Canonical polyadic decomposition of third-order tensors: Relaxed uniqueness conditions and algebraic algorithm

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• 作者：Ignat Domanov^a ; ^b ; ^{ignat.domanov@kuleuven.be" class="auth_mail" title="E-mail the corresponding author} ; Lieven De Lathauwer^a ; ^b ; ^{lieven.delathauwer@kuleuven.be" class="auth_mail" title="E-mail the corresponding author}
• 关键词：15A69 ; 15A23
• 刊名：Linear Algebra and its Applications
• 出版年：2017
• 出版时间：15 January 2017
• 年：2017
• 卷：513
• 期：Complete
• 页码：342-375
• 全文大小：655 K

文摘

Canonical Polyadic Decomposition (CPD) of a third-order tensor is a minimal decomposition into a sum of rank-1 tensors. We find new mild deterministic conditions for the uniqueness of individual rank-1 tensors in CPD and present an algorithm to recover them. We call the algorithm “algebraic” because it relies only on standard linear algebra. It does not involve more advanced procedures than the computation of the null space of a matrix and eigen/singular value decomposition. Simulations indicate that the new conditions for uniqueness and the working assumptions for the algorithm hold for a randomly generated class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002437951630492X&_mathId=si1.gif&_user=111111111&_pii=S002437951630492X&_rdoc=1&_issn=00243795&md5=f6de898797ba556d5fd0597b31bf6257" title="Click to view the MathML source">I×J×Kclass="mathContainer hidden">class="mathCode"> tensor of rank class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002437951630492X&_mathId=si2.gif&_user=111111111&_pii=S002437951630492X&_rdoc=1&_issn=00243795&md5=4d7c74b65baff1cdf1ad2719b01fc739" title="Click to view the MathML source">R≥K≥J≥I≥2class="mathContainer hidden">class="mathCode"> if R   is bounded as class="mathmlsrc">le="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002437951630492X&_mathId=si3.gif&_user=111111111&_pii=S002437951630492X&_rdoc=1&_issn=00243795&md5=46382bf2f5161f01eaf99696e8e05785">class="imgLazyJSB inlineImage" height="20" width="365" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S002437951630492X-si3.gif">cript>le="vertical-align:bottom" width="365" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S002437951630492X-si3.gif">cript>class="mathContainer hidden">class="mathCode"> at least for the dimensions that we have tested. This improves upon the famous Kruskal bound for uniqueness class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002437951630492X&_mathId=si4.gif&_user=111111111&_pii=S002437951630492X&_rdoc=1&_issn=00243795&md5=c007a50703b1af83fb3b936dd195b12e" title="Click to view the MathML source">R≤(I+J+K−2)/2class="mathContainer hidden">class="mathCode"> as soon as class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002437951630492X&_mathId=si5.gif&_user=111111111&_pii=S002437951630492X&_rdoc=1&_issn=00243795&md5=1c26b74db1d2f003d9a97fa282740de1" title="Click to view the MathML source">I≥3class="mathContainer hidden">class="mathCode">.

In the particular case class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002437951630492X&_mathId=si6.gif&_user=111111111&_pii=S002437951630492X&_rdoc=1&_issn=00243795&md5=ce7737c9ed7edde190c7645e4a21d2ec" title="Click to view the MathML source">R=Kclass="mathContainer hidden">class="mathCode">, the new bound above is equivalent to the bound class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002437951630492X&_mathId=si194.gif&_user=111111111&_pii=S002437951630492X&_rdoc=1&_issn=00243795&md5=d43df3f83f3ee7be3a942149e8bcf315" title="Click to view the MathML source">R≤(I−1)(J−1)class="mathContainer hidden">class="mathCode"> which is known to be necessary and sufficient for the generic uniqueness of the CPD. An existing algebraic algorithm (based on simultaneous diagonalization of a set of matrices) computes the CPD under the more restrictive constraint class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002437951630492X&_mathId=si8.gif&_user=111111111&_pii=S002437951630492X&_rdoc=1&_issn=00243795&md5=fc633c849a1f3849b8e7eb69ba9e9b03" title="Click to view the MathML source">R(R−1)≤I(I−1)J(J−1)/2class="mathContainer hidden">class="mathCode"> (implying that class="mathmlsrc">le="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002437951630492X&_mathId=si9.gif&_user=111111111&_pii=S002437951630492X&_rdoc=1&_issn=00243795&md5=2caf5463fa1b9b944ab5dd1f0871ed50">class="imgLazyJSB inlineImage" height="19" width="196" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S002437951630492X-si9.gif">cript>le="vertical-align:bottom" width="196" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S002437951630492X-si9.gif">cript>class="mathContainer hidden">class="mathCode">). We give an example of a low-dimensional but high-rank CPD that cannot be found by optimization-based algorithms in a reasonable amount of time while our approach takes less than a second. We demonstrate that, at least for class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002437951630492X&_mathId=si10.gif&_user=111111111&_pii=S002437951630492X&_rdoc=1&_issn=00243795&md5=486125734e02de25e433221b8d83af6f" title="Click to view the MathML source">R≤24class="mathContainer hidden">class="mathCode">, our algorithm can recover the rank-1 tensors in the CPD up to class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002437951630492X&_mathId=si194.gif&_user=111111111&_pii=S002437951630492X&_rdoc=1&_issn=00243795&md5=d43df3f83f3ee7be3a942149e8bcf315" title="Click to view the MathML source">R≤(I−1)(J−1)class="mathContainer hidden">class="mathCode">.