1,521 research outputs found
A chain rule for the expected suprema of Gaussian processes
The expected supremum of a Gaussian process indexed by the image of an index
set under a function class is bounded in terms of separate properties of the
index set and the function class. The bound is relevant to the estimation of
nonlinear transformations or the analysis of learning algorithms whenever
hypotheses are chosen from composite classes, as is the case for multi-layer
models
Maharam's problem
We construct an exhaustive submeasure that is not equivalent to a measure.
This solves problems of J. von Neumann (1937) and D. Maharam (1947)
Concentration of norms and eigenvalues of random matrices
We prove concentration results for operator norms of rectangular
random matrices and eigenvalues of self-adjoint random matrices. The random
matrices we consider have bounded entries which are independent, up to a
possible self-adjointness constraint. Our results are based on an isoperimetric
inequality for product spaces due to Talagrand.Comment: 15 pages; AMS-LaTeX; updated one referenc
Estimates on path delocalization for copolymers at selective interfaces
We consider a directed random walk model of a random heterogeneous polymer in
the proximity of an interface separating two selective solvents. This model
exhibits a localization/delocalization transition. A positive value of the free
energy corresponds to the localized regime and strong results on the polymer
path behavior are known in this case. We focus on the interior of the
delocalized phase, which is characterized by the free energy equal to zero, and
we show in particular that in this regime there are O(log N) monomers in the
unfavorable solvent (N is the length of the polymer). The previously known
result was o(N). Our approach is based on concentration bounds on suitably
restricted partition functions. The same idea allows also to interpolate
between different types of disorder in the weak coupling limit. In this way we
show the universal nature of this limit, previously considered only for binary
disorder.Comment: 17 pages, accepted for publication on Probab. Theory Rel. Field
A directed isoperimetric inequality with application to Bregman near neighbor lower bounds
Bregman divergences are a class of divergences parametrized by a
convex function and include well known distance functions like
and the Kullback-Leibler divergence. There has been extensive
research on algorithms for problems like clustering and near neighbor search
with respect to Bregman divergences, in all cases, the algorithms depend not
just on the data size and dimensionality , but also on a structure
constant that depends solely on and can grow without bound
independently.
In this paper, we provide the first evidence that this dependence on
might be intrinsic. We focus on the problem of approximate near neighbor search
for Bregman divergences. We show that under the cell probe model, any
non-adaptive data structure (like locality-sensitive hashing) for
-approximate near-neighbor search that admits probes must use space
. In contrast, for LSH under the best
bound is .
Our new tool is a directed variant of the standard boolean noise operator. We
show that a generalization of the Bonami-Beckner hypercontractivity inequality
exists "in expectation" or upon restriction to certain subsets of the Hamming
cube, and that this is sufficient to prove the desired isoperimetric inequality
that we use in our data structure lower bound.
We also present a structural result reducing the Hamming cube to a Bregman
cube. This structure allows us to obtain lower bounds for problems under
Bregman divergences from their analog. In particular, we get a
(weaker) lower bound for approximate near neighbor search of the form
for an -query non-adaptive data structure,
and new cell probe lower bounds for a number of other near neighbor questions
in Bregman space.Comment: 27 page
Typical entanglement of stabilizer states
How entangled is a randomly chosen bipartite stabilizer state? We show that
if the number of qubits each party holds is large the state will be close to
maximally entangled with probability exponentially close to one. We provide a
similar tight characterization of the entanglement present in the maximally
mixed state of a randomly chosen stabilizer code. Finally, we show that
typically very few GHZ states can be extracted from a random multipartite
stabilizer state via local unitary operations. Our main tool is a new
concentration inequality which bounds deviations from the mean of random
variables which are naturally defined on the Clifford group.Comment: Final version, to appear in PRA. 11 pages, 1 figur
Ultrametricity in the Edwards-Anderson Model
We test the property of ultrametricity for the spin glass three-dimensional
Edwards-Anderson model in zero magnetic field with numerical simulations up to
spins. We find an excellent agreement with the prediction of the mean
field theory. Since ultrametricity is not compatible with a trivial structure
of the overlap distribution our result contradicts the droplet theory.Comment: typos correcte
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