{ "id": "epbtnr7vfn", "slug": "tensor-factorization", "title": "Tensor factorization", "updated": "2026-09-30", "topic_path": [ { "id": "physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical" }, { "id": "physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math" }, { "id": "physical.math.statistics", "label": "Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics" }, { "id": "physical.math.statistics.multivariate-analysis", "label": "Multivariate association and dimension reduction", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics.multivariate-analysis" }, { "id": "physical.math.statistics.multivariate-analysis.alpha-mo-to-w", "label": "Titles Mo to W", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics.multivariate-analysis.alpha-mo-to-w" } ], "geo": [ { "id": "geo.nongeo.t1946.physical.math.statistics.multivariate-analysis", "label": "Non-geographic · 1946 to 2000: Multivariate association and dimension reduction", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics.multivariate-analysis", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1946", "label": "Non-geographic · 1946 to 2000", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946" }, { "id": "geo.nongeo.t1946.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical" }, { "id": "geo.nongeo.t1946.physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math" }, { "id": "geo.nongeo.t1946.physical.math.statistics", "label": "Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics" }, { "id": "geo.nongeo.t1946.physical.math.statistics.multivariate-analysis", "label": "Multivariate association and dimension reduction", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics.multivariate-analysis" } ] }, { "id": "geo.nongeo.t2001.physical.math.statistics", "label": "Non-geographic · 2001 to 2020: Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.physical.math.statistics", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t2001", "label": "Non-geographic · 2001 to 2020", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001" }, { "id": "geo.nongeo.t2001.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.physical" }, { "id": "geo.nongeo.t2001.physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.physical.math" }, { "id": "geo.nongeo.t2001.physical.math.statistics", "label": "Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.physical.math.statistics" } ] } ], "excerpt": "Tensor factorization decomposes a multiway array (tensor) into lower-dimensional component factors for dimensionality reduction and latent-structure recovery; its two central models, CP and Tucker, generalize the matrix singular value decomposition.", "snippet": "Tensor factorization decomposes a multiway array (tensor) into lower-dimensional component factors for dimensionality reduction and latent-structure recovery; its two central models, CP and Tucker, generalize the matrix singular value decomposition.", "node": "physical.math.statistics.multivariate-analysis.alpha-mo-to-w", "markdown": "# Tensor factorization\n\nTensor factorization decomposes a multiway array (a tensor) into a small number of lower-dimensional component factors, and is used for dimensionality reduction, estimation, and recovery of latent structure in high-dimensional data. The two central models are the CP (CANDECOMP/PARAFAC) decomposition, which writes a tensor as a sum of rank-one terms, and the [Tucker decomposition](https://www.edgepedia.org/tucker-decomposition), a higher-order form of principal component analysis; both generalize the matrix singular value decomposition (SVD) to arrays with three or more modes.