Spectral Methods for Matrix Rigidity with Applications

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 restricted variants of the rigidity problem over the complex numbers. Using spectral methods, we derive lower bounds on these variants. Two applications of such restricted variants are given. First, we show that our lower bound on a variant of rigidity implies lower bounds on size-depth tradeoos for arithmetic circuits with bounded coeecients computing linear transformations. These bounds generalize a result of Nisan and Wigderson. The second application is conditional; we show that it would suuce to prove lower bounds on certain restricted forms of rigidity to conclude several separation results such as separating the analogs of PH and PSPACE in communication complexity theory. Our results complement and strengthen a result of Razborov. We introduce a complexity measure, called AC 0-dimension, of families of boolean functions. While high rigidity implies large AC 0-dimension, large AC 0-dimension for explicit families would already give explicit languages outside the analog of PH in communication complexity. Moreover, the concept of AC 0-dimension allows us to formulate interesting combinatorial problems which may be easier than rigidity and which would still have consequences to separation questions in communication complexity.