Deformed Decomposition for Non-negative Tensors

Kazu Ghalamkari, Petr Taborsky, Morten Mørup
Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, PMLR 300:3025-3033, 2026.

Abstract

Non-negative tensor factorization finds widespread use in numerous applications, however, its global optimization has been a long-standing challenge. In particular, the Frobenius norm minimization, even in the rank-$1$ setting, is an NP-hard problem. We presently reformulate tensor decompositions using deformed algebra, which is associated with a generalized product such that the exponential law holds for generalized exponential functions, and show that the best rank-$1$ approximation thereby reduces to a convex optimization problem for the rich $\chi$-divergence family. Building on this foundation, we propose the deformed many-body approximation for non-negative tensors, which expands model capacity while maintaining global optimality by preserving the flatness of the model manifold. Introducing latent variables, for a subclass of $\chi$-divergences, we further develop an Expectation-Maximization-based framework for the deformed extension of traditional low-rank approximations as iterative convex subproblems. Through experiments on tensor-based probability mass function estimation, we show that the deformed decompositions provide implicit regularization and robustness against noise and mislabeled data. Beyond ordinary tensor algebra, our findings provide a factorization framework that enables us to leverage various divergences with convex rank-$1$ and many-body approximations.

Cite this Paper

BibTeX
@InProceedings{pmlr-v300-ghalamkari26a, title = { Deformed Decomposition for Non-negative Tensors }, author = {Ghalamkari, Kazu and Taborsky, Petr and M{\o}rup, Morten}, booktitle = {Proceedings of The 29th International Conference on Artificial Intelligence and Statistics}, pages = {3025--3033}, year = {2026}, editor = {Khan, Emtiyaz and Li, Yingzhen and Solin, Arno and Ramdas, Aaditya}, volume = {300}, series = {Proceedings of Machine Learning Research}, month = {02--05 May}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v300/main/assets/ghalamkari26a/ghalamkari26a.pdf}, url = {https://proceedings.mlr.press/v300/ghalamkari26a.html}, abstract = { Non-negative tensor factorization finds widespread use in numerous applications, however, its global optimization has been a long-standing challenge. In particular, the Frobenius norm minimization, even in the rank-$1$ setting, is an NP-hard problem. We presently reformulate tensor decompositions using deformed algebra, which is associated with a generalized product such that the exponential law holds for generalized exponential functions, and show that the best rank-$1$ approximation thereby reduces to a convex optimization problem for the rich $\chi$-divergence family. Building on this foundation, we propose the deformed many-body approximation for non-negative tensors, which expands model capacity while maintaining global optimality by preserving the flatness of the model manifold. Introducing latent variables, for a subclass of $\chi$-divergences, we further develop an Expectation-Maximization-based framework for the deformed extension of traditional low-rank approximations as iterative convex subproblems. Through experiments on tensor-based probability mass function estimation, we show that the deformed decompositions provide implicit regularization and robustness against noise and mislabeled data. Beyond ordinary tensor algebra, our findings provide a factorization framework that enables us to leverage various divergences with convex rank-$1$ and many-body approximations. } }
Endnote
%0 Conference Paper %T Deformed Decomposition for Non-negative Tensors %A Kazu Ghalamkari %A Petr Taborsky %A Morten Mørup %B Proceedings of The 29th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2026 %E Emtiyaz Khan %E Yingzhen Li %E Arno Solin %E Aaditya Ramdas %F pmlr-v300-ghalamkari26a %I PMLR %P 3025--3033 %U https://proceedings.mlr.press/v300/ghalamkari26a.html %V 300 %X Non-negative tensor factorization finds widespread use in numerous applications, however, its global optimization has been a long-standing challenge. In particular, the Frobenius norm minimization, even in the rank-$1$ setting, is an NP-hard problem. We presently reformulate tensor decompositions using deformed algebra, which is associated with a generalized product such that the exponential law holds for generalized exponential functions, and show that the best rank-$1$ approximation thereby reduces to a convex optimization problem for the rich $\chi$-divergence family. Building on this foundation, we propose the deformed many-body approximation for non-negative tensors, which expands model capacity while maintaining global optimality by preserving the flatness of the model manifold. Introducing latent variables, for a subclass of $\chi$-divergences, we further develop an Expectation-Maximization-based framework for the deformed extension of traditional low-rank approximations as iterative convex subproblems. Through experiments on tensor-based probability mass function estimation, we show that the deformed decompositions provide implicit regularization and robustness against noise and mislabeled data. Beyond ordinary tensor algebra, our findings provide a factorization framework that enables us to leverage various divergences with convex rank-$1$ and many-body approximations.
APA
Ghalamkari, K., Taborsky, P. & Mørup, M.. (2026). Deformed Decomposition for Non-negative Tensors . Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 300:3025-3033 Available from https://proceedings.mlr.press/v300/ghalamkari26a.html.

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