Using Elimination Theory to Construct Rigid Matrices

Abstract

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 [Val77], rigidity and similar rank-robustness functions of matrices have found numerous applications in circuit complexity, communication complexity, and learning complexity. Almost all n × n 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 = Ω(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 n × n matrix in this family are distinct primitive roots of unity of orders roughly exp(n 2 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 semicontinuous.

Matrix Rigidity and the Ill-Posedness of Robust PCA and Matrix Completion

“…due to elementary matrix properties [46]. Matrices which achieve this upper bound for every r are referred to as maximally rigid and it was only recently showed in [30] how to construct them explicitly, which was a long standing open question originally posed by Valiant in 1977. Matrix rigidity has important consequences for complexity of linear algebraic circuits but is also of interest for its mathematical properties.…”

“…It is shown in [30] that the matrix rigidity function might not be semicontinuous even for maximally rigid matrices. This translates into the set LS 3 It is easy to check that for a general choice of {a, .…”

“…It is well-known that solving these problems requires a low coherence between the low-rank matrix and the canonical basis, since in the extreme cases -when the low-rank matrix we wish to recover is also sparse -there is an inherent ambiguity. However, the well-posedness issue in both problems is an even more fundamental one: in some cases, both Robust PCA and matrix completion can fail to have any solutions due to the set of low-rank plus sparse matrices not being closed, which in turn is equivalent to the notion of the matrix rigidity function not being lower semicontinuous (Kumar et al, 2014). By constructing infinite families of matrices, we derive bounds on the rank and sparsity such that the set of low-rank plus sparse matrices is not closed.…”

On the geometry of tensor network states of 2 × N grids

“…In the proof of Theorem 1.10, we found a tensor which gave optimal border rank bounds via its N, N matrix flattening. It is still an open question to find an explicit tensor (explicit in the sense of complexity theory; see, e.g., [4]) in (C k ) ⊗2N where all N, N flattenings have maximal rank k N .…”

New applications of the polynomial method: The cap set conjecture and beyond

“…However, to date, the best lower bound on any explicit matrix A is R A (r) ≥ Ω( n 2 r log n r ) [54,120]. Other techniques that have been used to study rigid matrices include elimination theory [81], degree bounds [91,92], spectral methods [77], and algebraic geometry [58]. For a mostly up-to-date survey that also includes relations to other areas, see [93].…”