2,280 research outputs found
The Paulsen Problem, Continuous Operator Scaling, and Smoothed Analysis
The Paulsen problem is a basic open problem in operator theory: Given vectors
that are -nearly satisfying the
Parseval's condition and the equal norm condition, is it close to a set of
vectors that exactly satisfy the Parseval's
condition and the equal norm condition? Given , the squared
distance (to the set of exact solutions) is defined as where the infimum is over the set of exact solutions.
Previous results show that the squared distance of any -nearly
solution is at most and there are
-nearly solutions with squared distance at least .
The fundamental open question is whether the squared distance can be
independent of the number of vectors .
We answer this question affirmatively by proving that the squared distance of
any -nearly solution is . Our approach is based
on a continuous version of the operator scaling algorithm and consists of two
parts. First, we define a dynamical system based on operator scaling and use it
to prove that the squared distance of any -nearly solution is . Then, we show that by randomly perturbing the input vectors, the
dynamical system will converge faster and the squared distance of an
-nearly solution is when is large enough
and is small enough. To analyze the convergence of the dynamical
system, we develop some new techniques in lower bounding the operator capacity,
a concept introduced by Gurvits to analyze the operator scaling algorithm.Comment: Added Subsection 1.4; Incorporated comments and fixed typos; Minor
changes in various place
Marginal Release Under Local Differential Privacy
Many analysis and machine learning tasks require the availability of marginal
statistics on multidimensional datasets while providing strong privacy
guarantees for the data subjects. Applications for these statistics range from
finding correlations in the data to fitting sophisticated prediction models. In
this paper, we provide a set of algorithms for materializing marginal
statistics under the strong model of local differential privacy. We prove the
first tight theoretical bounds on the accuracy of marginals compiled under each
approach, perform empirical evaluation to confirm these bounds, and evaluate
them for tasks such as modeling and correlation testing. Our results show that
releasing information based on (local) Fourier transformations of the input is
preferable to alternatives based directly on (local) marginals
Foundation and empire : a critique of Hardt and Negri
In this article, Thompson complements recent critiques of Hardt and Negri's Empire (see Finn Bowring in Capital and Class, no. 83) using the tools of labour process theory to critique the political economy of Empire, and to note its unfortunate similarities to conventional theories of the knowledge economy
Global Solution to the Three-Dimensional Incompressible Flow of Liquid Crystals
The equations for the three-dimensional incompressible flow of liquid
crystals are considered in a smooth bounded domain. The existence and
uniqueness of the global strong solution with small initial data are
established. It is also proved that when the strong solution exists, all the
global weak solutions constructed in [16] must be equal to the unique strong
solution
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