55 research outputs found
On the power of homogeneous depth 4 arithmetic circuits
We prove exponential lower bounds on the size of homogeneous depth 4
arithmetic circuits computing an explicit polynomial in . Our results hold
for the {\it Iterated Matrix Multiplication} polynomial - in particular we show
that any homogeneous depth 4 circuit computing the entry in the product
of generic matrices of dimension must have size
.
Our results strengthen previous works in two significant ways.
Our lower bounds hold for a polynomial in . Prior to our work, Kayal et
al [KLSS14] proved an exponential lower bound for homogeneous depth 4 circuits
(over fields of characteristic zero) computing a poly in . The best known
lower bounds for a depth 4 homogeneous circuit computing a poly in was the
bound of by [LSS, KLSS14].Our exponential lower bounds
also give the first exponential separation between general arithmetic circuits
and homogeneous depth 4 arithmetic circuits. In particular they imply that the
depth reduction results of Koiran [Koi12] and Tavenas [Tav13] are tight even
for reductions to general homogeneous depth 4 circuits (without the restriction
of bounded bottom fanin).
Our lower bound holds over all fields. The lower bound of [KLSS14] worked
only over fields of characteristic zero. Prior to our work, the best lower
bound for homogeneous depth 4 circuits over fields of positive characteristic
was [LSS, KLSS14]
Sums of products of polynomials in few variables : lower bounds and polynomial identity testing
We study the complexity of representing polynomials as a sum of products of
polynomials in few variables. More precisely, we study representations of the
form such that each is
an arbitrary polynomial that depends on at most variables. We prove the
following results.
1. Over fields of characteristic zero, for every constant such that , we give an explicit family of polynomials , where
is of degree in variables, such that any
representation of the above type for with requires . This strengthens a recent result of Kayal and Saha
[KS14a] which showed similar lower bounds for the model of sums of products of
linear forms in few variables. It is known that any asymptotic improvement in
the exponent of the lower bounds (even for ) would separate VP
and VNP[KS14a].
2. We obtain a deterministic subexponential time blackbox polynomial identity
testing (PIT) algorithm for circuits computed by the above model when and
the individual degree of each variable in are at most and
for any constant . We get quasipolynomial running
time when . The PIT algorithm is obtained by combining our
lower bounds with the hardness-randomness tradeoffs developed in [DSY09, KI04].
To the best of our knowledge, this is the first nontrivial PIT algorithm for
this model (even for the case ), and the first nontrivial PIT algorithm
obtained from lower bounds for small depth circuits
Kakeya sets and the method of multiplicities
Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2009.Cataloged from PDF version of thesis.Includes bibliographical references (p. 51-53).We extend the "method of multiplicities" to get the following results, of interest in combinatorics and randomness extraction. 1. We show that every Kakeya set (a set of points that contains a line in every direction) in F' must be of size at least qn/2n. This bound is tight to within a 2 + o(1) factor for every n as q -- oc, compared to previous bounds that were off by exponential factors in n. 2. We give improved randomness extractors and "randomness mergers". Mergers are seeded functions that take as input A (possibly correlated) random variables in {0, 1}N and a short random seed and output a single random variable in {0, 1}N that is statistically close to having entropy (1 - 6) - N when one of the A input variables is distributed uniformly. The seed we require is only (1/6) - log A-bits long, which significantly improves upon previous construction of mergers. 3. Using our new mergers, we show how to construct randomness extractors that use logarithmic length seeds while extracting 1- o(1) fraction of the min-entropy of the source. The "method of multiplicities", as used in prior work, analyzed subsets of vector spaces over finite fields by constructing somewhat low degree interpolating polynomials that vanish on every point in the subset with high multiplicity. The typical use of this method involved showing that the interpolating polynomial also vanished on some points outside the subset, and then used simple bounds on the number of zeroes to complete the analysis. Our augmentation to this technique is that we prove, under appropriate conditions, that the interpolating polynomial vanishes with high multiplicity outside the set. This novelty leads to significantly tighter analyses.by Shubhangi Saraf.S.M
Acute and nonobtuse triangulations of polyhedral surfaces
In this paper, we prove the existence of acute triangulations for general polyhedral surfaces. We also show how to obtain nonobtuse subtriangulations of triangulated polyhedral surfaces.Massachusetts Institute of Technology (UROP Program
Improved rank bounds for design matrices and a new proof of Kelly's theorem
We study the rank of complex sparse matrices in which the supports of
different columns have small intersections. The rank of these matrices, called
design matrices, was the focus of a recent work by Barak et. al. (BDWY11) in
which they were used to answer questions regarding point configurations. In
this work we derive near-optimal rank bounds for these matrices and use them to
obtain asymptotically tight bounds in many of the geometric applications. As a
consequence of our improved analysis, we also obtain a new, linear algebraic,
proof of Kelly's theorem, which is the complex analog of the Sylvester-Gallai
theorem
Deterministic Factorization of Sparse Polynomials with Bounded Individual Degree
In this paper we study the problem of deterministic factorization of sparse
polynomials. We show that if is a
polynomial with monomials, with individual degrees of its variables bounded
by , then can be deterministically factored in time . Prior to our work, the only efficient factoring algorithms known for
this class of polynomials were randomized, and other than for the cases of
and , only exponential time deterministic factoring algorithms were
known.
A crucial ingredient in our proof is a quasi-polynomial sparsity bound for
factors of sparse polynomials of bounded individual degree. In particular we
show if is an -sparse polynomial in variables, with individual
degrees of its variables bounded by , then the sparsity of each factor of
is bounded by . This is the first nontrivial bound on
factor sparsity for . Our sparsity bound uses techniques from convex
geometry, such as the theory of Newton polytopes and an approximate version of
the classical Carath\'eodory's Theorem.
Our work addresses and partially answers a question of von zur Gathen and
Kaltofen (JCSS 1985) who asked whether a quasi-polynomial bound holds for the
sparsity of factors of sparse polynomials
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