Related papers: Fourier and Circulant Matrices are Not Rigid

The concept of matrix rigidity was introduced by Valiant(independently by Grigoriev) in the context of computing linear transformations. A matrix is rigid if it is far(in terms of Hamming distance) from any matrix of low rank. Although we…

Matrix rigidity is a notion put forth by Valiant as a means for proving arithmetic circuit lower bounds. A matrix is rigid if it is far, in Hamming distance, from any low rank matrix. Despite decades of efforts, no explicit matrix rigid…

The rigidity of a matrix $A$ for target rank $r$ is the minimum number of entries of $A$ that need to be changed in order to obtain a matrix of rank at most $r$. At MFCS'77, Valiant introduced matrix rigidity as a tool to prove circuit…

The rigidity of a matrix describes the minimal number of entries one has to change to reduce matrix's rank to r. We give very simple combinatorial proof of the lower bound for the rigidity of Sylvester (special case of Hadamard) matrix that…

We consider a notion of probabilistic rank and probabilistic sign-rank of a matrix, which measures the extent to which a matrix can be probabilistically represented by low-rank matrices. We demonstrate several connections with matrix…

For a matrix $M$ and a positive integer $r$, the rank $r$ rigidity of $M$ is the smallest number of entries of $M$ which one must change to make its rank at most $r$. There are many known applications of rigidity lower bounds to a variety…

The rigidity of a matrix measures how many of its entries need to be changed in order to reduce its rank to some value. Good lower bounds on the rigidity of an explicit matrix would imply good lower bounds for arithmetic circuits as well as…

For an $N \times N$ matrix $A$, its rank-$r$ rigidity, denoted $\mathcal{R}_A(r)$, is the minimum number of entries of $A$ that one must change to make its rank become at most $r$. Determining the rigidity of interesting explicit families…

A matrix is said to be {\it cyclic} if its characteristic polynomial is equal to its minimal polynomial. Cyclic matrices play an important role in some algorithms for matrix group computation, such as the Cyclic Meataxe developed by P. M.…

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…

Circulant matrices are an important tool widely used in coding theory and cryptography. A circulant matrix is a square matrix whose rows are the cyclic shifts of the first row. Such a matrix can be efficiently stored in memory because it is…

A ring is rigid if there is no nonzero locally nilpotent derivation on it. In terms of algebraic geometry, a rigid coordinate ring corresponds to an algebraic affine variety which does not allow any nontrivial algebraic additive group…

In a celebrated paper of 1893, Hadamard established the maximal determinant theorem, which establishes an upper bound on the determinant of a matrix with complex entries of norm at most $1$. His paper concludes with the suggestion that…

A $d$-dimensional (bar-and-joint) framework $(G,p)$ consists of a graph $G=(V,E)$ and a realisation $p:V\to \mathbb{R}^d$. It is rigid if every continuous motion of the vertices which preserves the lengths of the edges is induced by an…

For plane frameworks with reflection or rotational symmetries, where the group action is not necessarily free on the vertex set, we introduce a phase-symmetric orbit rigidity matrix for each irreducible representation of the group. We then…

We follow in this paper a recent line of work, consisting in characterizing the periodically rigid finitely generated groups, i.e., the groups for which every subshift of finite type which is weakly aperiodic is also strongly aperiodic. In…

Fourier matrices naturally appear in many applications and their stability is closely tied to performance guarantees of algorithms. The starting point of this article is a result that characterizes properties of an exponential system on a…

A bar-joint framework $(G,p)$ in $\mathbb{R}^d$ is rigid if the only edge-length preserving continuous motions of the vertices arise from isometries of $\mathbb{R}^d$. It is known that, when $(G,p)$ is generic, its rigidity depends only on…

Recently, Dvir, Golovnev, and Weinstein have shown that sufficiently strong lower bounds for linear data structures would imply new bounds for rigid matrices. However, their result utilizes an algorithm that requires an $NP$ oracle, and…

We give lower bounds on the largest singular value of arbitrary matrices, some of which are asymptotically tight for almost all matrices. To study when these bounds are exact, we introduce several combinatorial concepts. In particular, we…