Exact Recovery of Sparsely-Used Dictionaries

Daniel A. Spielman, Huan Wang, John Wright
Proceedings of the 25th Annual Conference on Learning Theory, PMLR 23:37.1-37.18, 2012.

Abstract

We consider the problem of learning sparsely used dictionaries with an arbitrary square dictionary and a random, sparse coefficient matrix. We prove that \emphO(n log \emphn) samples are sufficient to uniquely determine the coefficient matrix. Based on this proof, we design a polynomial-time algorithm, called Exact Recovery of Sparsely-Used Dictionaries (ER-SpUD), and prove that it probably recovers the dictionary and coefficient matrix when the coefficient matrix is sufficiently sparse. Simulation results show that ER-SpUD reveals the true dictionary as well as the coefficients with probability higher than many state-of-the-art algorithms.

Cite this Paper

BibTeX
@InProceedings{pmlr-v23-spielman12, title = {Exact Recovery of Sparsely-Used Dictionaries}, author = {Spielman, Daniel A. and Wang, Huan and Wright, John}, booktitle = {Proceedings of the 25th Annual Conference on Learning Theory}, pages = {37.1--37.18}, year = {2012}, editor = {Mannor, Shie and Srebro, Nathan and Williamson, Robert C.}, volume = {23}, series = {Proceedings of Machine Learning Research}, address = {Edinburgh, Scotland}, month = {25--27 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v23/spielman12/spielman12.pdf}, url = {https://proceedings.mlr.press/v23/spielman12.html}, abstract = {We consider the problem of learning sparsely used dictionaries with an arbitrary square dictionary and a random, sparse coefficient matrix. We prove that \emphO(n log \emphn) samples are sufficient to uniquely determine the coefficient matrix. Based on this proof, we design a polynomial-time algorithm, called Exact Recovery of Sparsely-Used Dictionaries (ER-SpUD), and prove that it probably recovers the dictionary and coefficient matrix when the coefficient matrix is sufficiently sparse. Simulation results show that ER-SpUD reveals the true dictionary as well as the coefficients with probability higher than many state-of-the-art algorithms.} }
Endnote
%0 Conference Paper %T Exact Recovery of Sparsely-Used Dictionaries %A Daniel A. Spielman %A Huan Wang %A John Wright %B Proceedings of the 25th Annual Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2012 %E Shie Mannor %E Nathan Srebro %E Robert C. Williamson %F pmlr-v23-spielman12 %I PMLR %P 37.1--37.18 %U https://proceedings.mlr.press/v23/spielman12.html %V 23 %X We consider the problem of learning sparsely used dictionaries with an arbitrary square dictionary and a random, sparse coefficient matrix. We prove that \emphO(n log \emphn) samples are sufficient to uniquely determine the coefficient matrix. Based on this proof, we design a polynomial-time algorithm, called Exact Recovery of Sparsely-Used Dictionaries (ER-SpUD), and prove that it probably recovers the dictionary and coefficient matrix when the coefficient matrix is sufficiently sparse. Simulation results show that ER-SpUD reveals the true dictionary as well as the coefficients with probability higher than many state-of-the-art algorithms.
RIS
TY - CPAPER TI - Exact Recovery of Sparsely-Used Dictionaries AU - Daniel A. Spielman AU - Huan Wang AU - John Wright BT - Proceedings of the 25th Annual Conference on Learning Theory DA - 2012/06/16 ED - Shie Mannor ED - Nathan Srebro ED - Robert C. Williamson ID - pmlr-v23-spielman12 PB - PMLR DP - Proceedings of Machine Learning Research VL - 23 SP - 37.1 EP - 37.18 L1 - http://proceedings.mlr.press/v23/spielman12/spielman12.pdf UR - https://proceedings.mlr.press/v23/spielman12.html AB - We consider the problem of learning sparsely used dictionaries with an arbitrary square dictionary and a random, sparse coefficient matrix. We prove that \emphO(n log \emphn) samples are sufficient to uniquely determine the coefficient matrix. Based on this proof, we design a polynomial-time algorithm, called Exact Recovery of Sparsely-Used Dictionaries (ER-SpUD), and prove that it probably recovers the dictionary and coefficient matrix when the coefficient matrix is sufficiently sparse. Simulation results show that ER-SpUD reveals the true dictionary as well as the coefficients with probability higher than many state-of-the-art algorithms. ER -
APA
Spielman, D.A., Wang, H. & Wright, J.. (2012). Exact Recovery of Sparsely-Used Dictionaries. Proceedings of the 25th Annual Conference on Learning Theory, in Proceedings of Machine Learning Research 23:37.1-37.18 Available from https://proceedings.mlr.press/v23/spielman12.html.

Related Material