60,783 research outputs found
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
Symbolic Algorithms for Language Equivalence and Kleene Algebra with Tests
We first propose algorithms for checking language equivalence of finite
automata over a large alphabet. We use symbolic automata, where the transition
function is compactly represented using a (multi-terminal) binary decision
diagrams (BDD). The key idea consists in computing a bisimulation by exploring
reachable pairs symbolically, so as to avoid redundancies. This idea can be
combined with already existing optimisations, and we show in particular a nice
integration with the disjoint sets forest data-structure from Hopcroft and
Karp's standard algorithm. Then we consider Kleene algebra with tests (KAT), an
algebraic theory that can be used for verification in various domains ranging
from compiler optimisation to network programming analysis. This theory is
decidable by reduction to language equivalence of automata on guarded strings,
a particular kind of automata that have exponentially large alphabets. We
propose several methods allowing to construct symbolic automata out of KAT
expressions, based either on Brzozowski's derivatives or standard automata
constructions. All in all, this results in efficient algorithms for deciding
equivalence of KAT expressions
Transition probability of Brownian motion in the octant and its application to default modeling
We derive a semi-analytic formula for the transition probability of
three-dimensional Brownian motion in the positive octant with absorption at the
boundaries. Separation of variables in spherical coordinates leads to an
eigenvalue problem for the resulting boundary value problem in the two angular
components. The main theoretical result is a solution to the original problem
expressed as an expansion into special functions and an eigenvalue which has to
be chosen to allow a matching of the boundary condition. We discuss and test
several computational methods to solve a finite-dimensional approximation to
this nonlinear eigenvalue problem. Finally, we apply our results to the
computation of default probabilities and credit valuation adjustments in a
structural credit model with mutual liabilities
Formal Verification of Nonlinear Inequalities with Taylor Interval Approximations
We present a formal tool for verification of multivariate nonlinear
inequalities. Our verification method is based on interval arithmetic with
Taylor approximations. Our tool is implemented in the HOL Light proof assistant
and it is capable to verify multivariate nonlinear polynomial and
non-polynomial inequalities on rectangular domains. One of the main features of
our work is an efficient implementation of the verification procedure which can
prove non-trivial high-dimensional inequalities in several seconds. We
developed the verification tool as a part of the Flyspeck project (a formal
proof of the Kepler conjecture). The Flyspeck project includes about 1000
nonlinear inequalities. We successfully tested our method on more than 100
Flyspeck inequalities and estimated that the formal verification procedure is
about 3000 times slower than an informal verification method implemented in
C++. We also describe future work and prospective optimizations for our method.Comment: 15 page
On Gaussian Random Supergravity
We study the distribution of metastable vacua and the likelihood of slow roll
inflation in high dimensional random landscapes. We consider two examples of
landscapes: a Gaussian random potential and an effective supergravity potential
defined via a Gaussian random superpotential and a trivial K\"ahler potential.
To examine these landscapes we introduce a random matrix model that describes
the correlations between various derivatives and we propose an efficient
algorithm that allows for a numerical study of high dimensional random fields.
Using these novel tools, we find that the vast majority of metastable critical
points in dimensional random supergravities are either approximately
supersymmetric with or supersymmetric. Such
approximately supersymmetric points are dynamical attractors in the landscape
and the probability that a randomly chosen critical point is metastable scales
as . We argue that random supergravities lead to potentially
interesting inflationary dynamics.Comment: 36 pages, 9 figure
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