367 research outputs found
Nonconventional Large Deviations Theorems
We obtain large deviations theorems for nonconventional sums with underlying
process being a Markov process satisfying the Doeblin condition or a dynamical
system such as subshift of finite type or hyperbolic or expanding
transformation
Resonances and Twist in Volume-Preserving Mappings
The phase space of an integrable, volume-preserving map with one action and
angles is foliated by a one-parameter family of -dimensional invariant
tori. Perturbations of such a system may lead to chaotic dynamics and
transport. We show that near a rank-one, resonant torus these mappings can be
reduced to volume-preserving "standard maps." These have twist only when the
image of the frequency map crosses the resonance curve transversely. We show
that these maps can be approximated---using averaging theory---by the usual
area-preserving twist or nontwist standard maps. The twist condition
appropriate for the volume-preserving setting is shown to be distinct from the
nondegeneracy condition used in (volume-preserving) KAM theory.Comment: Many typos fixed and notation simplified. New order
averaging theorem and volume-preserving variant. Numerical comparison with
averaging adde
Differentially Private Distributed Optimization
In distributed optimization and iterative consensus literature, a standard
problem is for agents to minimize a function over a subset of Euclidean
space, where the cost function is expressed as a sum . In this paper,
we study the private distributed optimization (PDOP) problem with the
additional requirement that the cost function of the individual agents should
remain differentially private. The adversary attempts to infer information
about the private cost functions from the messages that the agents exchange.
Achieving differential privacy requires that any change of an individual's cost
function only results in unsubstantial changes in the statistics of the
messages. We propose a class of iterative algorithms for solving PDOP, which
achieves differential privacy and convergence to the optimal value. Our
analysis reveals the dependence of the achieved accuracy and the privacy levels
on the the parameters of the algorithm. We observe that to achieve
-differential privacy the accuracy of the algorithm has the order of
Private Multiplicative Weights Beyond Linear Queries
A wide variety of fundamental data analyses in machine learning, such as
linear and logistic regression, require minimizing a convex function defined by
the data. Since the data may contain sensitive information about individuals,
and these analyses can leak that sensitive information, it is important to be
able to solve convex minimization in a privacy-preserving way.
A series of recent results show how to accurately solve a single convex
minimization problem in a differentially private manner. However, the same data
is often analyzed repeatedly, and little is known about solving multiple convex
minimization problems with differential privacy. For simpler data analyses,
such as linear queries, there are remarkable differentially private algorithms
such as the private multiplicative weights mechanism (Hardt and Rothblum, FOCS
2010) that accurately answer exponentially many distinct queries. In this work,
we extend these results to the case of convex minimization and show how to give
accurate and differentially private solutions to *exponentially many* convex
minimization problems on a sensitive dataset
Infinitely Many Stochastically Stable Attractors
Let f be a diffeomorphism of a compact finite dimensional boundaryless
manifold M exhibiting infinitely many coexisting attractors. Assume that each
attractor supports a stochastically stable probability measure and that the
union of the basins of attraction of each attractor covers Lebesgue almost all
points of M. We prove that the time averages of almost all orbits under random
perturbations are given by a finite number of probability measures. Moreover
these probability measures are close to the probability measures supported by
the attractors when the perturbations are close to the original map f.Comment: 14 pages, 2 figure
Fast-slow partially hyperbolic systems versus Freidlin-Wentzell random systems
We consider a simple class of fast-slow partially hyperbolic dynamical
systems and show that the (properly rescaled) behaviour of the slow variable is
very close to a Friedlin--Wentzell type random system for times that are rather
long, but much shorter than the metastability scale. Also, we show the
possibility of a "sink" with all the Lyapunov exponents positive, a phenomenon
that turns out to be related to the lack of absolutely continuity of the
central foliation.Comment: To appear in Journal of Statistical Physic
Stochastic stability at the boundary of expanding maps
We consider endomorphisms of a compact manifold which are expanding except
for a finite number of points and prove the existence and uniqueness of a
physical measure and its stochastical stability. We also characterize the
zero-noise limit measures for a model of the intermittent map and obtain
stochastic stability for some values of the parameter. The physical measures
are obtained as zero-noise limits which are shown to satisfy Pesin?s Entropy
Formula
Using a physics-informed neural network and fault zone acoustic monitoring to predict lab earthquakes
Predicting failure in solids has broad applications including earthquake prediction which remains an unattainable goal. However, recent machine learning work shows that laboratory earthquakes can be predicted using micro-failure events and temporal evolution of fault zone elastic properties. Remarkably, these results come from purely data-driven models trained with large datasets. Such data are equivalent to centuries of fault motion rendering application to tectonic faulting unclear. In addition, the underlying physics of such predictions is poorly understood. Here, we address scalability using a novel Physics-Informed Neural Network (PINN). Our model encodes fault physics in the deep learning loss function using time-lapse ultrasonic data. PINN models outperform data-driven models and significantly improve transfer learning for small training datasets and conditions outside those used in training. Our work suggests that PINN offers a promising path for machine learning-based failure prediction and, ultimately for improving our understanding of earthquake physics and prediction
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