A Provable Generalized Tensor Spectral Method for Uniform Hypergraph Partitioning

Debarghya Ghoshdastidar, Ambedkar Dukkipati
Proceedings of the 32nd International Conference on Machine Learning, PMLR 37:400-409, 2015.

Abstract

Matrix spectral methods play an important role in statistics and machine learning, and most often the word ‘matrix’ is dropped as, by default, one assumes that similarities or affinities are measured between two points, thereby resulting in similarity matrices. However, recent challenges in computer vision and text mining have necessitated the use of multi-way affinities in the learning methods, and this has led to a considerable interest in hypergraph partitioning methods in machine learning community. A plethora of “higher-order” algorithms have been proposed in the past decade, but their theoretical guarantees are not well-studied. In this paper, we develop a unified approach for partitioning uniform hypergraphs by means of a tensor trace optimization problem involving the affinity tensor, and a number of existing higher-order methods turn out to be special cases of the proposed formulation. We further propose an algorithm to solve the proposed trace optimization problem, and prove that it is consistent under a planted hypergraph model. We also provide experimental results to validate our theoretical findings.

Cite this Paper

BibTeX
@InProceedings{pmlr-v37-ghoshdastidar15, title = {A Provable Generalized Tensor Spectral Method for Uniform Hypergraph Partitioning}, author = {Ghoshdastidar, Debarghya and Dukkipati, Ambedkar}, booktitle = {Proceedings of the 32nd International Conference on Machine Learning}, pages = {400--409}, 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/ghoshdastidar15.pdf}, url = {https://proceedings.mlr.press/v37/ghoshdastidar15.html}, abstract = {Matrix spectral methods play an important role in statistics and machine learning, and most often the word ‘matrix’ is dropped as, by default, one assumes that similarities or affinities are measured between two points, thereby resulting in similarity matrices. However, recent challenges in computer vision and text mining have necessitated the use of multi-way affinities in the learning methods, and this has led to a considerable interest in hypergraph partitioning methods in machine learning community. A plethora of “higher-order” algorithms have been proposed in the past decade, but their theoretical guarantees are not well-studied. In this paper, we develop a unified approach for partitioning uniform hypergraphs by means of a tensor trace optimization problem involving the affinity tensor, and a number of existing higher-order methods turn out to be special cases of the proposed formulation. We further propose an algorithm to solve the proposed trace optimization problem, and prove that it is consistent under a planted hypergraph model. We also provide experimental results to validate our theoretical findings.} }
Endnote
%0 Conference Paper %T A Provable Generalized Tensor Spectral Method for Uniform Hypergraph Partitioning %A Debarghya Ghoshdastidar %A Ambedkar Dukkipati %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-ghoshdastidar15 %I PMLR %P 400--409 %U https://proceedings.mlr.press/v37/ghoshdastidar15.html %V 37 %X Matrix spectral methods play an important role in statistics and machine learning, and most often the word ‘matrix’ is dropped as, by default, one assumes that similarities or affinities are measured between two points, thereby resulting in similarity matrices. However, recent challenges in computer vision and text mining have necessitated the use of multi-way affinities in the learning methods, and this has led to a considerable interest in hypergraph partitioning methods in machine learning community. A plethora of “higher-order” algorithms have been proposed in the past decade, but their theoretical guarantees are not well-studied. In this paper, we develop a unified approach for partitioning uniform hypergraphs by means of a tensor trace optimization problem involving the affinity tensor, and a number of existing higher-order methods turn out to be special cases of the proposed formulation. We further propose an algorithm to solve the proposed trace optimization problem, and prove that it is consistent under a planted hypergraph model. We also provide experimental results to validate our theoretical findings.
RIS
TY - CPAPER TI - A Provable Generalized Tensor Spectral Method for Uniform Hypergraph Partitioning AU - Debarghya Ghoshdastidar AU - Ambedkar Dukkipati BT - Proceedings of the 32nd International Conference on Machine Learning DA - 2015/06/01 ED - Francis Bach ED - David Blei ID - pmlr-v37-ghoshdastidar15 PB - PMLR DP - Proceedings of Machine Learning Research VL - 37 SP - 400 EP - 409 L1 - http://proceedings.mlr.press/v37/ghoshdastidar15.pdf UR - https://proceedings.mlr.press/v37/ghoshdastidar15.html AB - Matrix spectral methods play an important role in statistics and machine learning, and most often the word ‘matrix’ is dropped as, by default, one assumes that similarities or affinities are measured between two points, thereby resulting in similarity matrices. However, recent challenges in computer vision and text mining have necessitated the use of multi-way affinities in the learning methods, and this has led to a considerable interest in hypergraph partitioning methods in machine learning community. A plethora of “higher-order” algorithms have been proposed in the past decade, but their theoretical guarantees are not well-studied. In this paper, we develop a unified approach for partitioning uniform hypergraphs by means of a tensor trace optimization problem involving the affinity tensor, and a number of existing higher-order methods turn out to be special cases of the proposed formulation. We further propose an algorithm to solve the proposed trace optimization problem, and prove that it is consistent under a planted hypergraph model. We also provide experimental results to validate our theoretical findings. ER -
APA
Ghoshdastidar, D. & Dukkipati, A.. (2015). A Provable Generalized Tensor Spectral Method for Uniform Hypergraph Partitioning. Proceedings of the 32nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 37:400-409 Available from https://proceedings.mlr.press/v37/ghoshdastidar15.html.

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