TY - GEN
T1 - Euclidean sections of ℓ1N with sublinear randomness and error-correction over the reals
AU - Guruswami, Venkatesan
AU - Lee, James R.
AU - Wigderson, Avi
PY - 2008
Y1 - 2008
N2 - It is well-known that ℝN has subspaces of dimension proportional to N on which the ℓ1 and ℓ2 norms are uniformly equivalent, but it is unknown how to construct them explicitly. We show that, for any δ > 0, such a subspace can be generated using only Nδ random bits. This improves over previous constructions of Artstein-Avidan and Milman, and of Lovett and Sodin, which require O(N logN), and O(N) random bits, respectively. Such subspaces are known to also yield error-correcting codes over the reals and compressed sensing matrices. Our subspaces are defined by the kernel of a relatively sparse matrix (with at most Nδ non-zero entries per row), and thus enable compressed sensing in near-linear O(N1+δ ) time. As in the work of Guruswami, Lee, and Razborov, our construction is the continuous analog of a Tanner code, and makes use of expander graphs to impose a collection of local linear constraints on vectors in the subspace. Our analysis is able to achieve uniform equivalence of the ℓ1 and ℓ2 norms (independent of the dimension). It has parallels to iterative decoding of Tanner codes, and leads to an analogous near-linear time algorithm for error-correction over reals.
AB - It is well-known that ℝN has subspaces of dimension proportional to N on which the ℓ1 and ℓ2 norms are uniformly equivalent, but it is unknown how to construct them explicitly. We show that, for any δ > 0, such a subspace can be generated using only Nδ random bits. This improves over previous constructions of Artstein-Avidan and Milman, and of Lovett and Sodin, which require O(N logN), and O(N) random bits, respectively. Such subspaces are known to also yield error-correcting codes over the reals and compressed sensing matrices. Our subspaces are defined by the kernel of a relatively sparse matrix (with at most Nδ non-zero entries per row), and thus enable compressed sensing in near-linear O(N1+δ ) time. As in the work of Guruswami, Lee, and Razborov, our construction is the continuous analog of a Tanner code, and makes use of expander graphs to impose a collection of local linear constraints on vectors in the subspace. Our analysis is able to achieve uniform equivalence of the ℓ1 and ℓ2 norms (independent of the dimension). It has parallels to iterative decoding of Tanner codes, and leads to an analogous near-linear time algorithm for error-correction over reals.
UR - http://www.scopus.com/inward/record.url?scp=51849090137&partnerID=8YFLogxK
U2 - 10.1007/978-3-540-85363-3_35
DO - 10.1007/978-3-540-85363-3_35
M3 - ???researchoutput.researchoutputtypes.contributiontobookanthology.conference???
AN - SCOPUS:51849090137
SN - 3540853626
SN - 9783540853626
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 444
EP - 454
BT - Approximation, Randomization and Combinatorial Optimization
T2 - 11th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2008 and 12th International Workshop on Randomization and Computation, RANDOM 2008
Y2 - 25 August 2008 through 27 August 2008
ER -