14 research outputs found
Using Elimination Theory to construct Rigid Matrices
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 n x 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
semi-continuous.Comment: 25 Pages, minor typos correcte
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 of which can only depend on
variables corresponding to *-entries in the i-th row of A. We conjecture that
no such system can have more than 2^{n-c\cdot mr(A)} solutions, where c>0 is an
absolute constant and mr(A) is the smallest rank over GF(2) of a completion of
A. The conjecture is related to an old problem of proving super-linear lower
bounds on the size of log-depth boolean circuits computing linear operators x
--> Mx. The conjecture is also a generalization of a classical question about
how much larger can non-linear codes be than linear ones. We prove some special
cases of the conjecture and establish some structural properties of solution
sets.Comment: 22 pages, to appear in: J. Comput.Syst.Sci
Matrix Rigidity from the Viewpoint of Parameterized Complexity
The rigidity of a matrix A for a target rank r over a field F is the minimum Hamming distance between A and a matrix of rank at most r. Rigidity is a classical concept in Computational Complexity Theory: constructions of rigid matrices are known to imply lower bounds of significant importance relating to arithmetic circuits. Yet, from the viewpoint of Parameterized Complexity, the study of central properties of matrices in general, and of the rigidity of a matrix in particular, has been neglected. In this paper, we conduct a comprehensive study of different aspects of the computation of the rigidity of general matrices in the framework of Parameterized Complexity. Naturally, given parameters r and k, the Matrix Rigidity problem asks whether the rigidity of A for the target rank r is at most k. We show that in case F equals the reals or F is any finite field, this problem is fixed-parameter tractable with respect to k+r. To this end, we present a dimension reduction procedure, which may be a valuable primitive in future studies of problems of this nature. We also employ central tools in Real Algebraic Geometry, which are not well known in Parameterized Complexity, as a black box. In particular, we view the output of our dimension reduction procedure as an algebraic variety. Our main results are complemented by a W[1]-hardness result and a subexponential-time parameterized algorithm for a special case of Matrix Rigidity, highlighting the different flavors of this problem
Complexity of linear circuits and geometry
We use algebraic geometry to study matrix rigidity, and more generally, the
complexity of computing a matrix-vector product, continuing a study initiated
by Kumar, et. al. We (i) exhibit many non-obvious equations testing for
(border) rigidity, (ii) compute degrees of varieties associated to rigidity,
(iii) describe algebraic varieties associated to families of matrices that are
expected to have super-linear rigidity, and (iv) prove results about the ideals
and degrees of cones that are of interest in their own right.Comment: 29 pages, final version to appear in FOC
Block Rigidity: Strong Multiplayer Parallel Repetition Implies Super-Linear Lower Bounds for Turing Machines
We prove that a sufficiently strong parallel repetition theorem for a special
case of multiplayer (multiprover) games implies super-linear lower bounds for
multi-tape Turing machines with advice. To the best of our knowledge, this is
the first connection between parallel repetition and lower bounds for time
complexity and the first major potential implication of a parallel repetition
theorem with more than two players.
Along the way to proving this result, we define and initiate a study of block
rigidity, a weakening of Valiant's notion of rigidity. While rigidity was
originally defined for matrices, or, equivalently, for (multi-output) linear
functions, we extend and study both rigidity and block rigidity for general
(multi-output) functions. Using techniques of Paul, Pippenger, Szemer\'edi and
Trotter, we show that a block-rigid function cannot be computed by multi-tape
Turing machines that run in linear (or slightly super-linear) time, even in the
non-uniform setting, where the machine gets an arbitrary advice tape.
We then describe a class of multiplayer games, such that, a sufficiently
strong parallel repetition theorem for that class of games implies an explicit
block-rigid function. The games in that class have the following property that
may be of independent interest: for every random string for the verifier
(which, in particular, determines the vector of queries to the players), there
is a unique correct answer for each of the players, and the verifier accepts if
and only if all answers are correct. We refer to such games as independent
games. The theorem that we need is that parallel repetition reduces the value
of games in this class from to , where is the number of
repetitions.
As another application of block rigidity, we show conditional size-depth
tradeoffs for boolean circuits, where the gates compute arbitrary functions
over large sets.Comment: 17 pages, ITCS 202
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Fourier and Circulant Matrices Are Not Rigid
The concept of matrix rigidity was first introduced by Valiant in [Friedman, 1993]. Roughly speaking, a matrix is rigid if its rank cannot be reduced significantly by changing a small number of entries. There has been extensive interest in rigid matrices as Valiant showed in [Friedman, 1993] that rigidity can be used to prove arithmetic circuit lower bounds.
In a surprising result, Alman and Williams showed that the (real valued) Hadamard matrix, which was conjectured to be rigid, is actually not very rigid. This line of work was extended by [Dvir and Edelman, 2017] to a family of matrices related to the Hadamard matrix, but over finite fields. In our work, we take another step in this direction and show that for any abelian group G and function f:G - > {C}, the matrix given by M_{xy} = f(x - y) for x,y in G is not rigid. In particular, we get that complex valued Fourier matrices, circulant matrices, and Toeplitz matrices are all not rigid and cannot be used to carry out Valiant\u27s approach to proving circuit lower bounds. This complements a recent result of Goldreich and Tal [Goldreich and Tal, 2016] who showed that Toeplitz matrices are nontrivially rigid (but not enough for Valiant\u27s method). Our work differs from previous non-rigidity results in that those works considered matrices whose underlying group of symmetries was of the form {F}_p^n with p fixed and n tending to infinity, while in the families of matrices we study, the underlying group of symmetries can be any abelian group and, in particular, the cyclic group {Z}_N, which has very different structure. Our results also suggest natural new candidates for rigidity in the form of matrices whose symmetry groups are highly non-abelian.
Our proof has four parts. The first extends the results of [Josh Alman and Ryan Williams, 2016; Dvir and Edelman, 2017] to generalized Hadamard matrices over the complex numbers via a new proof technique. The second part handles the N x N Fourier matrix when N has a particularly nice factorization that allows us to embed smaller copies of (generalized) Hadamard matrices inside of it. The third part uses results from number theory to bootstrap the non-rigidity for these special values of N and extend to all sufficiently large N. The fourth and final part involves using the non-rigidity of the Fourier matrix to show that the group algebra matrix, given by M_{xy} = f(x - y) for x,y in G, is not rigid for any function f and abelian group G
Matrix Rigidity Depends on the Target Field
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 (Valiant, 1977).
We study the dependence of rigidity on the target field. We consider especially two natural regimes: when one is allowed to make changes only from the field of definition of the matrix ("strict rigidity"), and when the changes are allowed to be in an arbitrary extension field ("absolute rigidity").
We demonstrate, apparently for the first time, a separation between these two concepts. We establish a gap of a factor of 3/2-o(1) between strict and absolute rigidities.
The question seems especially timely because of recent results by Dvir and Liu (Theory of Computing, 2020) where important families of matrices, previously expected to be rigid, are shown not to be absolutely rigid, while their strict rigidity remains open. Our lower-bound method combines elementary arguments from algebraic geometry with "untouched minors" arguments.
Finally, we point out that more families of long-time rigidity candidates fall as a consequence of the results of Dvir and Liu. These include the incidence matrices of projective planes over finite fields, proposed by Valiant as candidates for rigidity over ??