[1](https://epubs.siam.org/doi/10.1137/07070111X)\n\n| Key fact | Detail |\n|---|---|\n| CP model | A tensor is written as a sum of R rank-one tensors, each an outer product of one vector per mode[1](https://epubs.siam.org/doi/10.1137/07070111X) |\n| Tucker model | A core tensor multiplied by a matrix along each mode[1](https://epubs.siam.org/doi/10.1137/07070111X) |\n| Parameter savings | CP reduces a \\( D_{\\mathrm{th}} \\)-order tensor from \\( \\prod_{d=1}^{D} p_d \\) free parameters to \\( R \\cdot \\sum_{d=1}^{D} p_d \\)[2](https://web.ics.purdue.edu/~sun244/Tensor_survey.pdf) |\n| Uniqueness | CP is essentially unique under mild conditions; Kruskal's sufficient condition is stated in terms of the k-ranks of the factor matrices[3](https://doi.org/10.1016/0024-3795%2877%2990069-6) |\n| Workhorse algorithm | Alternating least squares: fix all factor matrices but one, solve a linear least-squares subproblem, repeat[1](https://epubs.siam.org/doi/10.1137/07070111X) |\n| Main failure mode | The best rank-R approximation may not exist, producing degenerate solutions with diverging components[4](https://pmc.ncbi.nlm.nih.gov/articles/PMC2792364/) |\n| Rank choice | Tensor rank is NP-hard to compute, so in practice one fits several ranks and compares fit[5](https://ar5iv.labs.arxiv.org/html/1711.10781) |\n\n## How it works\n\nA rank-one third-order tensor of size \\( I \\times J \\times K \\) is the outer product of three vectors, \\( X(i,j,k) = a(i)\\,b(j)\\,c(k) \\).[6](http://www.cs.ucr.edu/~epapalex/papers/tsp17-tensors.pdf) The rank of a tensor is the smallest number of such rank-one terms that sum to it. Unlike matrix rank, tensor rank can differ depending on whether the entries are treated as real or complex, and computing tensor rank is NP-hard, so exact determination can be intractable for large or general inputs.[1](https://epubs.siam.org/doi/10.1137/07070111X)\n\nThe CP decomposition approximates a tensor as\n\n\\[ X \\approx \\sum_{r=1}^{R} a_r \\circ b_r \\circ c_r, \\]\n\nwhere each triple of vectors forms one component and \\( R \\) is the chosen rank. The Tucker decomposition instead writes\n\n\\[ X \\approx G \\times_1 A \\times_2 B \\times_3 C, \\]\n\na core tensor \\( G \\) transformed by a factor matrix along each mode; it is a form of higher-order PCA.[1](https://epubs.siam.org/doi/10.1137/07070111X)\n\nUniqueness is where tensors depart most sharply from matrices. A matrix factorization is unique only under extra assumptions such as the orthogonality imposed by the SVD, and matrix factor analysis suffers from a rotation problem; tensor decompositions are unique under much milder conditions.[7](https://oar.princeton.edu/bitstream/88435/pr1w814/1/TensorDecompositionsUniquenessThreshold.pdf) Harshman's 1970 theorem gives uniqueness, with a polynomial-time algorithm, when two factor matrices are linearly independent and no two columns of the third are collinear.[7](https://oar.princeton.edu/bitstream/88435/pr1w814/1/TensorDecompositionsUniquenessThreshold.pdf) Kruskal's 1977 theorem gives uniqueness conditions expressed in terms of the k-ranks of the factor matrices, where the k-rank of a matrix is the largest k such that any k columns are linearly independent.[3](https://doi.org/10.1016/0024-3795%2877%2990069-6) The Tucker decomposition, by contrast, is generally not unique because its core tensor can be arbitrarily structured.[5](https://ar5iv.labs.arxiv.org/html/1711.10781)\n\n## How it is done\n\nFor CP, the workhorse is alternating least squares (ALS): fix two factor matrices, solve a linear least-squares problem for the third (the update can be written \\( T_{(n)} \\approx A_{(n)} \\cdot V_{\\{n\\}}^{\\mathrm{T}} \\) via Khatri–Rao products of the other factors), and cycle through the modes until a convergence criterion is met.[1](https://epubs.siam.org/doi/10.1137/07070111X) • [8](https://ftp.esat.kuleuven.be/pub/stadius/sistakulak/reports/algorithms_for_tensor_decompositions.pdf) ALS is simple to implement but can take many iterations, is not guaranteed to converge even to a stationary point, and depends heavily on the starting guess, so practitioners run it from multiple random initializations.