TR09-106 Authors: Abhinav Kumar, Satyanarayana V. Lokam, Vijay M. Patankar, Jayalal Sarma

Publication: 28th October 2009 09:21

Downloads: 3160

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The rigidity of a matrix A for target rank r is the minimum number of entries

of A that must be changed to ensure that the rank of the altered matrix is at

most r. Since its introduction by Valiant (1977), rigidity and similar

rank-robustness functions of matrices have found numerous applications in

circuit complexity, communication complexity, and learning complexity. Almost

all nxn matrices over an infinite field have a rigidity of (n-r)^2. It is a

long-standing open question to construct infinite families of explicit matrices

even with superlinear rigidity when r=Omega(n).

In this paper, we construct an infinite family of complex matrices with the

largest possible, i.e., (n-r)^2, rigidity. The entries of an nxn matrix in this

family are distinct primitive roots of unity of orders roughly exp(n^4 log n).

To the best of our knowledge, this is the first family of concrete (but not

entirely explicit) matrices having maximal rigidity and a succinct algebraic

description.

Our construction is based on elimination theory of polynomial ideals. In

particular, we use results on the existence of polynomials in elimination

ideals with effective degree upper bounds (effective Nullstellensatz). Using

elementary algebraic geometry, we prove that the dimension of the affine

variety of matrices of rigidity at most k is exactly n^2-(n-r)^2+k$. Finally,

we use elimination theory to examine whether the rigidity function is

semi-continuous.