7,499 research outputs found
A lower bound for the determinantal complexity of a hypersurface
We prove that the determinantal complexity of a hypersurface of degree is bounded below by one more than the codimension of the singular locus,
provided that this codimension is at least . As a result, we obtain that the
determinantal complexity of the permanent is . We also prove
that for , there is no nonsingular hypersurface in of
degree that has an expression as a determinant of a matrix of
linear forms while on the other hand for , a general determinantal
expression is nonsingular. Finally, we answer a question of Ressayre by showing
that the determinantal complexity of the unique (singular) cubic surface
containing a single line is .Comment: 7 pages, 0 figure
A Quadratic Lower Bound for Homogeneous Algebraic Branching Programs
An algebraic branching program (ABP) is a directed acyclic graph, with a start vertex s, and end vertex t and each edge having a weight which is an affine form in variables x_1, x_2, ..., x_n over an underlying field. An ABP computes a polynomial in a natural way, as the sum of weights of all paths from s to t, where the weight of a path is the product of the weights of the edges in the path. An ABP is said to be homogeneous if the polynomial computed at every vertex is homogeneous. In this paper, we show that any homogeneous algebraic branching program which computes the polynomial x_1^n + x_2^n + ... + x_n^n has at least Omega(n^2) vertices (and edges).
To the best of our knowledge, this seems to be the first non-trivial super-linear lower bound on the number of vertices for a general homogeneous ABP and slightly improves the known lower bound of Omega(n log n) on the number of edges in a general (possibly non-homogeneous) ABP, which follows from the classical results of Strassen (1973) and Baur--Strassen (1983).
On the way, we also get an alternate and unified proof of an Omega(n log n) lower bound on the size of a homogeneous arithmetic circuit (follows from [Strassen, 1973] and [Baur-Strassen, 1983]), and an n/2 lower bound (n over reals) on the determinantal complexity of an explicit polynomial [Mignon-Ressayre, 2004], [Cai, Chen, Li, 2010], [Yabe, 2015]. These are currently the best lower bounds known for these problems for any explicit polynomial, and were originally proved nearly two decades apart using seemingly different proof techniques
A Lower Bound on Determinantal Complexity
The determinantal complexity of a polynomial over a field is the dimension of the smallest matrix
whose entries are affine functions in such that
. We prove that the determinantal complexity of the polynomial
is at least .
For every -variate polynomial of degree , the determinantal complexity
is trivially at least , and it is a long standing open problem to prove a
lower bound which is super linear in . Our result is the first
lower bound for any explicit polynomial which is bigger by a constant factor
than , and improves upon the prior best bound of , proved
by Alper, Bogart and Velasco [ABV17] for the same polynomial.Comment: v2: corrected a few typos and added reference
Arithmetic Circuits and the Hadamard Product of Polynomials
Motivated by the Hadamard product of matrices we define the Hadamard product
of multivariate polynomials and study its arithmetic circuit and branching
program complexity. We also give applications and connections to polynomial
identity testing. Our main results are the following. 1. We show that
noncommutative polynomial identity testing for algebraic branching programs
over rationals is complete for the logspace counting class \ceql, and over
fields of characteristic the problem is in \ModpL/\Poly. 2.We show an
exponential lower bound for expressing the Raz-Yehudayoff polynomial as the
Hadamard product of two monotone multilinear polynomials. In contrast the
Permanent can be expressed as the Hadamard product of two monotone multilinear
formulas of quadratic size.Comment: 20 page
An Almost Cubic Lower Bound for Depth Three Arithmetic Circuits
We show an almost cubic lower bound on the size of any depth three arithmetic circuit computing an explicit multilinear polynomial in n variables over any field. This improves upon the previously known quadratic lower bound by Shpilka and Wigderson [CCC, 1999]
Depth-4 Lower Bounds, Determinantal Complexity : A Unified Approach
Tavenas has recently proved that any n^{O(1)}-variate and degree n polynomial
in VP can be computed by a depth-4 circuit of size 2^{O(\sqrt{n}\log n)}. So to
prove VP not equal to VNP, it is sufficient to show that an explicit polynomial
in VNP of degree n requires 2^{\omega(\sqrt{n}\log n)} size depth-4 circuits.
Soon after Tavenas's result, for two different explicit polynomials, depth-4
circuit size lower bounds of 2^{\Omega(\sqrt{n}\log n)} have been proved Kayal
et al. and Fournier et al. In particular, using combinatorial design Kayal et
al.\ construct an explicit polynomial in VNP that requires depth-4 circuits of
size 2^{\Omega(\sqrt{n}\log n)} and Fournier et al.\ show that iterated matrix
multiplication polynomial (which is in VP) also requires 2^{\Omega(\sqrt{n}\log
n)} size depth-4 circuits.
In this paper, we identify a simple combinatorial property such that any
polynomial f that satisfies the property would achieve similar circuit size
lower bound for depth-4 circuits. In particular, it does not matter whether f
is in VP or in VNP. As a result, we get a very simple unified lower bound
analysis for the above mentioned polynomials.
Another goal of this paper is to compare between our current knowledge of
depth-4 circuit size lower bounds and determinantal complexity lower bounds. We
prove the that the determinantal complexity of iterated matrix multiplication
polynomial is \Omega(dn) where d is the number of matrices and n is the
dimension of the matrices. So for d=n, we get that the iterated matrix
multiplication polynomial achieves the current best known lower bounds in both
fronts: depth-4 circuit size and determinantal complexity. To the best of our
knowledge, a \Theta(n) bound for the determinantal complexity for the iterated
matrix multiplication polynomial was known only for constant d>1 by Jansen.Comment: Extension of the previous uploa
Permanent v. determinant: an exponential lower bound assumingsymmetry and a potential path towards Valiant's conjecture
International audienceWe initiate a study of determinantal representations with symmetry. We show that Grenet's determinantal representation for the permanent is optimal among determinantal representations respecting left multiplication by permutation and diagonal matrices (roughly half the symmetry group of the permanent). In particular, if any optimal determinantal representation of the permanent must be polynomially related to one with such symmetry, then Valiant's conjecture on permanent v. determinant is true
Geometric lower bounds for generalized ranks
We revisit a geometric lower bound for Waring rank of polynomials (symmetric
rank of symmetric tensors) of Landsberg and Teitler and generalize it to a
lower bound for rank with respect to arbitrary varieties, improving the bound
given by the "non-Abelian" catalecticants recently introduced by Landsberg and
Ottaviani. This is applied to give lower bounds for ranks of multihomogeneous
polynomials (partially symmetric tensors); a special case is the simultaneous
Waring decomposition problem for a linear system of polynomials. We generalize
the classical Apolarity Lemma to multihomogeneous polynomials and give some
more general statements. Finally we revisit the lower bound of Ranestad and
Schreyer, and again generalize it to multihomogeneous polynomials and some more
general settings.Comment: 43 pages. v2: minor change
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