TY - GEN

T1 - Learning compressed sensing

AU - Weiss, Yair

AU - Chang, Hyun Sung

AU - Freeman, William T.

PY - 2007

Y1 - 2007

N2 - Compressed sensing [7], [6] is a recent set of mathematical results showing that sparse signals can be exactly reconstructed from a small number of linear measurements. Interestingly, for ideal sparse signals with no measurement noise, random measurements allow perfect reconstruction while measurements based on principal component analysis (PCA) or independent component analysis (ICA) do not At the same time, for other signal and noise distributions, PCA and ICA can significantly outperform random projections in terms of enabling reconstruction from a small number of measurements. In this paper we ask: given a training set typical of the signals we wish to measure, what are the optimal set of linear projections for compressed sensing ? We show that the optimal projections are in general not the principal components nor the independent components of the data, but rather a seemingly novel set of projections that capture what is still uncertain about the signal, given the training set. We also show that the projections onto the learned uncertain components may far outperform random projections. This is particularly true in the case of natural images, where random projections have vanishingly small signal to noise ratio as the number of pixels becomes large.

AB - Compressed sensing [7], [6] is a recent set of mathematical results showing that sparse signals can be exactly reconstructed from a small number of linear measurements. Interestingly, for ideal sparse signals with no measurement noise, random measurements allow perfect reconstruction while measurements based on principal component analysis (PCA) or independent component analysis (ICA) do not At the same time, for other signal and noise distributions, PCA and ICA can significantly outperform random projections in terms of enabling reconstruction from a small number of measurements. In this paper we ask: given a training set typical of the signals we wish to measure, what are the optimal set of linear projections for compressed sensing ? We show that the optimal projections are in general not the principal components nor the independent components of the data, but rather a seemingly novel set of projections that capture what is still uncertain about the signal, given the training set. We also show that the projections onto the learned uncertain components may far outperform random projections. This is particularly true in the case of natural images, where random projections have vanishingly small signal to noise ratio as the number of pixels becomes large.

UR - http://www.scopus.com/inward/record.url?scp=84940656044&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84940656044

T3 - 45th Annual Allerton Conference on Communication, Control, and Computing 2007

SP - 535

EP - 541

BT - 45th Annual Allerton Conference on Communication, Control, and Computing 2007

PB - University of Illinois at Urbana-Champaign, Coordinated Science Laboratory and Department of Computer and Electrical Engineering

T2 - 45th Annual Allerton Conference on Communication, Control, and Computing 2007

Y2 - 26 September 2007 through 28 September 2007

ER -