[1](https://epubs.siam.org/doi/10.1137/07070111X)\n\nBecause tensor rank cannot be computed directly, one fits models at several ranks and picks the best approximation; the CORCONDIA consistency diagnostic compares different numbers of components.[5](https://ar5iv.labs.arxiv.org/html/1711.10781) • [1](https://epubs.siam.org/doi/10.1137/07070111X) Spectral initialization from the [HOSVD](https://www.edgepedia.org/higher-order-singular-value-decomposition), computed from leading singular vectors of mode matricizations (unfolding a tensor into a matrix), is widely used, and truncating to the HOSVD before ALS improves convergence speed provided the reduced dimensions are not smaller than the tensor rank; the HOOI algorithm refines an HOSVD initialization by ALS.[9](https://arxiv.org/html/2405.18412) • [10](https://www.gipsa-lab.grenoble-inp.fr/~pierre.comon/FichiersPdf/ComoLA09-jchemo.pdf) • [5](https://ar5iv.labs.arxiv.org/html/1711.10781) Numerical comparisons find nonlinear least-squares methods (Gauss–Newton, Levenberg–Marquardt) less sensitive to initialization, more efficient, and less prone to swamps, at higher memory and time cost per iteration; all-at-once optimizers (CP-OPT, CP-NLS) and direct computation via simultaneous diagonalization are the main alternatives.[8](https://ftp.esat.kuleuven.be/pub/stadius/sistakulak/reports/algorithms_for_tensor_decompositions.pdf) • [1](https://epubs.siam.org/doi/10.1137/07070111X) • [11](https://users.wfu.edu/ballard/pdfs/tensor_textbook.pdf) A hybrid initialization, TASD, compresses the tensor by Tucker decomposition (via HOOI) before applying simultaneous diagonalization to the low-dimensional core, improving robustness to noise.[12](https://doi.org/10.48550/arxiv.2505.23046) Degeneracy is monitored by watching the smallest singular values of the factor matrices together with the factor norms.[4](https://pmc.ncbi.nlm.nih.gov/articles/PMC2792364/) Standard software includes the N-way Toolbox, Tensor Toolbox, and Multilinear Engine.[1](https://epubs.siam.org/doi/10.1137/07070111X)\n\n## Origin\n\nHitchcock proposed expressing a tensor as a finite sum of rank-one terms, the polyadic form, in 1927.[13](https://doi.org/10.1002/sapm192761164) Cattell stated the principle of parallel proportional profiles for choosing factors in 1944.[14](https://doi.org/10.1007/bf02288739) Tucker introduced the three-mode factor analysis model, with a core matrix relating three factor matrices, in work culminating in his 1966 paper.[15](https://doi.org/10.1007/bf02289464) In 1970 the same trilinear model was proposed twice independently: Harshman presented PARAFAC, generalizing Cattell's principle and proving uniqueness under stated conditions,[16](https://psychology.uwo.ca/faculty/harshman/wpppfac0.pdf) while Carroll and Chang presented CANDECOMP, an N-way generalization of the Eckart–Young decomposition, from which they developed INDSCAL.[17](https://doi.org/10.1007/bf02310791) Kruskal supplied the classical uniqueness theorem in 1977.[3](https://doi.org/10.1016/0024-3795%2877%2990069-6)\n\n## Variants\n\nThe Tucker model is computed in its SVD-like constrained form by the HOSVD, which takes the mode-q factor matrices to be the left singular matrices of the mode-q matricization; truncating this constrained version usually gives a good approximation, though not the optimal one.[9](https://arxiv.org/html/2405.18412) • [18](https://ftp.esat.kuleuven.be/pub/stadius/delathauwer/reports/ldl-09-34.pdf) The block term decomposition, introduced by De Lathauwer in 2008, unifies the two models: with a single term it is a Tucker decomposition, and with scalar cores it is CP.