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Electronic Colloquium on Computational Complexity

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Reports tagged with Matrix Rigidity:
TR97-043 | 25th September 1997
Bruno Codenotti, Pavel Pudlak, Giovanni Resta

Some structural properties of low rank matrices related to computational complexity

Revisions: 1 , Comments: 1

We consider the conjecture stating that a matrix with rank
$o(n)$ and ones on the main diagonal must contain nonzero
entries on a $2\times 2$ submatrix with one entry on the main
diagonal. We show that a slightly stronger conjecture implies
that ... more >>>

TR98-075 | 9th December 1998
Adam Klivans, Dieter van Melkebeek

Graph Nonisomorphism has Subexponential Size Proofs Unless the Polynomial-Time Hierarchy Collapses.

We establish hardness versus randomness trade-offs for a
broad class of randomized procedures. In particular, we create efficient
nondeterministic simulations of bounded round Arthur-Merlin games using
a language in exponential time that cannot be decided by polynomial
size oracle circuits with access to satisfiability. We show that every
language with ... more >>>

TR05-070 | 6th July 2005
Mahdi Cheraghchi

On Matrix Rigidity and the Complexity of Linear Forms

The rigidity function of a matrix is defined as the minimum number of its entries that need to be changed in order to reduce the rank of the matrix to below a given parameter. Proving a strong enough lower bound on the rigidity of a matrix implies a nontrivial lower ... more >>>

TR08-065 | 11th July 2008
Noga Alon, Rina Panigrahy, Sergey Yekhanin

Deterministic Approximation Algorithms for the Nearest Codeword Problem

The Nearest Codeword Problem (NCP) is a basic algorithmic question in the theory of error-correcting codes. Given a point v in an n-dimensional space over F_2 and a linear subspace L in F_2^n of dimension k NCP asks to find a point l in L that minimizes the (Hamming) distance ... more >>>

TR09-008 | 15th January 2009
Stasys Jukna, Georg Schnitger

Min-Rank Conjecture for Log-Depth Circuits

A completion of an m-by-n matrix A with entries in {0,1,*} is obtained
by setting all *-entries to constants 0 or 1. A system of semi-linear
equations over GF(2) has the form Mx=f(x), where M is a completion of
A and f:{0,1}^n --> {0,1}^m is an operator, the i-th coordinate ... more >>>

TR09-084 | 24th September 2009
Arkadev Chattopadhyay, Avi Wigderson

Linear systems over composite moduli

We study solution sets to systems of generalized linear equations of the following form:
$\ell_i (x_1, x_2, \cdots , x_n)\, \in \,A_i \,\, (\text{mod } m)$,
where $\ell_1, \ldots ,\ell_t$ are linear forms in $n$ Boolean variables, each $A_i$ is an arbitrary subset of $\mathbb{Z}_m$, and $m$ is a composite ... more >>>

TR10-086 | 17th May 2010
Henning Wunderlich

On a Theorem of Razborov

In an unpublished Russian manuscript Razborov proved that a matrix family with high
rigidity over a finite field would yield a language outside the polynomial hierarchy
in communication complexity.

We present an alternative proof that strengthens the original result in several ways.
In particular, we replace rigidity by the strictly ... more >>>

TR10-149 | 22nd September 2010
Boaz Barak, Zeev Dvir, Avi Wigderson, Amir Yehudayoff

Rank Bounds for Design Matrices with Applications to Combinatorial Geometry and Locally Correctable Codes

Revisions: 1

A $(q,k,t)$-design matrix is an m x n matrix whose pattern of zeros/non-zeros satisfies the following design-like condition: each row has at most $q$ non-zeros, each column has at least $k$ non-zeros and the supports of every two columns intersect in at most t rows. We prove that the rank ... more >>>

TR11-044 | 25th March 2011
Shubhangi Saraf, Sergey Yekhanin

Noisy Interpolation of Sparse Polynomials, and Applications

Let $f\in F_q[x]$ be a polynomial of degree $d\leq q/2.$ It is well-known that $f$ can be uniquely recovered from its values at some $2d$ points even after some small fraction of the values are corrupted. In this paper we establish a similar result for sparse polynomials. We show that ... more >>>

TR12-133 | 21st October 2012
Noga Alon, Gil Cohen

On Rigid Matrices and Subspace Polynomials

Revisions: 1

We introduce a class of polynomials, which we call \emph{subspace polynomials} and show that the problem of explicitly constructing a rigid matrix can be reduced to the problem of explicitly constructing a small hitting set for this class. We prove that small-bias sets are hitting sets for the class of ... more >>>

TR13-043 | 25th March 2013
Oded Goldreich, Avi Wigderson

On the Size of Depth-Three Boolean Circuits for Computing Multilinear Functions

Revisions: 1

We propose that multi-linear functions of relatively low degree
over GF(2) may be good candidates for obtaining exponential
lower bounds on the size of constant-depth Boolean circuits
(computing explicit functions).
Specifically, we propose to move gradually from linear functions
to multilinear ones, and conjecture that, for any $t\geq2$,
more >>>

TR14-172 | 12th December 2014
Alex Samorodnitsky, Ilya Shkredov, Sergey Yekhanin

Kolmogorov Width of Discrete Linear Spaces: an Approach to Matrix Rigidity

A square matrix $V$ is called rigid if every matrix $V^\prime$ 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 ... more >>>

TR15-079 | 7th May 2015
Oded Goldreich, Avishay Tal

Matrix Rigidity of Random Toeplitz Matrices

We prove that random $n$-by-$n$ Toeplitz (alternatively Hankel) matrices over $GF(2)$ have rigidity $\Omega(\frac{n^3}{r^2\log n})$ for rank $r \ge \sqrt{n}$, with high probability. This improves, for $r = o(n/\log n \log\log n)$, over the $\Omega(\frac{n^2}{r} \cdot\log(\frac{n}{r}))$ bound that is known for many explicit matrices.

Our result implies that the explicit ... more >>>

TR16-093 | 4th June 2016
Cyrus Rashtchian

Bounded Matrix Rigidity and John's Theorem

Using John's Theorem, we prove a lower bound on the bounded rigidity of a sign matrix, defined as the Hamming distance between this matrix and the set of low-rank, real-valued matrices with entries bounded in absolute value. For Hadamard matrices, our asymptotic leading constant is tighter than known results by ... more >>>

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