Estimating Unknown Sparsity in Compressed Sensing

Miles Lopes
Proceedings of the 30th International Conference on Machine Learning, PMLR 28(3):217-225, 2013.

Abstract

In the theory of compressed sensing (CS), the sparsity \|x\|_0 of the unknown signal x∈\R^p is commonly assumed to be a known parameter. However, it is typically unknown in practice. Due to the fact that many aspects of CS depend on knowing \|x\|_0, it is important to estimate this parameter in a data-driven way. A second practical concern is that \|x\|_0 is a highly unstable function of x. In particular, for real signals with entries not exactly equal to 0, the value \|x\|_0=p is not a useful description of the effective number of coordinates. In this paper, we propose to estimate a stable measure of sparsity s(x):=\|x\|_1^2/\|x\|_2^2, which is a sharp lower bound on \|x\|_0. Our estimation procedure uses only a small number of linear measurements, does not rely on any sparsity assumptions, and requires very little computation. A confidence interval for s(x) is provided, and its width is shown to have no dependence on the signal dimension p. Moreover, this result extends naturally to the matrix recovery setting, where a soft version of matrix rank can be estimated with analogous guarantees. Finally, we show that the use of randomized measurements is essential to estimating s(x). This is accomplished by proving that the minimax risk for estimating s(x) with deterministic measurements is large when n≪p.

Cite this Paper

BibTeX
@InProceedings{pmlr-v28-lopes13, title = {Estimating Unknown Sparsity in Compressed Sensing}, author = {Lopes, Miles}, booktitle = {Proceedings of the 30th International Conference on Machine Learning}, pages = {217--225}, year = {2013}, editor = {Dasgupta, Sanjoy and McAllester, David}, volume = {28}, number = {3}, series = {Proceedings of Machine Learning Research}, address = {Atlanta, Georgia, USA}, month = {17--19 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v28/lopes13.pdf}, url = {https://proceedings.mlr.press/v28/lopes13.html}, abstract = {In the theory of compressed sensing (CS), the sparsity \|x\|_0 of the unknown signal x∈\R^p is commonly assumed to be a known parameter. However, it is typically unknown in practice. Due to the fact that many aspects of CS depend on knowing \|x\|_0, it is important to estimate this parameter in a data-driven way. A second practical concern is that \|x\|_0 is a highly unstable function of x. In particular, for real signals with entries not exactly equal to 0, the value \|x\|_0=p is not a useful description of the effective number of coordinates. In this paper, we propose to estimate a stable measure of sparsity s(x):=\|x\|_1^2/\|x\|_2^2, which is a sharp lower bound on \|x\|_0. Our estimation procedure uses only a small number of linear measurements, does not rely on any sparsity assumptions, and requires very little computation. A confidence interval for s(x) is provided, and its width is shown to have no dependence on the signal dimension p. Moreover, this result extends naturally to the matrix recovery setting, where a soft version of matrix rank can be estimated with analogous guarantees. Finally, we show that the use of randomized measurements is essential to estimating s(x). This is accomplished by proving that the minimax risk for estimating s(x) with deterministic measurements is large when n≪p.} }
Endnote
%0 Conference Paper %T Estimating Unknown Sparsity in Compressed Sensing %A Miles Lopes %B Proceedings of the 30th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2013 %E Sanjoy Dasgupta %E David McAllester %F pmlr-v28-lopes13 %I PMLR %P 217--225 %U https://proceedings.mlr.press/v28/lopes13.html %V 28 %N 3 %X In the theory of compressed sensing (CS), the sparsity \|x\|_0 of the unknown signal x∈\R^p is commonly assumed to be a known parameter. However, it is typically unknown in practice. Due to the fact that many aspects of CS depend on knowing \|x\|_0, it is important to estimate this parameter in a data-driven way. A second practical concern is that \|x\|_0 is a highly unstable function of x. In particular, for real signals with entries not exactly equal to 0, the value \|x\|_0=p is not a useful description of the effective number of coordinates. In this paper, we propose to estimate a stable measure of sparsity s(x):=\|x\|_1^2/\|x\|_2^2, which is a sharp lower bound on \|x\|_0. Our estimation procedure uses only a small number of linear measurements, does not rely on any sparsity assumptions, and requires very little computation. A confidence interval for s(x) is provided, and its width is shown to have no dependence on the signal dimension p. Moreover, this result extends naturally to the matrix recovery setting, where a soft version of matrix rank can be estimated with analogous guarantees. Finally, we show that the use of randomized measurements is essential to estimating s(x). This is accomplished by proving that the minimax risk for estimating s(x) with deterministic measurements is large when n≪p.
RIS
TY - CPAPER TI - Estimating Unknown Sparsity in Compressed Sensing AU - Miles Lopes BT - Proceedings of the 30th International Conference on Machine Learning DA - 2013/05/26 ED - Sanjoy Dasgupta ED - David McAllester ID - pmlr-v28-lopes13 PB - PMLR DP - Proceedings of Machine Learning Research VL - 28 IS - 3 SP - 217 EP - 225 L1 - http://proceedings.mlr.press/v28/lopes13.pdf UR - https://proceedings.mlr.press/v28/lopes13.html AB - In the theory of compressed sensing (CS), the sparsity \|x\|_0 of the unknown signal x∈\R^p is commonly assumed to be a known parameter. However, it is typically unknown in practice. Due to the fact that many aspects of CS depend on knowing \|x\|_0, it is important to estimate this parameter in a data-driven way. A second practical concern is that \|x\|_0 is a highly unstable function of x. In particular, for real signals with entries not exactly equal to 0, the value \|x\|_0=p is not a useful description of the effective number of coordinates. In this paper, we propose to estimate a stable measure of sparsity s(x):=\|x\|_1^2/\|x\|_2^2, which is a sharp lower bound on \|x\|_0. Our estimation procedure uses only a small number of linear measurements, does not rely on any sparsity assumptions, and requires very little computation. A confidence interval for s(x) is provided, and its width is shown to have no dependence on the signal dimension p. Moreover, this result extends naturally to the matrix recovery setting, where a soft version of matrix rank can be estimated with analogous guarantees. Finally, we show that the use of randomized measurements is essential to estimating s(x). This is accomplished by proving that the minimax risk for estimating s(x) with deterministic measurements is large when n≪p. ER -
APA
Lopes, M.. (2013). Estimating Unknown Sparsity in Compressed Sensing. Proceedings of the 30th International Conference on Machine Learning, in Proceedings of Machine Learning Research 28(3):217-225 Available from https://proceedings.mlr.press/v28/lopes13.html.

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