[19](https://doi.org/10.1137/070690729) Nonnegative variants constrain the factor matrices to be nonnegative, following the interpretability arguments made for nonnegative matrix factorization; unlike unconstrained CP, nonnegative CP is always well defined.[20](https://psycnet.apa.org/doi/10.1145/2915921) • [21](https://ar5iv.labs.arxiv.org/html/1412.1885) Other named models in the family include PARAFAC2, INDSCAL, CANDELINC, DEDICOM, and PARATUCK2.[1](https://epubs.siam.org/doi/10.1137/07070111X) For very high-order tensors, the Tensor-Train decomposition, proposed by Oseledets in 2011, imposes a parsimonious model to counter the curse of dimensionality.[22](https://doi.org/10.1137/090752286) A 2024 algorithm based on Koszul–Young flattenings decomposes generic \\( n \\times n \\times n \\) tensors and certifies uniqueness up to rank \\( (2-\\epsilon)n \\). A 2014 paper proved identifiability of a specific \\( 5 \\times 5 \\times 5 \\) tensor of rank 7; this exceeds Kruskal's exact sufficient bound for those dimensions, but not the simplified \\( r \\le 3n/2 \\) bound.[23](https://arxiv.org/html/2411.14344v1) • [25](https://arxiv.org/abs/1403.4157)\n\n## Applications\n\nPsychometrics and chemometrics historically drove the theory and algorithms; signal processing followed in the 1990s, and computer science adopted tensors roughly a decade before 2017.[6](http://www.cs.ucr.edu/~epapalex/papers/tsp17-tensors.pdf) [Signal processing](https://www.edgepedia.org/signal-processing) uses CP for unsupervised separation of speech mixtures, code-division communication signals without knowledge of the codes, and emitter localization for radar and passive sensing.[6](http://www.cs.ucr.edu/~epapalex/papers/tsp17-tensors.pdf) In machine learning, tensor decompositions support temporal pattern discovery, knowledge-graph relation inference, and method-of-moments estimation for Gaussian mixture and topic models.[5](https://ar5iv.labs.arxiv.org/html/1711.10781) Tucker approximation is used for dimensionality reduction of large tensor datasets, subspace estimation, harmonic retrieval, and classification.[18](https://ftp.esat.kuleuven.be/pub/stadius/delathauwer/reports/ldl-09-34.pdf) Neural-network compression applies tensor-train, block-term, and Tucker decompositions to fully connected layers, which hold about 80% of VGG-19's parameters.[2](https://web.ics.purdue.edu/~sun244/Tensor_survey.pdf)\n\n## Limitations and alternatives\n\nThe sharpest contrast with matrices concerns existence. For matrices, the Eckart–Young theorem gives the best low-rank approximation by truncated SVD; for tensors no such theorem exists, and the best rank-k approximation may not exist at all, a prevalent phenomenon rather than a pathological exception.[10](https://www.gipsa-lab.grenoble-inp.fr/~pierre.comon/FichiersPdf/ComoLA09-jchemo.pdf) • [9](https://arxiv.org/html/2405.18412) De Silva and Lim showed that for \\( 2 \\le R \\le \\min(I,J,K) \\) there always exists a three-way array of rank \\( R+1 \\) with no optimal CP solution.[4](https://pmc.ncbi.nlm.nih.gov/articles/PMC2792364/) When no optimum exists, any sequence of factors decreasing the CP criterion to its infimum becomes nearly rank deficient while some factor norms tend to infinity, the signature of degeneracy; constraining one factor matrix to be column-wise orthonormal, or imposing nonnegativity on all factors, guarantees an optimal solution.[4](https://pmc.ncbi.nlm.nih.gov/articles/PMC2792364/) For generic \\( I \\times J \\times 2 \\) arrays, diverging components can be handled by computing a generalized [Schur decomposition](https://www.edgepedia.org/schur-decomposition) instead, which always exists and splits into a nondiverging CP part plus a sparse Tucker3 part.