35,236 research outputs found
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
Existence of Local Covariant Time Ordered Products of Quantum Fields in Curved Spacetime
We establish the existence of local, covariant time ordered products of local
Wick polynomials for a free scalar field in curved spacetime. Our time ordered
products satisfy all of the hypotheses of our previous uniqueness theorem, so
our construction essentially completes the analysis of the existence,
uniqueness and renormalizability of the perturbative expansion for nonlinear
quantum field theories in curved spacetime. As a byproduct of our analysis, we
derive a scaling expansion of the time ordered products about the total
diagonal that expresses them as a sum of products of polynomials in the
curvature times Lorentz invariant distributions, plus a remainder term of
arbitrary low scaling degree.Comment: Minor revisions, 2 references added. 43 pages, Latex format, no
figure
On the Limits of Depth Reduction at Depth 3 Over Small Finite Fields
Recently, Gupta et.al. [GKKS2013] proved that over Q any -variate
and -degree polynomial in VP can also be computed by a depth three
circuit of size . Over fixed-size
finite fields, Grigoriev and Karpinski proved that any
circuit that computes (or ) must be of size
[GK1998]. In this paper, we prove that over fixed-size finite fields, any
circuit for computing the iterated matrix multiplication
polynomial of generic matrices of size , must be of size
. The importance of this result is that over fixed-size
fields there is no depth reduction technique that can be used to compute all
the -variate and -degree polynomials in VP by depth 3 circuits of
size . The result [GK1998] can only rule out such a possibility
for depth 3 circuits of size .
We also give an example of an explicit polynomial () in
VNP (not known to be in VP), for which any circuit computing
it (over fixed-size fields) must be of size . The
polynomial we consider is constructed from the combinatorial design. An
interesting feature of this result is that we get the first examples of two
polynomials (one in VP and one in VNP) such that they have provably stronger
circuit size lower bounds than Permanent in a reasonably strong model of
computation.
Next, we prove that any depth 4
circuit computing
(over any field) must be of size . To the best of our knowledge, the polynomial is the
first example of an explicit polynomial in VNP such that it requires
size depth four circuits, but no known matching
upper bound
Algebraic statistical models
Many statistical models are algebraic in that they are defined in terms of
polynomial constraints, or in terms of polynomial or rational parametrizations.
The parameter spaces of such models are typically semi-algebraic subsets of the
parameter space of a reference model with nice properties, such as for example
a regular exponential family. This observation leads to the definition of an
`algebraic exponential family'. This new definition provides a unified
framework for the study of statistical models with algebraic structure. In this
paper we review the ingredients to this definition and illustrate in examples
how computational algebraic geometry can be used to solve problems arising in
statistical inference in algebraic models
Functional lower bounds for arithmetic circuits and connections to boolean circuit complexity
We say that a circuit over a field functionally computes an
-variate polynomial if for every we have that . This is in contrast to syntactically computing , when as
formal polynomials. In this paper, we study the question of proving lower
bounds for homogeneous depth- and depth- arithmetic circuits for
functional computation. We prove the following results :
1. Exponential lower bounds homogeneous depth- arithmetic circuits for a
polynomial in .
2. Exponential lower bounds for homogeneous depth- arithmetic circuits
with bounded individual degree for a polynomial in .
Our main motivation for this line of research comes from our observation that
strong enough functional lower bounds for even very special depth-
arithmetic circuits for the Permanent imply a separation between and
. Thus, improving the second result to get rid of the bounded individual
degree condition could lead to substantial progress in boolean circuit
complexity. Besides, it is known from a recent result of Kumar and Saptharishi
[KS15] that over constant sized finite fields, strong enough average case
functional lower bounds for homogeneous depth- circuits imply
superpolynomial lower bounds for homogeneous depth- circuits.
Our proofs are based on a family of new complexity measures called shifted
evaluation dimension, and might be of independent interest
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