TY - GEN
T1 - On sparsity, redundancy and quality of frame representations
AU - Akçakaya, Mehmet
AU - Tarokh, Vahid
PY - 2007/12/1
Y1 - 2007/12/1
N2 - We consider approximations of signals by the elements of a frame in a complex vector space of dimension N and formulate both the noiseless and the noisy sparse representation problems. The noiseless representation problem is to find sparse representations of a signal r given that such representations exist. In this case, we explicitly construct a frame, referred to as the Vandermonde frame, for which the noiseless sparse representation problem can be solved uniquely using O(N2) operations, as long as the number of non-zero coefficients in the sparse representation of r is ∈N for some 0 ≤ ∈ ≤ 0.5, thus improving on a result of Candès and Tao [3]. We also show that ∈ ≤ 0.5 cannot be relaxed without violating uniqueness. The noisy sparse representation problem is to find sparse representations of a signal r satisfying a distortion criterion. In this case, we establish a lower bound on the trade-off between the sparsity of the representation, the underlying distortion and the redundancy of any given frame.
AB - We consider approximations of signals by the elements of a frame in a complex vector space of dimension N and formulate both the noiseless and the noisy sparse representation problems. The noiseless representation problem is to find sparse representations of a signal r given that such representations exist. In this case, we explicitly construct a frame, referred to as the Vandermonde frame, for which the noiseless sparse representation problem can be solved uniquely using O(N2) operations, as long as the number of non-zero coefficients in the sparse representation of r is ∈N for some 0 ≤ ∈ ≤ 0.5, thus improving on a result of Candès and Tao [3]. We also show that ∈ ≤ 0.5 cannot be relaxed without violating uniqueness. The noisy sparse representation problem is to find sparse representations of a signal r satisfying a distortion criterion. In this case, we establish a lower bound on the trade-off between the sparsity of the representation, the underlying distortion and the redundancy of any given frame.
UR - http://www.scopus.com/inward/record.url?scp=51649094858&partnerID=8YFLogxK
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U2 - 10.1109/ISIT.2007.4557114
DO - 10.1109/ISIT.2007.4557114
M3 - Conference contribution
AN - SCOPUS:51649094858
SN - 1424414296
SN - 9781424414291
T3 - IEEE International Symposium on Information Theory - Proceedings
SP - 951
EP - 955
BT - Proceedings - 2007 IEEE International Symposium on Information Theory, ISIT 2007
T2 - 2007 IEEE International Symposium on Information Theory, ISIT 2007
Y2 - 24 June 2007 through 29 June 2007
ER -