On the complexity of matrix product
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
ALT '02 Proceedings of the 13th International Conference on Algorithmic Learning Theory
Lower bounds on the bounded coefficient complexity of bilinear maps
Journal of the ACM (JACM)
On relations between counting communication complexity classes
Journal of Computer and System Sciences
Theoretical Computer Science - Algorithmic learning theory(ALT 2002)
Elusive functions and lower bounds for arithmetic circuits
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Rigidity of a simple extended lower triangular matrix
Information Processing Letters
Learning complexity vs communication complexity
Combinatorics, Probability and Computing
Linear time approximation schemes for the Gale-Berlekamp game and related minimization problems
Proceedings of the forty-first annual ACM symposium on Theory of computing
On covering and rank problems for boolean matrices and their applications
COCOON'99 Proceedings of the 5th annual international conference on Computing and combinatorics
Sparse and low-rank matrix decompositions
Allerton'09 Proceedings of the 47th annual Allerton conference on Communication, control, and computing
Lower bounds on matrix rigidity via a quantum argument
ICALP'06 Proceedings of the 33rd international conference on Automata, Languages and Programming - Volume Part I
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The rigidity of a matrix measures the number of entries that must be changed in order to reduce its rank below a certain value. The known lower bounds on the rigidity of explicit matrices are very weak. It is known that stronger lower bounds would have implications to complexity theory. We consider weaker forms of the rigidity problem over the complex numbers. Using spectral methods, we derive lower bounds on these variants. We then give two applications of such weaker forms. First, we show that our lower bound on a variant of rigidity implies lower bounds on size-depth tradeoffs for arithmetic circuits with bounded coefficients computing linear transformations. These bounds generalize a recent result of Nisan and Wigderson. The second application is conditional; we show that it would suffice to prove lower bounds on certain weaker forms of rigidity to conclude several separation results in communication complexity theory. Our results complement and strengthen a result of Razborov.