4,172 research outputs found
Arithmetic Circuit Lower Bounds via MaxRank
We introduce the polynomial coefficient matrix and identify maximum rank of
this matrix under variable substitution as a complexity measure for
multivariate polynomials. We use our techniques to prove super-polynomial lower
bounds against several classes of non-multilinear arithmetic circuits. In
particular, we obtain the following results :
As our main result, we prove that any homogeneous depth-3 circuit for
computing the product of matrices of dimension requires
size. This improves the lower bounds by Nisan and
Wigderson(1995) when .
There is an explicit polynomial on variables and degree at most
for which any depth-3 circuit of product dimension at most
(dimension of the space of affine forms feeding into each
product gate) requires size . This generalizes the lower bounds
against diagonal circuits proved by Saxena(2007). Diagonal circuits are of
product dimension 1.
We prove a lower bound on the size of product-sparse
formulas. By definition, any multilinear formula is a product-sparse formula.
Thus, our result extends the known super-polynomial lower bounds on the size of
multilinear formulas by Raz(2006).
We prove a lower bound on the size of partitioned arithmetic
branching programs. This result extends the known exponential lower bound on
the size of ordered arithmetic branching programs given by Jansen(2008).Comment: 22 page
Parallel Repetition of Entangled Games with Exponential Decay via the Superposed Information Cost
In a two-player game, two cooperating but non communicating players, Alice
and Bob, receive inputs taken from a probability distribution. Each of them
produces an output and they win the game if they satisfy some predicate on
their inputs/outputs. The entangled value of a game is the
maximum probability that Alice and Bob can win the game if they are allowed to
share an entangled state prior to receiving their inputs.
The -fold parallel repetition of consists of instances of
where the players receive all the inputs at the same time and produce all
the outputs at the same time. They win if they win each instance of .
In this paper we show that for any game such that , decreases exponentially in . First, for
any game on the uniform distribution, we show that , where and are the sizes of the input
and output sets. From this result, we show that for any entangled game ,
where is the input distribution of and
. This implies parallel
repetition with exponential decay as long as for
general games. To prove this parallel repetition, we introduce the concept of
\emph{Superposed Information Cost} for entangled games which is inspired from
the information cost used in communication complexity.Comment: In the first version of this paper we presented a different, stronger
Corollary 1 but due to an error in the proof we had to modify it in the
second version. This third version is a minor update. We correct some typos
and re-introduce a proof accidentally commented out in the second versio
Quantum Networks for Concentrating Entanglement
If two parties, Alice and Bob, share some number, n, of partially entangled
pairs of qubits, then it is possible for them to concentrate these pairs into
some smaller number of maximally entangled states. We present a simplified
version of the algorithm for such entanglement concentration, and we describe
efficient networks for implementing these operations.Comment: 15 pages, 2 figure
Superpolynomial lower bounds for general homogeneous depth 4 arithmetic circuits
In this paper, we prove superpolynomial lower bounds for the class of
homogeneous depth 4 arithmetic circuits. We give an explicit polynomial in VNP
of degree in variables such that any homogeneous depth 4 arithmetic
circuit computing it must have size .
Our results extend the works of Nisan-Wigderson [NW95] (which showed
superpolynomial lower bounds for homogeneous depth 3 circuits),
Gupta-Kamath-Kayal-Saptharishi and Kayal-Saha-Saptharishi [GKKS13, KSS13]
(which showed superpolynomial lower bounds for homogeneous depth 4 circuits
with bounded bottom fan-in), Kumar-Saraf [KS13a] (which showed superpolynomial
lower bounds for homogeneous depth 4 circuits with bounded top fan-in) and
Raz-Yehudayoff and Fournier-Limaye-Malod-Srinivasan [RY08, FLMS13] (which
showed superpolynomial lower bounds for multilinear depth 4 circuits). Several
of these results in fact showed exponential lower bounds.
