1,048 research outputs found
A generalization of Hausdorff dimension applied to Hilbert cubes and Wasserstein spaces
A Wasserstein spaces is a metric space of sufficiently concentrated
probability measures over a general metric space. The main goal of this paper
is to estimate the largeness of Wasserstein spaces, in a sense to be precised.
In a first part, we generalize the Hausdorff dimension by defining a family of
bi-Lipschitz invariants, called critical parameters, that measure largeness for
infinite-dimensional metric spaces. Basic properties of these invariants are
given, and they are estimated for a naturel set of spaces generalizing the
usual Hilbert cube. In a second part, we estimate the value of these new
invariants in the case of some Wasserstein spaces, as well as the dynamical
complexity of push-forward maps. The lower bounds rely on several embedding
results; for example we provide bi-Lipschitz embeddings of all powers of any
space inside its Wasserstein space, with uniform bound and we prove that the
Wasserstein space of a d-manifold has "power-exponential" critical parameter
equal to d.Comment: v2 Largely expanded version, as reflected by the change of title; all
part I on generalized Hausdorff dimension is new, as well as the embedding of
Hilbert cubes into Wasserstein spaces. v3 modified according to the referee
final remarks ; to appear in Journal of Topology and Analysi
Clustering as an example of optimizing arbitrarily chosen objective functions
This paper is a reflection upon a common practice of solving various types of learning problems by optimizing arbitrarily chosen criteria in the hope that they are well correlated with the criterion actually used for assessment of the results. This issue has been investigated using clustering as an example, hence a unified view of clustering as an optimization problem is first proposed, stemming from the belief that typical design choices in clustering, like the number of clusters or similarity measure can be, and often are suboptimal, also from the point of view of clustering quality measures later used for algorithm comparison and ranking. In order to illustrate our point we propose a generalized clustering framework and provide a proof-of-concept using standard benchmark datasets and two popular clustering methods for comparison
Ordered Measurements of Permutationally-Symmetric Qubit Strings
We show that any sequence of measurements on a permutationally-symmetric
(pure or mixed) multi-qubit string leaves the unmeasured qubit substring also
permutationally-symmetric. In addition, we show that the measurement
probabilities for an arbitrary sequence of single-qubit measurements are
independent of how many unmeasured qubits have been lost prior to the
measurement. Our results are valuable for quantum information processing of
indistinguishable particles by post-selection, e.g. in cases where the results
of an experiment are discarded conditioned upon the occurrence of a given event
such as particle loss. Furthermore, our results are important for the design of
adaptive-measurement strategies, e.g. a series of measurements where for each
measurement instance, the measurement basis is chosen depending on prior
measurement results.Comment: 13 page
Numerical Analysis of Boosting Scheme for Scalable NMR Quantum Computation
Among initialization schemes for ensemble quantum computation beginning at
thermal equilibrium, the scheme proposed by Schulman and Vazirani [L. J.
Schulman and U. V. Vazirani, in Proceedings of the 31st ACM Symposium on Theory
of Computing (STOC'99) (ACM Press, New York, 1999), pp. 322-329] is known for
the simple quantum circuit to redistribute the biases (polarizations) of qubits
and small time complexity. However, our numerical simulation shows that the
number of qubits initialized by the scheme is rather smaller than expected from
the von Neumann entropy because of an increase in the sum of the binary
entropies of individual qubits, which indicates a growth in the total classical
correlation. This result--namely, that there is such a significant growth in
the total binary entropy--disagrees with that of their analysis.Comment: 14 pages, 18 figures, RevTeX4, v2,v3: typos corrected, v4: minor
changes in PROGRAM 1, conforming it to the actual programs used in the
simulation, v5: correction of a typographical error in the inequality sign in
PROGRAM 1, v6: this version contains a new section on classical correlations,
v7: correction of a wrong use of terminology, v8: Appendix A has been added,
v9: published in PR
Failure mode and effect analysis in interhospital transport of critically ill patients: a tool to design checklists for handovers in the mobile intensive care unit
FDTD Simulation of Thermal Noise in Open Cavities
A numerical model based on the finite-difference time-domain (FDTD) method is
developed to simulate thermal noise in open cavities owing to output coupling.
The absorbing boundary of the FDTD grid is treated as a blackbody, whose
thermal radiation penetrates the cavity in the grid. The calculated amount of
thermal noise in a one-dimensional dielectric cavity recovers the standard
result of the quantum Langevin equation in the Markovian regime. Our FDTD
simulation also demonstrates that in the non-Markovian regime the buildup of
the intracavity noise field depends on the ratio of the cavity photon lifetime
to the coherence time of thermal radiation. The advantage of our numerical
method is that the thermal noise is introduced in the time domain without prior
knowledge of cavity modes.Comment: 8 pages, 7 figure
A Number-Theoretic Error-Correcting Code
In this paper we describe a new error-correcting code (ECC) inspired by the
Naccache-Stern cryptosystem. While by far less efficient than Turbo codes, the
proposed ECC happens to be more efficient than some established ECCs for
certain sets of parameters. The new ECC adds an appendix to the message. The
appendix is the modular product of small primes representing the message bits.
The receiver recomputes the product and detects transmission errors using
modular division and lattice reduction
Stacking Gravitational Wave Signals from Soft Gamma Repeater Bursts
Soft gamma repeaters (SGRs) have unique properties that make them intriguing
targets for gravitational wave (GW) searches. They are nearby, their burst
emission mechanism may involve neutron star crust fractures and excitation of
quasi-normal modes, and they burst repeatedly and sometimes spectacularly. A
recent LIGO search for transient GW from these sources placed upper limits on a
set of almost 200 individual SGR bursts. These limits were within the
theoretically predicted range of some models. We present a new search strategy
which builds upon the method used there by "stacking" potential GW signals from
multiple SGR bursts. We assume that variation in the time difference between
burst electromagnetic emission and burst GW emission is small relative to the
GW signal duration, and we time-align GW excess power time-frequency tilings
containing individual burst triggers to their corresponding electromagnetic
emissions. Using Monte Carlo simulations, we confirm that gains in GW energy
sensitivity of N^{1/2} are possible, where N is the number of stacked SGR
bursts. Estimated sensitivities for a mock search for gravitational waves from
the 2006 March 29 storm from SGR 1900+14 are also presented, for two GW
emission models, "fluence-weighted" and "flat" (unweighted).Comment: 17 pages, 16 figures, submitted to PR
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