203 research outputs found
Information recovery in the Hayden-Preskill protocol
We revisit information retrieval from evaporating black holes in the
Hayden-Preskill protocol, treating the black hole dynamics as Haar-random. We
compute, down to the first exponentially suppressed terms, all integer-indexed
R\'enyi mutual informations between a black hole, its radiation, and a
reference that catalogues Alice's diaries. We find that dropping a diary into a
young black hole effectively delays the Page time. We also compute the
radiation : diary reflected R\'enyi entropies, and identify a technical reason
why they cannot be continued to the reflected entropy by the replica trick.Comment: 24 pages plus appendice
Scalable Hash-Based Estimation of Divergence Measures
We propose a scalable divergence estimation method based on hashing. Consider
two continuous random variables and whose densities have bounded
support. We consider a particular locality sensitive random hashing, and
consider the ratio of samples in each hash bin having non-zero numbers of Y
samples. We prove that the weighted average of these ratios over all of the
hash bins converges to f-divergences between the two samples sets. We show that
the proposed estimator is optimal in terms of both MSE rate and computational
complexity. We derive the MSE rates for two families of smooth functions; the
H\"{o}lder smoothness class and differentiable functions. In particular, it is
proved that if the density functions have bounded derivatives up to the order
, where is the dimension of samples, the optimal parametric MSE rate
of can be achieved. The computational complexity is shown to be
, which is optimal. To the best of our knowledge, this is the first
empirical divergence estimator that has optimal computational complexity and
achieves the optimal parametric MSE estimation rate.Comment: 11 pages, Proceedings of the 21st International Conference on
Artificial Intelligence and Statistics (AISTATS) 2018, Lanzarote, Spai
Epidemics on contact networks: a general stochastic approach
Dynamics on networks is considered from the perspective of Markov stochastic
processes. We partially describe the state of the system through network motifs
and infer any missing data using the available information. This versatile
approach is especially well adapted for modelling spreading processes and/or
population dynamics. In particular, the generality of our systematic framework
and the fact that its assumptions are explicitly stated suggests that it could
be used as a common ground for comparing existing epidemics models too complex
for direct comparison, such as agent-based computer simulations. We provide
many examples for the special cases of susceptible-infectious-susceptible (SIS)
and susceptible-infectious-removed (SIR) dynamics (e.g., epidemics propagation)
and we observe multiple situations where accurate results may be obtained at
low computational cost. Our perspective reveals a subtle balance between the
complex requirements of a realistic model and its basic assumptions.Comment: Main document: 16 pages, 7 figures. Electronic Supplementary Material
(included): 6 pages, 1 tabl
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