528 research outputs found
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
Eroding dipoles and vorticity growth for Euler flows in R 3 : Axisymmetric flow without swirl
A review of analyses based upon anti-parallel vortex structures suggests that structurally stable
dipoles with eroding circulation may offer a path to the study of vorticity growth in solutions of
Euler’s equations in R3
. We examine here the possible formation of such a structure in axisymmetric
flow without swirl, leading to maximal growth of vorticity as t
4/3
. Our study suggests that the
optimizing flow giving the t
4/3 growth mimics an exact solution of Euler’s equations representing
an eroding toroidal vortex dipole which locally conserves kinetic energy. The dipole cross-section
is a perturbation of the classical Sadovskii dipole having piecewise constant vorticity, which
breaks the symmetry of closed streamlines. The structure of this perturbed Sadovskii dipole is
analyzed asymptotically at large times, and its predicted properties are verified numerically. We
also show numerically that if mirror symmetry of the dipole is not imposed but axial symmetry
maintained, an instability leads to breakup into smaller vortical structures
Bandit Online Optimization Over the Permutahedron
The permutahedron is the convex polytope with vertex set consisting of the
vectors for all permutations (bijections) over
. We study a bandit game in which, at each step , an
adversary chooses a hidden weight weight vector , a player chooses a
vertex of the permutahedron and suffers an observed loss of
.
A previous algorithm CombBand of Cesa-Bianchi et al (2009) guarantees a
regret of for a time horizon of . Unfortunately,
CombBand requires at each step an -by- matrix permanent approximation to
within improved accuracy as grows, resulting in a total running time that
is super linear in , making it impractical for large time horizons.
We provide an algorithm of regret with total time
complexity . The ideas are a combination of CombBand and a recent
algorithm by Ailon (2013) for online optimization over the permutahedron in the
full information setting. The technical core is a bound on the variance of the
Plackett-Luce noisy sorting process's "pseudo loss". The bound is obtained by
establishing positive semi-definiteness of a family of 3-by-3 matrices
generated from rational functions of exponentials of 3 parameters
Stochastic Streams: Sample Complexity vs. Space Complexity
We address the trade-off between the computational resources needed to process a large data set and the number of samples available from the data set. Specifically, we consider the following abstraction: we receive a potentially infinite stream of IID samples from some unknown distribution D, and are tasked with computing some function f(D). If the stream is observed for time t, how much memory, s, is required to estimate f(D)? We refer to t as the sample complexity and s as the space complexity. The main focus of this paper is investigating the trade-offs between the space and sample complexity. We study these trade-offs for several canonical problems studied in the data stream model: estimating the collision probability, i.e., the second moment of a distribution, deciding if a graph is connected, and approximating the dimension of an unknown subspace. Our results are based on techniques for simulating different classical sampling procedures in this model, emulating random walks given a sequence of IID samples, as well as leveraging a characterization between communication bounded protocols and statistical query algorithms
Efficient solvability of Hamiltonians and limits on the power of some quantum computational models
We consider quantum computational models defined via a Lie-algebraic theory.
In these models, specified initial states are acted on by Lie-algebraic quantum
gates and the expectation values of Lie algebra elements are measured at the
end. We show that these models can be efficiently simulated on a classical
computer in time polynomial in the dimension of the algebra, regardless of the
dimension of the Hilbert space where the algebra acts. Similar results hold for
the computation of the expectation value of operators implemented by a
gate-sequence. We introduce a Lie-algebraic notion of generalized mean-field
Hamiltonians and show that they are efficiently ("exactly") solvable by means
of a Jacobi-like diagonalization method. Our results generalize earlier ones on
fermionic linear optics computation and provide insight into the source of the
power of the conventional model of quantum computation.Comment: 6 pages; no figure
Static Data Structure Lower Bounds Imply Rigidity
We show that static data structure lower bounds in the group (linear) model
imply semi-explicit lower bounds on matrix rigidity. In particular, we prove
that an explicit lower bound of on the cell-probe
complexity of linear data structures in the group model, even against
arbitrarily small linear space , would already imply a
semi-explicit () construction of rigid matrices with
significantly better parameters than the current state of art (Alon, Panigrahy
and Yekhanin, 2009). Our results further assert that polynomial () data structure lower bounds against near-optimal space, would
imply super-linear circuit lower bounds for log-depth linear circuits (a
four-decade open question). In the succinct space regime , we show
that any improvement on current cell-probe lower bounds in the linear model
would also imply new rigidity bounds. Our results rely on a new connection
between the "inner" and "outer" dimensions of a matrix (Paturi and Pudlak,
2006), and on a new reduction from worst-case to average-case rigidity, which
is of independent interest
Fast Discovery of Reliable k-terminal Subgraphs
Peer reviewe
Improved bounds for reduction to depth 4 and depth 3
Koiran showed that if a -variate polynomial of degree (with
) is computed by a circuit of size , then it is also computed by
a homogeneous circuit of depth four and of size
. Using this result, Gupta, Kamath, Kayal and
Saptharishi gave a upper bound for the
size of the smallest depth three circuit computing a -variate polynomial of
degree given by a circuit of size .
We improve here Koiran's bound. Indeed, we show that if we reduce an
arithmetic circuit to depth four, then the size becomes
. Mimicking Gupta, Kamath, Kayal and
Saptharishi's proof, it also implies the same upper bound for depth three
circuits.
This new bound is not far from optimal in the sense that Gupta, Kamath, Kayal
and Saptharishi also showed a lower bound for the size
of homogeneous depth four circuits such that gates at the bottom have fan-in at
most . Finally, we show that this last lower bound also holds if the
fan-in is at least
Classical simulation of noninteracting-fermion quantum circuits
We show that a class of quantum computations that was recently shown to be
efficiently simulatable on a classical computer by Valiant corresponds to a
physical model of noninteracting fermions in one dimension. We give an
alternative proof of his result using the language of fermions and extend the
result to noninteracting fermions with arbitrary pairwise interactions, where
gates can be conditioned on outcomes of complete von Neumann measurements in
the computational basis on other fermionic modes in the circuit. This last
result is in remarkable contrast with the case of noninteracting bosons where
universal quantum computation can be achieved by allowing gates to be
conditioned on classical bits (quant-ph/0006088).Comment: 26 pages, 1 figure, uses wick.sty; references added to recent results
by E. Knil
- …