Abstract
A square matrix V is called rigid if every matrix $${V^\prime}$$V? obtained by altering a small number of entries of V has sufficiently high rank. While random matrices are rigid with high probability, no explicit constructions of rigid matrices are known to date. Obtaining such explicit matrices would have major implications in computational complexity theory. One approach to establishing rigidity of a matrix V is to come up with a property that is satisfied by any collection of vectors arising from a low-dimensional space, but is not satisfied by the rows of V even after alterations. In this paper, we propose such a candidate property that has the potential of establishing rigidity of combinatorial design matrices over the field $${\mathbb{F}_2.}$$F2. Stated informally, we conjecture that under a suitable embedding of $${\mathbb{F}_2^n}$$F2n into $${\mathbb{R}^n,}$$Rn, vectors arising from a low-dimensional $${\mathbb{F}_2}$$F2-linear space always have somewhat small Kolmogorov width, i.e., admit a non-trivial simultaneous approximation by a low-dimensional Euclidean space. This implies rigidity of combinatorial designs, as their rows do not admit such an approximation even after alterations. Our main technical contribution is a collection of results establishing weaker forms and special cases of the conjecture above.
Highlights
The notion of matrix rigidity was introduced by Leslie Valiant in 1977 [23]
In order to establish rigidity of matrices Vm over the field F2 we propose a certain property that is not satisfied by the rows of Vm even after alterations, yet that we conjecture to hold for any collection of vectors arising from a low-dimensional F2-linear space
In this paper we suggested a new path to establishing rigidity of design matrices over the field F2
Summary
The notion of matrix rigidity was introduced by Leslie Valiant in 1977 [23]. In this paper we say that an n × n matrix A defined over a field is (r, d)-rigid, if it is not possible to reduce the rank of A below r by arbitrarily altering each row of A in up to d coordinates. Explicit rigid matrices are known to imply lower bounds for computational complexity of explicit functions. The most prominent reduction of this nature is due to Valiant [23] who showed that for each (Ω(n), n )-rigid matrix A ∈ Fn×n the linear transformation induced by A cannot be computed by a linear circuit that simultaneously has size O(n) and depth O(log n). Two other reductions that call for explicit (r, d)-rigid matrices with a sub-linear value of the remaining rank r, are given in [15, 18]. 25] to show that after up to d arbitrary changes per row there remains a somewhat large minor that has not been touched This yields explicit nn r, Ω log rr rigid matrices over fields of size Ω(n), when log n ≤ r ≤ n/2. It is not hard to show that a random matrix over any field is at least n , Ω(n) rigid with a very high probability
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