47,841 research outputs found
Estimating operator norms using covering nets
We present several polynomial- and quasipolynomial-time approximation schemes
for a large class of generalized operator norms. Special cases include the
norm of matrices for , the support function of the set of
separable quantum states, finding the least noisy output of
entanglement-breaking quantum channels, and approximating the injective tensor
norm for a map between two Banach spaces whose factorization norm through
is bounded.
These reproduce and in some cases improve upon the performance of previous
algorithms by Brand\~ao-Christandl-Yard and followup work, which were based on
the Sum-of-Squares hierarchy and whose analysis used techniques from quantum
information such as the monogamy principle of entanglement. Our algorithms, by
contrast, are based on brute force enumeration over carefully chosen covering
nets. These have the advantage of using less memory, having much simpler proofs
and giving new geometric insights into the problem. Net-based algorithms for
similar problems were also presented by Shi-Wu and Barak-Kelner-Steurer, but in
each case with a run-time that is exponential in the rank of some matrix. We
achieve polynomial or quasipolynomial runtimes by using the much smaller nets
that exist in spaces. This principle has been used in learning theory,
where it is known as Maurey's empirical method.Comment: 24 page
Reproducing Kernels of Generalized Sobolev Spaces via a Green Function Approach with Distributional Operators
In this paper we introduce a generalized Sobolev space by defining a
semi-inner product formulated in terms of a vector distributional operator
consisting of finitely or countably many distributional operators
, which are defined on the dual space of the Schwartz space. The types of
operators we consider include not only differential operators, but also more
general distributional operators such as pseudo-differential operators. We
deduce that a certain appropriate full-space Green function with respect to
now becomes a conditionally positive
definite function. In order to support this claim we ensure that the
distributional adjoint operator of is
well-defined in the distributional sense. Under sufficient conditions, the
native space (reproducing-kernel Hilbert space) associated with the Green
function can be isometrically embedded into or even be isometrically
equivalent to a generalized Sobolev space. As an application, we take linear
combinations of translates of the Green function with possibly added polynomial
terms and construct a multivariate minimum-norm interpolant to data
values sampled from an unknown generalized Sobolev function at data sites
located in some set . We provide several examples, such
as Mat\'ern kernels or Gaussian kernels, that illustrate how many
reproducing-kernel Hilbert spaces of well-known reproducing kernels are
isometrically equivalent to a generalized Sobolev space. These examples further
illustrate how we can rescale the Sobolev spaces by the vector distributional
operator . Introducing the notion of scale as part of the
definition of a generalized Sobolev space may help us to choose the "best"
kernel function for kernel-based approximation methods.Comment: Update version of the publish at Num. Math. closed to Qi Ye's Ph.D.
thesis (\url{http://mypages.iit.edu/~qye3/PhdThesis-2012-AMS-QiYe-IIT.pdf}
MapReduce and Streaming Algorithms for Diversity Maximization in Metric Spaces of Bounded Doubling Dimension
Given a dataset of points in a metric space and an integer , a diversity
maximization problem requires determining a subset of points maximizing
some diversity objective measure, e.g., the minimum or the average distance
between two points in the subset. Diversity maximization is computationally
hard, hence only approximate solutions can be hoped for. Although its
applications are mainly in massive data analysis, most of the past research on
diversity maximization focused on the sequential setting. In this work we
present space and pass/round-efficient diversity maximization algorithms for
the Streaming and MapReduce models and analyze their approximation guarantees
for the relevant class of metric spaces of bounded doubling dimension. Like
other approaches in the literature, our algorithms rely on the determination of
high-quality core-sets, i.e., (much) smaller subsets of the input which contain
good approximations to the optimal solution for the whole input. For a variety
of diversity objective functions, our algorithms attain an
-approximation ratio, for any constant , where
is the best approximation ratio achieved by a polynomial-time,
linear-space sequential algorithm for the same diversity objective. This
improves substantially over the approximation ratios attainable in Streaming
and MapReduce by state-of-the-art algorithms for general metric spaces. We
provide extensive experimental evidence of the effectiveness of our algorithms
on both real world and synthetic datasets, scaling up to over a billion points.Comment: Extended version of
http://www.vldb.org/pvldb/vol10/p469-ceccarello.pdf, PVLDB Volume 10, No. 5,
January 201
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