Abstract
In sparse approximation theory, the fundamental problem is to reconstruct a signal A ∈ ℝn from linear measurements 〈A, ψ〉 with respect to a dictionary of ψi's. Recently, there is focus on the novel direction of Compressed Sensing [1] where the reconstruction can be done with very few - O(k log n) - linear measurements over a modified dictionary if the signal is compressible, that is, its information is concentrated in k coefficients with the original dictionary. In particular, these results [1], [2], [3] prove that there exists a single O(k log n) × n measurement matrix such that any such signal can be reconstructed from these measurements, with error at most O(1) times the worst case error for the class of such signals. Compressed sensing has generated tremendous excitement both because of the sophisticated underlying Mathematics and because of its potential applications. In this paper, we address outstanding open problems in Compressed Sensing. Our main result is an explicit construction of a non-adaptive measurement matrix and the corresponding reconstruction algorithm so that with a number of measurements polynomial in k, log n, 1/ε, we can reconstruct compressible signals. This is the first known polynomial time explicit construction of any such measurement matrix. In addition, our result improves the error guarantee from O(1) to 1 + ε and improves the reconstruction time from poly(n) to poly(k log n). Our second result is a randomized construction of O(k polylog(n)) measurements that work for each signal with high probability and gives per-instance approximation guarantees rather than over the class of all signals. Previous work on Compressed Sensing does not provide such per-instance approximation guarantees; our result improves the best known number of measurements known from prior work in other areas including Learning Theory [4], [5], Streaming algorithms [6], [7], [8] and Complexity Theory [9] for this case. Our approach is combinatorial. In particular, we use two parallel sets of group tests, one to filter and the other to certify and estimate; the resulting algorithms are quite simple to implement.
| Original language | English (US) |
|---|---|
| Title of host publication | 2006 IEEE Conference on Information Sciences and Systems, CISS 2006 - Proceedings |
| Pages | 198-201 |
| Number of pages | 4 |
| DOIs | |
| State | Published - Dec 1 2007 |
| Event | 2006 40th Annual Conference on Information Sciences and Systems, CISS 2006 - Princeton, NJ, United States Duration: Mar 22 2006 → Mar 24 2006 |
Publication series
| Name | 2006 IEEE Conference on Information Sciences and Systems, CISS 2006 - Proceedings |
|---|
Other
| Other | 2006 40th Annual Conference on Information Sciences and Systems, CISS 2006 |
|---|---|
| Country | United States |
| City | Princeton, NJ |
| Period | 3/22/06 → 3/24/06 |
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ASJC Scopus subject areas
- Computer Science(all)
Cite this
Combinatorial algorithms for compressed sensing. / Cormode, Graham; Muthukrishnan, Shanmugavelayutham.
2006 IEEE Conference on Information Sciences and Systems, CISS 2006 - Proceedings. 2007. p. 198-201 4067802 (2006 IEEE Conference on Information Sciences and Systems, CISS 2006 - Proceedings).Research output: Chapter in Book/Report/Conference proceeding › Conference contribution
}
TY - GEN
T1 - Combinatorial algorithms for compressed sensing
AU - Cormode, Graham
AU - Muthukrishnan, Shanmugavelayutham
PY - 2007/12/1
Y1 - 2007/12/1
N2 - In sparse approximation theory, the fundamental problem is to reconstruct a signal A ∈ ℝn from linear measurements 〈A, ψ〉 with respect to a dictionary of ψi's. Recently, there is focus on the novel direction of Compressed Sensing [1] where the reconstruction can be done with very few - O(k log n) - linear measurements over a modified dictionary if the signal is compressible, that is, its information is concentrated in k coefficients with the original dictionary. In particular, these results [1], [2], [3] prove that there exists a single O(k log n) × n measurement matrix such that any such signal can be reconstructed from these measurements, with error at most O(1) times the worst case error for the class of such signals. Compressed sensing has generated tremendous excitement both because of the sophisticated underlying Mathematics and because of its potential applications. In this paper, we address outstanding open problems in Compressed Sensing. Our main result is an explicit construction of a non-adaptive measurement matrix and the corresponding reconstruction algorithm so that with a number of measurements polynomial in k, log n, 1/ε, we can reconstruct compressible signals. This is the first known polynomial time explicit construction of any such measurement matrix. In addition, our result improves the error guarantee from O(1) to 1 + ε and improves the reconstruction time from poly(n) to poly(k log n). Our second result is a randomized construction of O(k polylog(n)) measurements that work for each signal with high probability and gives per-instance approximation guarantees rather than over the class of all signals. Previous work on Compressed Sensing does not provide such per-instance approximation guarantees; our result improves the best known number of measurements known from prior work in other areas including Learning Theory [4], [5], Streaming algorithms [6], [7], [8] and Complexity Theory [9] for this case. Our approach is combinatorial. In particular, we use two parallel sets of group tests, one to filter and the other to certify and estimate; the resulting algorithms are quite simple to implement.
AB - In sparse approximation theory, the fundamental problem is to reconstruct a signal A ∈ ℝn from linear measurements 〈A, ψ〉 with respect to a dictionary of ψi's. Recently, there is focus on the novel direction of Compressed Sensing [1] where the reconstruction can be done with very few - O(k log n) - linear measurements over a modified dictionary if the signal is compressible, that is, its information is concentrated in k coefficients with the original dictionary. In particular, these results [1], [2], [3] prove that there exists a single O(k log n) × n measurement matrix such that any such signal can be reconstructed from these measurements, with error at most O(1) times the worst case error for the class of such signals. Compressed sensing has generated tremendous excitement both because of the sophisticated underlying Mathematics and because of its potential applications. In this paper, we address outstanding open problems in Compressed Sensing. Our main result is an explicit construction of a non-adaptive measurement matrix and the corresponding reconstruction algorithm so that with a number of measurements polynomial in k, log n, 1/ε, we can reconstruct compressible signals. This is the first known polynomial time explicit construction of any such measurement matrix. In addition, our result improves the error guarantee from O(1) to 1 + ε and improves the reconstruction time from poly(n) to poly(k log n). Our second result is a randomized construction of O(k polylog(n)) measurements that work for each signal with high probability and gives per-instance approximation guarantees rather than over the class of all signals. Previous work on Compressed Sensing does not provide such per-instance approximation guarantees; our result improves the best known number of measurements known from prior work in other areas including Learning Theory [4], [5], Streaming algorithms [6], [7], [8] and Complexity Theory [9] for this case. Our approach is combinatorial. In particular, we use two parallel sets of group tests, one to filter and the other to certify and estimate; the resulting algorithms are quite simple to implement.
UR - http://www.scopus.com/inward/record.url?scp=44049103219&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=44049103219&partnerID=8YFLogxK
U2 - 10.1109/CISS.2006.286461
DO - 10.1109/CISS.2006.286461
M3 - Conference contribution
AN - SCOPUS:44049103219
SN - 1424403502
SN - 9781424403509
T3 - 2006 IEEE Conference on Information Sciences and Systems, CISS 2006 - Proceedings
SP - 198
EP - 201
BT - 2006 IEEE Conference on Information Sciences and Systems, CISS 2006 - Proceedings
ER -