The Sample Complexity of Dictionary Learning

Daniel Vainsencher, Shie Mannor, Alfred M. Bruckstein
Proceedings of the 24th Annual Conference on Learning Theory, PMLR 19:773-788, 2011.

Abstract

A large set of signals can sometimes be described sparsely using a dictionary, that is, every element can be represented as a linear combination of few elements from the dictionary. Algorithms for various signal processing applications, including classification, denoising and signal separation, learn a dictionary from a given set of signals to be represented. Can we expect that the error in representing by such a dictionary a previously unseen signal from the same source will be of similar magnitude as those for the given examples? We assume signals are generated from a fixed distribution, and study these questions from a statistical learning theory perspective. We develop generalization bounds on the quality of the learned dictionary for two types of constraints on the coefficient selection, as measured by the expected $L_2$ error in representation when the dictionary is used. For the case of $l_1$ regularized coefficient selection we provide a generalization bound of the order of $O\left(\sqrt{np\ln(m λ)/m}\right)$, where $n$ is the dimension, $p$ is the number of elements in the dictionary, λis a bound on the $l_1$ norm of the coefficient vector and m is the number of samples, which complements existing results. For the case of representing a new signal as a combination of at most $k$ dictionary elements, we provide a bound ofthe order $O(\sqrt{np\ln(m k)/m})$ under an assumption on the closeness to orthogonality of the dictionary (low Babel function). We further show that this assumption holds for most dictionaries in high dimensions in a strong probabilistic sense. Our results also include bounds that converge as $1/m$, not previously known for this problem. We provide similar results in a general setting using kernels with weak smoothness requirements.

