90,459 research outputs found
Approximate probabilistic verification of hybrid systems
Hybrid systems whose mode dynamics are governed by non-linear ordinary
differential equations (ODEs) are often a natural model for biological
processes. However such models are difficult to analyze. To address this, we
develop a probabilistic analysis method by approximating the mode transitions
as stochastic events. We assume that the probability of making a mode
transition is proportional to the measure of the set of pairs of time points
and value states at which the mode transition is enabled. To ensure a sound
mathematical basis, we impose a natural continuity property on the non-linear
ODEs. We also assume that the states of the system are observed at discrete
time points but that the mode transitions may take place at any time between
two successive discrete time points. This leads to a discrete time Markov chain
as a probabilistic approximation of the hybrid system. We then show that for
BLTL (bounded linear time temporal logic) specifications the hybrid system
meets a specification iff its Markov chain approximation meets the same
specification with probability . Based on this, we formulate a sequential
hypothesis testing procedure for verifying -approximately- that the Markov
chain meets a BLTL specification with high probability. Our case studies on
cardiac cell dynamics and the circadian rhythm indicate that our scheme can be
applied in a number of realistic settings
Sparse Recovery via Differential Inclusions
In this paper, we recover sparse signals from their noisy linear measurements
by solving nonlinear differential inclusions, which is based on the notion of
inverse scale space (ISS) developed in applied mathematics. Our goal here is to
bring this idea to address a challenging problem in statistics, \emph{i.e.}
finding the oracle estimator which is unbiased and sign-consistent using
dynamics. We call our dynamics \emph{Bregman ISS} and \emph{Linearized Bregman
ISS}. A well-known shortcoming of LASSO and any convex regularization
approaches lies in the bias of estimators. However, we show that under proper
conditions, there exists a bias-free and sign-consistent point on the solution
paths of such dynamics, which corresponds to a signal that is the unbiased
estimate of the true signal and whose entries have the same signs as those of
the true signs, \emph{i.e.} the oracle estimator. Therefore, their solution
paths are regularization paths better than the LASSO regularization path, since
the points on the latter path are biased when sign-consistency is reached. We
also show how to efficiently compute their solution paths in both continuous
and discretized settings: the full solution paths can be exactly computed piece
by piece, and a discretization leads to \emph{Linearized Bregman iteration},
which is a simple iterative thresholding rule and easy to parallelize.
Theoretical guarantees such as sign-consistency and minimax optimal -error
bounds are established in both continuous and discrete settings for specific
points on the paths. Early-stopping rules for identifying these points are
given. The key treatment relies on the development of differential inequalities
for differential inclusions and their discretizations, which extends the
previous results and leads to exponentially fast recovering of sparse signals
before selecting wrong ones.Comment: In Applied and Computational Harmonic Analysis, 201
Marginal Release Under Local Differential Privacy
Many analysis and machine learning tasks require the availability of marginal
statistics on multidimensional datasets while providing strong privacy
guarantees for the data subjects. Applications for these statistics range from
finding correlations in the data to fitting sophisticated prediction models. In
this paper, we provide a set of algorithms for materializing marginal
statistics under the strong model of local differential privacy. We prove the
first tight theoretical bounds on the accuracy of marginals compiled under each
approach, perform empirical evaluation to confirm these bounds, and evaluate
them for tasks such as modeling and correlation testing. Our results show that
releasing information based on (local) Fourier transformations of the input is
preferable to alternatives based directly on (local) marginals
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