Generalized matrix completion and algebraic natural proofs
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Generalized matrix completion and algebraic natural proofs. / Bläser, Markus; Jindal, Gorav; Ikenmeyer, Christian; Lysikov, Vladimir.
STOC 2018: Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing. New York : Association for Computing Machinery, 2018. s. 17-29 (Proceedings of the Annual ACM Symposium on Theory of Computing).Publikation: Bidrag til bog/antologi/rapport › Konferencebidrag i proceedings › Forskning › fagfællebedømt
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TY - GEN
T1 - Generalized matrix completion and algebraic natural proofs
AU - Bläser, Markus
AU - Jindal, Gorav
AU - Ikenmeyer, Christian
AU - Lysikov, Vladimir
PY - 2018/6/20
Y1 - 2018/6/20
N2 - Algebraic natural proofs were recently introduced by Forbes, Shpilka and Volk (Proc. of the 49th Annual ACM SIGACT Symposium on Theory of Computing (STOC), pages 653–664, 2017) and independently by Grochow, Kumar, Saks and Saraf (CoRR, abs/1701.01717, 2017) as an attempt to transfer Razborov and Rudich’s famous barrier result (J. Comput. Syst. Sci., 55(1): 24–35, 1997) for Boolean circuit complexity to algebraic complexity theory. Razborov and Rudich’s barrier result relies on a widely believed assumption, namely, the existence of pseudo-random generators. Unfortunately, there is no known analogous theory of pseudo-randomness in the algebraic setting. Therefore, Forbes et al. use a concept called succinct hitting sets instead. This assumption is related to polynomial identity testing, but it is currently not clear how plausible this assumption is. Forbes et al. are only able to construct succinct hitting sets against rather weak models of arithmetic circuits. Generalized matrix completion is the following problem: Given a matrix with affine linear forms as entries, find an assignment to the variables in the linear forms such that the rank of the resulting matrix is minimal. We call this rank the completion rank. Computing the completion rank is an NP-hard problem. As our first main result, we prove that it is also NP-hard to determine whether a given matrix can be approximated by matrices of completion rank ≤ b. The minimum quantity b for which this is possible is called border completion rank (similar to the border rank of tensors). Naturally, algebraic natural proofs can only prove lower bounds for such border complexity measures. Furthermore, these border complexity measures play an important role in the geometric complexity program.
AB - Algebraic natural proofs were recently introduced by Forbes, Shpilka and Volk (Proc. of the 49th Annual ACM SIGACT Symposium on Theory of Computing (STOC), pages 653–664, 2017) and independently by Grochow, Kumar, Saks and Saraf (CoRR, abs/1701.01717, 2017) as an attempt to transfer Razborov and Rudich’s famous barrier result (J. Comput. Syst. Sci., 55(1): 24–35, 1997) for Boolean circuit complexity to algebraic complexity theory. Razborov and Rudich’s barrier result relies on a widely believed assumption, namely, the existence of pseudo-random generators. Unfortunately, there is no known analogous theory of pseudo-randomness in the algebraic setting. Therefore, Forbes et al. use a concept called succinct hitting sets instead. This assumption is related to polynomial identity testing, but it is currently not clear how plausible this assumption is. Forbes et al. are only able to construct succinct hitting sets against rather weak models of arithmetic circuits. Generalized matrix completion is the following problem: Given a matrix with affine linear forms as entries, find an assignment to the variables in the linear forms such that the rank of the resulting matrix is minimal. We call this rank the completion rank. Computing the completion rank is an NP-hard problem. As our first main result, we prove that it is also NP-hard to determine whether a given matrix can be approximated by matrices of completion rank ≤ b. The minimum quantity b for which this is possible is called border completion rank (similar to the border rank of tensors). Naturally, algebraic natural proofs can only prove lower bounds for such border complexity measures. Furthermore, these border complexity measures play an important role in the geometric complexity program.
KW - Algebraic natural proofs
KW - Completion rank
KW - Geometric complexity theory
KW - Matrix completion
KW - Tensor rank
UR - http://www.scopus.com/inward/record.url?scp=85049907649&partnerID=8YFLogxK
U2 - 10.1145/3188745.3188832
DO - 10.1145/3188745.3188832
M3 - Article in proceedings
AN - SCOPUS:85049907649
T3 - Proceedings of the Annual ACM Symposium on Theory of Computing
SP - 17
EP - 29
BT - STOC 2018
PB - Association for Computing Machinery
CY - New York
T2 - 50th Annual ACM Symposium on Theory of Computing, STOC 2018
Y2 - 25 June 2018 through 29 June 2018
ER -
ID: 232711612