Cite this Paper

BibTeX
@InProceedings{pmlr-v19-vainsencher11a, title = {The Sample Complexity of Dictionary Learning}, author = {Vainsencher, Daniel and Mannor, Shie and Bruckstein, Alfred M.}, booktitle = {Proceedings of the 24th Annual Conference on Learning Theory}, pages = {773--788}, year = {2011}, editor = {Kakade, Sham M. and von Luxburg, Ulrike}, volume = {19}, series = {Proceedings of Machine Learning Research}, address = {Budapest, Hungary}, month = {09--11 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v19/vainsencher11a/vainsencher11a.pdf}, url = {https://proceedings.mlr.press/v19/vainsencher11a.html}, abstract = {A large set of signals can sometimes be described sparsely using a dictionary, that is, every element can be represented as a linear combination of few elements from the dictionary. Algorithms for various signal processing applications, including classification, denoising and signal separation, learn a dictionary from a given set of signals to be represented. Can we expect that the error in representing by such a dictionary a previously unseen signal from the same source will be of similar magnitude as those for the given examples? We assume signals are generated from a fixed distribution, and study these questions from a statistical learning theory perspective. We develop generalization bounds on the quality of the learned dictionary for two types of constraints on the coefficient selection, as measured by the expected $L_2$ error in representation when the dictionary is used. For the case of $l_1$ regularized coefficient selection we provide a generalization bound of the order of $O\left(\sqrt{np\ln(m λ)/m}\right)$, where $n$ is the dimension, $p$ is the number of elements in the dictionary, λis a bound on the $l_1$ norm of the coefficient vector and m is the number of samples, which complements existing results. For the case of representing a new signal as a combination of at most $k$ dictionary elements, we provide a bound ofthe order $O(\sqrt{np\ln(m k)/m})$ under an assumption on the closeness to orthogonality of the dictionary (low Babel function). We further show that this assumption holds for most dictionaries in high dimensions in a strong probabilistic sense. Our results also include bounds that converge as $1/m$, not previously known for this problem. We provide similar results in a general setting using kernels with weak smoothness requirements.} }
Endnote
%0 Conference Paper %T The Sample Complexity of Dictionary Learning %A Daniel Vainsencher %A Shie Mannor %A Alfred M. Bruckstein %B Proceedings of the 24th Annual Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2011 %E Sham M. Kakade %E Ulrike von Luxburg %F pmlr-v19-vainsencher11a %I PMLR %P 773--788 %U https://proceedings.mlr.press/v19/vainsencher11a.html %V 19 %X A large set of signals can sometimes be described sparsely using a dictionary, that is, every element can be represented as a linear combination of few elements from the dictionary. Algorithms for various signal processing applications, including classification, denoising and signal separation, learn a dictionary from a given set of signals to be represented. Can we expect that the error in representing by such a dictionary a previously unseen signal from the same source will be of similar magnitude as those for the given examples? We assume signals are generated from a fixed distribution, and study these questions from a statistical learning theory perspective. We develop generalization bounds on the quality of the learned dictionary for two types of constraints on the coefficient selection, as measured by the expected $L_2$ error in representation when the dictionary is used. For the case of $l_1$ regularized coefficient selection we provide a generalization bound of the order of $O\left(\sqrt{np\ln(m λ)/m}\right)$, where $n$ is the dimension, $p$ is the number of elements in the dictionary, λis a bound on the $l_1$ norm of the coefficient vector and m is the number of samples, which complements existing results. For the case of representing a new signal as a combination of at most $k$ dictionary elements, we provide a bound ofthe order $O(\sqrt{np\ln(m k)/m})$ under an assumption on the closeness to orthogonality of the dictionary (low Babel function). We further show that this assumption holds for most dictionaries in high dimensions in a strong probabilistic sense. Our results also include bounds that converge as $1/m$, not previously known for this problem. We provide similar results in a general setting using kernels with weak smoothness requirements.
RIS
TY - CPAPER TI - The Sample Complexity of Dictionary Learning AU - Daniel Vainsencher AU - Shie Mannor AU - Alfred M. Bruckstein BT - Proceedings of the 24th Annual Conference on Learning Theory DA - 2011/12/21 ED - Sham M. Kakade ED - Ulrike von Luxburg ID - pmlr-v19-vainsencher11a PB - PMLR DP - Proceedings of Machine Learning Research VL - 19 SP - 773 EP - 788 L1 - http://proceedings.mlr.press/v19/vainsencher11a/vainsencher11a.pdf UR - https://proceedings.mlr.press/v19/vainsencher11a.html AB - A large set of signals can sometimes be described sparsely using a dictionary, that is, every element can be represented as a linear combination of few elements from the dictionary. Algorithms for various signal processing applications, including classification, denoising and signal separation, learn a dictionary from a given set of signals to be represented. Can we expect that the error in representing by such a dictionary a previously unseen signal from the same source will be of similar magnitude as those for the given examples? We assume signals are generated from a fixed distribution, and study these questions from a statistical learning theory perspective. We develop generalization bounds on the quality of the learned dictionary for two types of constraints on the coefficient selection, as measured by the expected $L_2$ error in representation when the dictionary is used. For the case of $l_1$ regularized coefficient selection we provide a generalization bound of the order of $O\left(\sqrt{np\ln(m λ)/m}\right)$, where $n$ is the dimension, $p$ is the number of elements in the dictionary, λis a bound on the $l_1$ norm of the coefficient vector and m is the number of samples, which complements existing results. For the case of representing a new signal as a combination of at most $k$ dictionary elements, we provide a bound ofthe order $O(\sqrt{np\ln(m k)/m})$ under an assumption on the closeness to orthogonality of the dictionary (low Babel function). We further show that this assumption holds for most dictionaries in high dimensions in a strong probabilistic sense. Our results also include bounds that converge as $1/m$, not previously known for this problem. We provide similar results in a general setting using kernels with weak smoothness requirements. ER -
APA
Vainsencher, D., Mannor, S. & Bruckstein, A.M.. (2011). The Sample Complexity of Dictionary Learning. Proceedings of the 24th Annual Conference on Learning Theory, in Proceedings of Machine Learning Research 19:773-788 Available from https://proceedings.mlr.press/v19/vainsencher11a.html.

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