparse cores: The \(3\times 3\times 3\) case, In memoriam Joseph B. Kruskal 1928--2010, A singular value decomposition of a \(k\)-way array for a principal component analysis of multiway data, \(\text{PTA-}k\), Computing the unique CANDECOMP/PARAFAC decomposition of unbalanced tensors by homotopy method, Symmetric tensor decomposition, PARAFAC-based channel estimation and data recovery in nonlinear MIMO spread spectrum communication systems, Three-factor association models for three-way contingency tables, Exact probabilities for typical ranks of \(2 \times 2 \times 2\) and \(3 \times 3 \times 2\) tensors, Kruskal's uniqueness inequality is sharp, Some convergence results on the regularized alternating least-squares method for tensor decomposition, Tensor decompositions with banded matrix factors, Typical ranks for \(m\times n\times (m-1)n\) tensors with \(m \leq n\), Grassmann secants, identifiability, and linear systems of tensors, Bounds on the ranks of some 3-tensors, Bayesian multi-tensor factorization, A contemporary linear representation theory for ordinary differential equations: multilinear algebra in folded arrays (folarrs) perspective and its use in multidimensional case, Nonnegative tensor factorization as an alternative Csiszar-Tusnady procedure: algorithms, convergence, probabilistic interpretations and novel probabilistic tensor latent variable analysis algorithms, Effective identifiability criteria for tensors and polynomials, Identifiability of large phylogenetic mixture models, Cayley's hyperdeterminant: A combinatorial approach via representation theory, Parameter identifiability in a class of random graph mixture models, Simplicity and typical rank results for three-way arrays, Uniqueness of \(N\)-way \(N\)-mode hierarchical classes models., The tensor rank of tensor product of two three-qubit W states is eight, A probabilistic foundation for dynamical systems: theoretical background and mathematical formulation, Two-dimensional DOA estimation for generalized coprime planar arrays: a fast-convergence trilinear decomposition approach, Blind identification of multiuser nonlinear channels using tensor decomposition and precoding, Hierarchical multilinear models for multiway data, On the generic and typical ranks of 3-tensors, A contribution to the visualisation of three-way arrays, On uniqueness in CANDECOMP/PARAFAC, The \(K\)-INDSCAL model for heterogeneous three-way dissimilarity data, The \(3\times 3\times 3\) hyperdeterminant as a polynomial in the fundamental invariants for \(\mathrm{SL}_3(\mathbb C)\times\mathrm{SL}_3(\mathbb C)\times\mathrm{SL}_3(\mathbb C)\), On the non-existence of optimal solutions and the occurrence of ``degeneracy in the CANDECOMP/PARAFAC model, Hyperspectral image classification based on mathematical morphology and tensor decomposition, Decomposition-by-normalization (DBN): leveraging approximate functional dependencies for efficient CP and Tucker decompositions, Weighted Moore-Penrose inverses of arbitrary-order tensors, Some uniqueness results for PARAFAC2, Uniqueness proof for a family of models sharing features of Tucker's three-mode factor analysis and PARAFAC/CANDECOMP, On a geometric method for the identifiability of forms, Efficient semiparametric estimation and model selection for multidimensional mixtures, Multilayer tensor factorization with applications to recommender systems, Kruskal's condition for uniqueness in Candecomp/Parafac when ranks and \(k\)-ranks coincide, The Carroll and Chang conjecture of equal Indscal components when Candecomp/Parafac gives perfect fit, Bounds on the tensor rank, Kruskal's polynomial for \(2 \times{}2 \times{}2\) arrays and a generalization to \(2 \times{}n \times{}n\) arrays, Beyond principal component analysis: A trilinear decomposition model and least squares estimation, Computing the spark: mixed-integer programming for the (vector) matroid girth problem, Direction finding via biquaternion matrix diagonalization with vector-sensors, Iterative methods for the canonical decomposition of multi-way arrays: application to blind underdetermined mixture identification, Bilinear factorizations subject to monomial equality constraints via tensor decompositions, PARAFAC: Parallel factor analysis, Tensor ranks for the inversion of tensor-product binomials, Polynomial meta-models with canonical low-rank approximations: numerical insights and comparison to sparse polynomial chaos expansions, On the optimal evaluation of a set of bilinear forms, Nonparametric estimation of a latent variable model, Simplicity transformations for three-way arrays with symmetric slices, and applications to Tucker-3 models with sparse core arrays, Global properties of tensor rank, Blind equalization of nonlinear channels using a tensor decomposition with code/space/time diversities, Identifiability of rank-3 tensors, On uniqueness conditions for Candecomp/Parafac and Indscal with full column rank in one mode, Matrix inversion cases with size-independent tensor rank estimates, On large-scale dynamic topic modeling with nonnegative CP tensor decomposition, Identifiability of parameters in latent structure models with many observed variables, Hadamard powers of rank two, doubly nonnegative matrices, Key-frame detection and super-resolution of hyperspectral video via sparse-based cumulative tensor factorization, Spectrum blind reconstruction and direction of arrival estimation of multi-band signals at sub-Nyquist sampling rates, Rank and optimal computation of generic tensors, On commutativity and approximation, Polystability in positive characteristic and degree lower bounds for invariant rings, On the maximum rank of a tensor product, Identifiability of latent class models with covariates, Rank-metric lattices, Inertial accelerated SGD algorithms for solving large-scale lower-rank tensor CP decomposition problems, Low-CP-rank tensor completion via practical regularization

Cites Work

Retrieved from "https://portal.mardi4nfdi.de/w/index.php?title=Publication:1240771&oldid=13322629"