8,037 research outputs found
Martingale Problem under Nonlinear Expectations
We formulate and solve the martingale problem in a nonlinear expectation
space. Unlike the classical work of Stroock and Varadhan (1969) where the
linear operator in the associated PDE is naturally defined from the
corresponding diffusion process, the main difficulty in the nonlinear setting
is to identify an appropriate class of nonlinear operators for the associated
fully nonlinear PDEs.
Based on the analysis of the martingale problem, we introduce the notion of
weak solution for stochastic differential equations under nonlinear
expectations and obtain an existence theorem under the H\"older continuity
condition of the coefficients. The approach to establish the existence of weak
solutions generalizes the classical Girsanov transformation method in that it
no longer requires the two (probability) measures to be absolutely continuous.Comment: The new version simplifies some proofs for the main theorems and
generalizes some result
Learning to Prune Deep Neural Networks via Layer-wise Optimal Brain Surgeon
How to develop slim and accurate deep neural networks has become crucial for
real- world applications, especially for those employed in embedded systems.
Though previous work along this research line has shown some promising results,
most existing methods either fail to significantly compress a well-trained deep
network or require a heavy retraining process for the pruned deep network to
re-boost its prediction performance. In this paper, we propose a new layer-wise
pruning method for deep neural networks. In our proposed method, parameters of
each individual layer are pruned independently based on second order
derivatives of a layer-wise error function with respect to the corresponding
parameters. We prove that the final prediction performance drop after pruning
is bounded by a linear combination of the reconstructed errors caused at each
layer. Therefore, there is a guarantee that one only needs to perform a light
retraining process on the pruned network to resume its original prediction
performance. We conduct extensive experiments on benchmark datasets to
demonstrate the effectiveness of our pruning method compared with several
state-of-the-art baseline methods
The effects of massive graviton on the equilibrium between the black hole and radiation gas in an isolated box
It is well known that the black hole can has temperature and radiate the
particles with black body spectrum, i.e. Hawking radiation. Therefore, if the
black hole is surrounded by an isolated box, there is a thermal equilibrium
between the black hole and radiation gas. A simple case considering the thermal
equilibrium between the Schwarzschild black hole and radiation gas in an
isolated box has been well investigated previously in detail, i.e. taking the
conservation of energy and principle of maximal entropy for the isolated system
into account. In this paper, following the above spirit, the effects of massive
graviton on the thermal equilibrium will be investigated. For the gravity with
massive graviton, we will use the de Rham-Gabadadze-Tolley (dRGT) massive
gravity which has been proven to be ghost free. Because the graviton mass
depends on two parameters in the dRGT massive gravity, here we just investigate
two simple cases related to the two parameters, respectively. Our results show
that in the first case the massive graviton can suppress or increase the
condensation of black hole in the radiation gas although the diagram is
similar like the Schwarzschild black hole case. For the second case, a new
diagram has been obtained. Moreover, an interesting and important
prediction is that the condensation of black hole just increases from the zero
radius of horizon in this case, which is very different from the Schwarzschild
black hole case.Comment: 9 pages, 4 figure
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