2,323 research outputs found
Communication Lower Bounds for Statistical Estimation Problems via a Distributed Data Processing Inequality
We study the tradeoff between the statistical error and communication cost of
distributed statistical estimation problems in high dimensions. In the
distributed sparse Gaussian mean estimation problem, each of the machines
receives data points from a -dimensional Gaussian distribution with
unknown mean which is promised to be -sparse. The machines
communicate by message passing and aim to estimate the mean . We
provide a tight (up to logarithmic factors) tradeoff between the estimation
error and the number of bits communicated between the machines. This directly
leads to a lower bound for the distributed \textit{sparse linear regression}
problem: to achieve the statistical minimax error, the total communication is
at least , where is the number of observations that
each machine receives and is the ambient dimension. These lower results
improve upon [Sha14,SD'14] by allowing multi-round iterative communication
model. We also give the first optimal simultaneous protocol in the dense case
for mean estimation.
As our main technique, we prove a \textit{distributed data processing
inequality}, as a generalization of usual data processing inequalities, which
might be of independent interest and useful for other problems.Comment: To appear at STOC 2016. Fixed typos in theorem 4.5 and incorporated
reviewers' suggestion
A finite analog of the AGT relation I: finite W-algebras and quasimaps' spaces
Recently Alday, Gaiotto and Tachikawa proposed a conjecture relating
4-dimensional super-symmetric gauge theory for a gauge group G with certain
2-dimensional conformal field theory. This conjecture implies the existence of
certain structures on the (equivariant) intersection cohomology of the
Uhlenbeck partial compactification of the moduli space of framed G-bundles on
P^2. More precisely, it predicts the existence of an action of the
corresponding W-algebra on the above cohomology, satisfying certain properties.
We propose a "finite analog" of the (above corollary of the) AGT conjecture.
Namely, we replace the Uhlenbeck space with the space of based quasi-maps from
P^1 to any partial flag variety G/P of G and conjecture that its equivariant
intersection cohomology carries an action of the finite W-algebra U(g,e)
associated with the principal nilpotent element in the Lie algebra of the Levi
subgroup of P; this action is expected to satisfy some list of natural
properties. This conjecture generalizes the main result of arXiv:math/0401409
when P is the Borel subgroup. We prove our conjecture for G=GL(N), using the
works of Brundan and Kleshchev interpreting the algebra U(g,e) in terms of
certain shifted Yangians.Comment: minor change
Tannakian categories, linear differential algebraic groups, and parameterized linear differential equations
We provide conditions for a category with a fiber functor to be equivalent to
the category of representations of a linear differential algebraic group. This
generalizes the notion of a neutral Tannakian category used to characterize the
category of representations of a linear algebraic group.Comment: 26 pages; corrected misprints; simplified Definition 2; more
references adde
Surface Operators in N=2 Abelian Gauge Theory
We generalise the analysis in [arXiv:0904.1744] to superspace, and explicitly
prove that for any embedding of surface operators in a general, twisted N=2
pure abelian theory on an arbitrary four-manifold, the parameters transform
naturally under the SL(2,Z) duality of the theory. However, for
nontrivially-embedded surface operators, exact S-duality holds if and only if
the "quantum" parameter effectively vanishes, while the overall SL(2,Z) duality
holds up to a c-number at most, regardless. Nevertheless, this observation sets
the stage for a physical proof of a remarkable mathematical result by
Kronheimer and Mrowka--that expresses a "ramified" analog of the Donaldson
invariants solely in terms of the ordinary Donaldson invariants--which, will
appear, among other things, in forthcoming work. As a prelude to that, the
effective interaction on the corresponding u-plane will be computed. In
addition, the dependence on second Stiefel-Whitney classes and the appearance
of a Spin^c structure in the associated low-energy Seiberg-Witten theory with
surface operators, will also be demonstrated. In the process, we will stumble
upon an interesting phase factor that is otherwise absent in the "unramified"
case.Comment: 46 pages. Minor refinemen
Costs and benefits of agricultural price stabilization in Brazil
In recent years, agricultural price stabilization policies have been recommended in Brazil as a way to reduce government intervention and open the sector for international trade without internalizing the instability of world prices. The proposal discussed (and eventually implemented in 1987) was to establish a system of price bands around a moving average of past prices, with the government relying on stocks to defend the bands. The authors evaluated the"band proposal"for six commodities, using historical data and posing this question: what would have happened if price bands had been adopted in the past six to ten years (compared with free trade)? There were two major findings. First, the implications of adopting a band-rule policy depend heavily on the specific characteristics of the commodities. Second, the welfare gains for risk reduction through agricultural price stabilization are unlikely to be large relative to the welfare gains from price reform that reduces market distortions for these six agricultural commodities. More research into the macroeconomic implications of price stabilization policies is necessary, particularly in countries with unstable but moderate rates of inflation.Environmental Economics&Policies,Economic Theory&Research,Markets and Market Access,Access to Markets,Insurance&Risk Mitigation
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