The main ingredient in our proof is a new complexity measure of {\it bounded
support} shifted partial derivatives. This measure allows us to prove
exponential lower bounds for homogeneous depth 4 circuits where all the
monomials computed at the bottom layer have {\it bounded support} (but possibly
unbounded degree/fan-in), strengthening the results of Gupta et al and Kayal et
al [GKKS13, KSS13]. This new lower bound combined with a careful "random
restriction" procedure (that transforms general depth 4 homogeneous circuits to
depth 4 circuits with bounded support) gives us our final result
The Spatial Self: Location-Based Identity Performance on Social Media
This is the author's final manuscript. Copyright 2014 SAGE PublicationsAs a growing number of social media platforms now include location information from their users, researchers are confronted with new online representations of individuals, social networks, and the places they inhabit. To better understand these representations and their implications, we introduce the concept of the âspatial selfâ: a theoretical framework encapsulating the process of online self-presentation based on the display of offline physical activities. Building on previous studies in social science, humanities, and computer and information science, we analyze the ways offline experiences are harnessed and performed online. We first provide an encompassing interdisciplinary survey of research that investigates the relationships between location, information technology, and identity performance. Then, we identify and characterize the spatial self as well as examine its occurrences through three case studies of popular social media sites: Instagram, Facebook, and Foursquare. Finally, we offer possible research directions and methodological considerations for the analysis of geocoded social media data
Monomial Testing and Applications
In this paper, we devise two algorithms for the problem of testing
-monomials of degree in any multivariate polynomial represented by a
circuit, regardless of the primality of . One is an time
randomized algorithm. The other is an time deterministic
algorithm for the same -monomial testing problem but requiring the
polynomials to be represented by tree-like circuits. Several applications of
-monomial testing are also given, including a deterministic
upper bound for the -set -packing problem.Comment: 17 pages, 4 figures, submitted FAW-AAIM 2013. arXiv admin note:
substantial text overlap with arXiv:1302.5898; and text overlap with
arXiv:1007.2675, arXiv:1007.2678, arXiv:1007.2673 by other author
Wide bandwidth LFM transmission through a wavelength-controlled photonic true time delay device
We demonstrate for the first time high quality wide bandwidth (600 MHz) LFM transmission through a wavelength-controlled photonic true time delay device with bandwidth-limited resolution and peak sidelobe level below -37 dB at 5 GHz
Counting dependent and independent strings
The paper gives estimations for the sizes of the the following sets: (1) the
set of strings that have a given dependency with a fixed string, (2) the set of
strings that are pairwise \alpha independent, (3) the set of strings that are
mutually \alpha independent. The relevant definitions are as follows: C(x) is
the Kolmogorov complexity of the string x. A string y has \alpha -dependency
with a string x if C(y) - C(y|x) \geq \alpha. A set of strings {x_1, \ldots,
x_t} is pairwise \alpha-independent if for all i different from j, C(x_i) -
C(x_i | x_j) \leq \alpha. A tuple of strings (x_1, \ldots, x_t) is mutually
\alpha-independent if C(x_{\pi(1)} \ldots x_{\pi(t)}) \geq C(x_1) + \ldots +
C(x_t) - \alpha, for every permutation \pi of [t]
Simple extractors via constructions of cryptographic pseudo-random generators
Trevisan has shown that constructions of pseudo-random generators from hard
functions (the Nisan-Wigderson approach) also produce extractors. We show that
constructions of pseudo-random generators from one-way permutations (the
Blum-Micali-Yao approach) can be used for building extractors as well. Using
this new technique we build extractors that do not use designs and
polynomial-based error-correcting codes and that are very simple and efficient.
For example, one extractor produces each output bit separately in
time. These extractors work for weak sources with min entropy , for
arbitrary constant , have seed length , and their
output length is .Comment: 21 pages, an extended abstract will appear in Proc. ICALP 2005; small
corrections, some comments and references adde
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