[24](https://epubs.siam.org/doi/10.1137/070692121)\n\nCP algorithms also converge to local solutions, especially when bottlenecks exist in a factor matrix or the model is not unique, and the loss surface has exponentially many local minima when the signal-to-noise ratio is weak.[21](https://ar5iv.labs.arxiv.org/html/1412.1885) • [12](https://doi.org/10.48550/arxiv.2505.23046) Spurious local minima appear when the signal strength \\( \\lambda \\) is much smaller than the dimension \\( d \\), while power iteration initialized by the HOSVD yields consistent estimation when \\( \\lambda \\gtrsim d^{3/4} \\).[9](https://arxiv.org/html/2405.18412) Tang, Chhor, Klopp, and Zhang established non-asymptotic, minimax-optimal error bounds for ALS under general CP models with arbitrary rank and order, matched by lower bounds, and proved a two-phase convergence behavior: a quadratic phase under low coherence followed by a linear phase near the truth.[12](https://doi.org/10.48550/arxiv.2505.23046) Choosing between the models is a trade-off: CP gives the most compact, essentially unique representation but can be ill-posed and fit poorly, while Tucker is not unique but often fits better and has been shown generally more effective at estimating missing values.[21](https://ar5iv.labs.arxiv.org/html/1412.1885) • [20](https://psycnet.apa.org/doi/10.1145/2915921) A common rule of thumb is CP for latent parameter estimation and Tucker for subspace estimation, compression, and dimensionality reduction.[5](https://ar5iv.labs.arxiv.org/html/1711.10781) Flattening a tensor and applying PCA is statistically inefficient and lacks interpretability, and the matrix PCA/SVD equivalence does not carry over to multiway PCA.[9](https://arxiv.org/html/2405.18412)\n\n## References\n\n1. [Tensor Decompositions and Applications (Kolda & Bader, SIAM Review 2009)](https://epubs.siam.org/doi/10.1137/07070111X)\n2. [An Overview of Tensor Analysis in Modern Statistical Learning (survey, author-hosted)](https://web.ics.purdue.edu/~sun244/Tensor_survey.pdf)\n3. [Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics (Linear Algebra and its Applications, 1977)](https://doi.org/10.1016/0024-3795%2877%2990069-6)\n4. [On the Non-Existence of Optimal Solutions and the Occurrence of \"Degeneracy\" in the CANDECOMP/PARAFAC Model (Psychometrika; PMC copy)](https://pmc.ncbi.nlm.nih.gov/articles/PMC2792364/)\n5. [Introduction to Tensor Decompositions and their Applications in Machine Learning (Rabanser, Shchur, Günnemann)](https://ar5iv.labs.arxiv.org/html/1711.10781)\n6. [Tensor Decomposition for Signal Processing and Machine Learning (Sidiropoulos, De Lathauwer, Fu, Huang, Papalexakis, Faloutsos; IEEE TSP 2017)](http://www.cs.ucr.edu/~epapalex/papers/tsp17-tensors.pdf)\n7. [Open Problem: Tensor Decompositions: Algorithms up to the Uniqueness Threshold? (Bhaskara, Charikar, Moitra, Vijayaraghavan, COLT 2014)](https://oar.princeton.edu/bitstream/88435/pr1w814/1/TensorDecompositionsUniquenessThreshold.pdf)\n8. [Algorithms for the Computation of the Canonical Polyadic and Block Term Decompositions (SIAM Optimization / KU Leuven report)](https://ftp.esat.kuleuven.be/pub/stadius/sistakulak/reports/algorithms_for_tensor_decompositions.pdf)\n9. [Tensor Methods in High Dimensional Data Analysis: Opportunities and Challenges (2024 survey)](https://arxiv.org/html/2405.18412)\n10. [Tensor Decompositions, Alternating Least Squares and other Tales (Comon, Luciani & De Almeida, J. Chemometrics 2009)](https://www.gipsa-lab.grenoble-inp.fr/~pierre.comon/FichiersPdf/ComoLA09-jchemo.pdf)\n11. [Tensor Decompositions: a textbook (Ballard, Wake Forest)](https://users.wfu.edu/ballard/pdfs/tensor_textbook.pdf)\n12. [Tang, Runshi and colleagues (2025). Revisit CP Tensor Decomposition: Statistical Optimality and Fast Convergence. arXiv (Cornell