Guaranteed Tensor Decomposition: A Moment Approach

Gongguo Tang, Parikshit Shah
Proceedings of the 32nd International Conference on Machine Learning, PMLR 37:1491-1500, 2015.

Abstract

We develop a theoretical and computational framework to perform guaranteed tensor decomposition, which also has the potential to accomplish other tensor tasks such as tensor completion and denoising. We formulate tensor decomposition as a problem of measure estimation from moments. By constructing a dual polynomial, we demonstrate that measure optimization returns the correct CP decomposition under an incoherence condition on the rank-one factors. To address the computational challenge, we present a hierarchy of semidefinite programs based on sums-of-squares relaxations of the measure optimization problem. By showing that the constructed dual polynomial is a sum-of-squares modulo the sphere, we prove that the smallest SDP in the relaxation hierarchy is exact and the decomposition can be extracted from the solution under the same incoherence condition. One implication is that the tensor nuclear norm can be computed exactly using the smallest SDP as long as the rank-one factors of the tensor are incoherent. Numerical experiments are conducted to test the performance of the moment approach.

Cite this Paper


BibTeX
@InProceedings{pmlr-v37-tanga15, title = {Guaranteed Tensor Decomposition: A Moment Approach}, author = {Tang, Gongguo and Shah, Parikshit}, booktitle = {Proceedings of the 32nd International Conference on Machine Learning}, pages = {1491--1500}, year = {2015}, editor = {Bach, Francis and Blei, David}, volume = {37}, series = {Proceedings of Machine Learning Research}, address = {Lille, France}, month = {07--09 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v37/tanga15.pdf}, url = { http://proceedings.mlr.press/v37/tanga15.html }, abstract = {We develop a theoretical and computational framework to perform guaranteed tensor decomposition, which also has the potential to accomplish other tensor tasks such as tensor completion and denoising. We formulate tensor decomposition as a problem of measure estimation from moments. By constructing a dual polynomial, we demonstrate that measure optimization returns the correct CP decomposition under an incoherence condition on the rank-one factors. To address the computational challenge, we present a hierarchy of semidefinite programs based on sums-of-squares relaxations of the measure optimization problem. By showing that the constructed dual polynomial is a sum-of-squares modulo the sphere, we prove that the smallest SDP in the relaxation hierarchy is exact and the decomposition can be extracted from the solution under the same incoherence condition. One implication is that the tensor nuclear norm can be computed exactly using the smallest SDP as long as the rank-one factors of the tensor are incoherent. Numerical experiments are conducted to test the performance of the moment approach.} }
Endnote
%0 Conference Paper %T Guaranteed Tensor Decomposition: A Moment Approach %A Gongguo Tang %A Parikshit Shah %B Proceedings of the 32nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2015 %E Francis Bach %E David Blei %F pmlr-v37-tanga15 %I PMLR %P 1491--1500 %U http://proceedings.mlr.press/v37/tanga15.html %V 37 %X We develop a theoretical and computational framework to perform guaranteed tensor decomposition, which also has the potential to accomplish other tensor tasks such as tensor completion and denoising. We formulate tensor decomposition as a problem of measure estimation from moments. By constructing a dual polynomial, we demonstrate that measure optimization returns the correct CP decomposition under an incoherence condition on the rank-one factors. To address the computational challenge, we present a hierarchy of semidefinite programs based on sums-of-squares relaxations of the measure optimization problem. By showing that the constructed dual polynomial is a sum-of-squares modulo the sphere, we prove that the smallest SDP in the relaxation hierarchy is exact and the decomposition can be extracted from the solution under the same incoherence condition. One implication is that the tensor nuclear norm can be computed exactly using the smallest SDP as long as the rank-one factors of the tensor are incoherent. Numerical experiments are conducted to test the performance of the moment approach.
RIS
TY - CPAPER TI - Guaranteed Tensor Decomposition: A Moment Approach AU - Gongguo Tang AU - Parikshit Shah BT - Proceedings of the 32nd International Conference on Machine Learning DA - 2015/06/01 ED - Francis Bach ED - David Blei ID - pmlr-v37-tanga15 PB - PMLR DP - Proceedings of Machine Learning Research VL - 37 SP - 1491 EP - 1500 L1 - http://proceedings.mlr.press/v37/tanga15.pdf UR - http://proceedings.mlr.press/v37/tanga15.html AB - We develop a theoretical and computational framework to perform guaranteed tensor decomposition, which also has the potential to accomplish other tensor tasks such as tensor completion and denoising. We formulate tensor decomposition as a problem of measure estimation from moments. By constructing a dual polynomial, we demonstrate that measure optimization returns the correct CP decomposition under an incoherence condition on the rank-one factors. To address the computational challenge, we present a hierarchy of semidefinite programs based on sums-of-squares relaxations of the measure optimization problem. By showing that the constructed dual polynomial is a sum-of-squares modulo the sphere, we prove that the smallest SDP in the relaxation hierarchy is exact and the decomposition can be extracted from the solution under the same incoherence condition. One implication is that the tensor nuclear norm can be computed exactly using the smallest SDP as long as the rank-one factors of the tensor are incoherent. Numerical experiments are conducted to test the performance of the moment approach. ER -
APA
Tang, G. & Shah, P.. (2015). Guaranteed Tensor Decomposition: A Moment Approach. Proceedings of the 32nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 37:1491-1500 Available from http://proceedings.mlr.press/v37/tanga15.html .

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