Higher-order singular value decomposition: Difference between revisions

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{{Short description|Tensor decomposition}}

In [[multilinear algebra]], the '''higher-order singular value decomposition''' ('''HOSVD''') is a misnomer since thethere resultingdoes not exist a single tensor decomposition does notthat retainretains all the defining properties of the matrix SVD. The matrix SVD simultaneously yields a rank-𝑅 decomposition and computes orthonormal subspaces for the row and column spaces and computes a diagonal matrix. These properties are not realized within a single algorithm for higher-order tensors, but are instead realized by two distinct algorithmic developments and the efforts ofrepresent two distinct research directions. Harshman, as well as, the team of Carol and Chang proposed [[Canonical polyadic decomposition]] (CPD), which is a variant of the [[tensor rank decomposition]], in which thea tensor is approximated as a sum of ''K rank-1'' tensors for a user-specified ''K''. [[L. R. Tucker]] proposed a strategy for computing orthonormal subspaces for third order tensors. Aspecsts of these algorithms can be traced as far back as [[F. L. Hitchcock]] in 1928.<ref name=":0">{{Cite journal|last=Hitchcock|first=Frank L|date=1928-04-01|title=Multiple Invariants and Generalized Rank of a M-Way Array or Tensor|journal=Journal of Mathematics and Physics|language=en|volume=7|issue=1–4|pages=39–79|doi=10.1002/sapm19287139|issn=1467-9590}}</ref>

[[Lieven De Lathauwer |De Lathauwer]] ''et al.''<ref name=":2">{{Cite journal|last1=De Lathauwer|first1=L.|last2=De Moor|first2=B.|last3=Vandewalle|first3=J.|date=2000-01-01|title=On the Best Rank-1 and Rank-(R1 ,R2 ,. . .,RN) Approximation of Higher-Order Tensors|journal=SIAM Journal on Matrix Analysis and Applications|volume=21|issue=4|pages=1324–1342|doi=10.1137/S0895479898346995|issn=0895-4798|citeseerx=10.1.1.102.9135}}</ref><ref name="DeLathauwer00">{{Cite journal|last1=De Lathauwer|first1=L.|last2=De Moor|first2=B.|last3=Vandewalle|first3=J.|date=2000-01-01|title=A Multilinear Singular Value Decomposition|journal=SIAM Journal on Matrix Analysis and Applications|volume=21|issue=4|pages=1253–1278|doi=10.1137/s0895479896305696|issn=0895-4798|citeseerx=10.1.1.102.9135}}</ref> introduced clarity to Tucker's concepts with two highly influential papers, while [[Vasilescu]] and [[Demetri Terzopoulos| Terzopoulos]]<ref name=":Vasilescu2002">{{cite conference

|author=M. A. O. Vasilescu, D. Terzopoulos

|title=Multilinear Analysis of Image Ensembles: TensorFaces