University).](https://doi.org/10.48550/arxiv.2505.23046)\n13. [Frank L. Hitchcock (1927). The Expression of a Tensor or a Polyadic as a Sum of Products. Studies in Applied Mathematics.](https://doi.org/10.1002/sapm192761164)\n14. [Raymond B. Cattell (1944). “Parallel Proportional Profiles” and other Principles for Determining the Choice of Factors by Rotation. Psychometrika.](https://doi.org/10.1007/bf02288739)\n15. [Ledyard R Tucker (1966). Some Mathematical Notes on Three-Mode Factor Analysis. Psychometrika.](https://doi.org/10.1007/bf02289464)\n16. [Foundations of the PARAFAC procedure: Models and conditions for an 'explanatory' multi-modal factor analysis (Harshman, UCLA Working Papers in Phonetics 16, 1970)](https://psychology.uwo.ca/faculty/harshman/wpppfac0.pdf)\n17. [J. Douglas Carroll, Jih-Jie Chang (1970). Analysis of Individual Differences in Multidimensional Scaling Via an N-way Generalization of “Eckart-Young” Decomposition. Psychometrika.](https://doi.org/10.1007/bf02310791)\n18. [A Survey of Tensor Methods (De Lathauwer, KU Leuven report)](https://ftp.esat.kuleuven.be/pub/stadius/delathauwer/reports/ldl-09-34.pdf)\n19. [Lieven De Lathauwer (2008). Decompositions of a Higher-Order Tensor in Block Terms, Part II: Definitions and Uniqueness. SIAM Journal on Matrix Analysis and Applications.](https://doi.org/10.1137/070690729)\n20. [Tensors for Data Mining and Data Fusion: Models, Applications, and Scalable Algorithms (Papalexakis, Faloutsos & Sidiropoulos, ACM TIST 2016/2017; author copy excerpts from cs.ucr.edu merged here)](https://psycnet.apa.org/doi/10.1145/2915921)\n21. [Decomposition of Big Tensors With Low Multilinear Rank (Zhang, Zhou, et al.)](https://ar5iv.labs.arxiv.org/html/1412.1885)\n22. [I. V. Oseledets (2011). Tensor-Train Decomposition. SIAM Journal on Scientific Computing.](https://doi.org/10.1137/090752286)\n23. [Overcomplete Tensor Decomposition via Koszul–Young Flattenings (2024)](https://arxiv.org/html/2411.14344v1)\n24. [A Method to Avoid Diverging Components in the Candecomp/Parafac Model for Generic I×J×2 Arrays (SIAM J. Matrix Anal. Appl.)](https://epubs.siam.org/doi/10.1137/070692121)\n25. [An algorithm for generic and low-rank specific identifiability of complex tensors](https://arxiv.org/abs/1403.4157)\n\n---\n*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Multivariate association and dimension reduction › Titles Mo to W*\n\n*Initially written Sep 29, 2026 · Reviewed: Sep 30, 2026 · Edited: Sep 30, 2026 · Last review: Sep 30, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [ "https://web.ics.purdue.edu/~sun244/Tensor_survey.pdf", "http://www.cs.ucr.edu/~epapalex/papers/tsp17-tensors.pdf" ], "url": "https://www.edgepedia.org/tensor-factorization", "markdown_url": "https://www.edgepedia.org/tensor-factorization.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. For other uses, organizations over USD 100M in revenue or 100M monthly users license separately.", "spdx": "LicenseRef-Edgepedia-Community-1.0" }, "credit": "\"Tensor factorization\", Edgepedia (EdgeChat), https://www.edgepedia.org/tensor-factorization. Edgepedia Community License 1.0.", "credit_md": "\"[Tensor factorization](https://www.edgepedia.org/tensor-factorization)\", Edgepedia (EdgeChat), [https://www.edgepedia.org/tensor-factorization](https://www.edgepedia.org/tensor-factorization). [Edgepedia Community License 1.0](https://www.edgepedia.org/edgepedia/license).", "credit_html": "\"Tensor factorization\", Edgepedia (EdgeChat), https://www.edgepedia.org/tensor-factorization. Edgepedia Community License 1.0.", "speakable": "Tensor factorization decomposes a multiway array into lower-dimensional component factors for dimensionality reduction and latent-structure recovery; its two central models, CP and Tucker, generalize the matrix singular value decomposition." }