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. W e consider weaker forms of the rigidity problem over the complex numbers. Using spectral methods, we derive lower bounds on these variants. W e 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 coeficients computing linear transformations. These bounds generalize a recent result of Nisan and Wigderson. The second application is conditional; we show that it would sufice 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.
Lower bounds on matrix rigidity have applications to circuit and communication complexity REF .
Spectral methods for matrix rigidity with applications to size-depth tradeoffs and communication complexity
Proceedings of IEEE 36th Annual Foundations of Computer Science
Proceedings of IEEE 36th Annual Foundations of Computer Science
Mathematics Computer Science