108 research outputs found
Automatic Markov Chain Monte Carlo Procedures for Sampling from Multivariate Distributions
Generating samples from multivariate distributions efficiently is an important task in Monte Carlo integration and many other stochastic simulation problems. Markov chain Monte Carlo has been shown to be very efficient compared to "conventional methods", especially when many dimensions are involved. In this article we propose a Hit-and-Run sampler in combination with the Ratio-of-Uniforms method. We show that it is well suited for an algorithm to generate points from quite arbitrary distributions, which include all log-concave distributions. The algorithm works automatically in the sense that only the mode (or an approximation of it) and an oracle is required, i.e., a subroutine that returns the value of the density function at any point x. We show that the number of evaluations of the density increases slowly with dimension. (author's abstract)Series: Preprint Series / Department of Applied Statistics and Data Processin
Convex set of quantum states with positive partial transpose analysed by hit and run algorithm
The convex set of quantum states of a composite system with
positive partial transpose is analysed. A version of the hit and run algorithm
is used to generate a sequence of random points covering this set uniformly and
an estimation for the convergence speed of the algorithm is derived. For this algorithm works faster than sampling over the entire set of states and
verifying whether the partial transpose is positive. The level density of the
PPT states is shown to differ from the Marchenko-Pastur distribution, supported
in [0,4] and corresponding asymptotically to the entire set of quantum states.
Based on the shifted semi--circle law, describing asymptotic level density of
partially transposed states, and on the level density for the Gaussian unitary
ensemble with constraints for the spectrum we find an explicit form of the
probability distribution supported in [0,3], which describes well the level
density obtained numerically for PPT states.Comment: 11 pages, 4 figure
Near-Optimal Evasion of Convex-Inducing Classifiers
Classifiers are often used to detect miscreant activities. We study how an
adversary can efficiently query a classifier to elicit information that allows
the adversary to evade detection at near-minimal cost. We generalize results of
Lowd and Meek (2005) to convex-inducing classifiers. We present algorithms that
construct undetected instances of near-minimal cost using only polynomially
many queries in the dimension of the space and without reverse engineering the
decision boundary.Comment: 8 pages; to appear at AISTATS'201
On largest volume simplices and sub-determinants
We show that the problem of finding the simplex of largest volume in the
convex hull of points in can be approximated with a factor
of in polynomial time. This improves upon the previously best
known approximation guarantee of by Khachiyan. On the other hand,
we show that there exists a constant such that this problem cannot be
approximated with a factor of , unless . % This improves over the
inapproximability that was previously known. Our hardness result holds
even if , in which case there exists a \bar c\,^{d}-approximation
algorithm that relies on recent sampling techniques, where is again a
constant. We show that similar results hold for the problem of finding the
largest absolute value of a subdeterminant of a matrix
Generalizing Informed Sampling for Asymptotically Optimal Sampling-based Kinodynamic Planning via Markov Chain Monte Carlo
Asymptotically-optimal motion planners such as RRT* have been shown to
incrementally approximate the shortest path between start and goal states. Once
an initial solution is found, their performance can be dramatically improved by
restricting subsequent samples to regions of the state space that can
potentially improve the current solution. When the motion planning problem lies
in a Euclidean space, this region , called the informed set, can be
sampled directly. However, when planning with differential constraints in
non-Euclidean state spaces, no analytic solutions exists to sampling
directly.
State-of-the-art approaches to sampling in such domains such as
Hierarchical Rejection Sampling (HRS) may still be slow in high-dimensional
state space. This may cause the planning algorithm to spend most of its time
trying to produces samples in rather than explore it. In this paper,
we suggest an alternative approach to produce samples in the informed set
for a wide range of settings. Our main insight is to recast this
problem as one of sampling uniformly within the sub-level-set of an implicit
non-convex function. This recasting enables us to apply Monte Carlo sampling
methods, used very effectively in the Machine Learning and Optimization
communities, to solve our problem. We show for a wide range of scenarios that
using our sampler can accelerate the convergence rate to high-quality solutions
in high-dimensional problems
Improved mixing rates of directed cycles by added connection
We investigate the mixing rate of a Markov chain where a combination of long
distance edges and non-reversibility is introduced: as a first step, we focus
here on the following graphs: starting from the cycle graph, we select random
nodes and add all edges connecting them. We prove a square factor improvement
of the mixing rate compared to the reversible version of the Markov chain
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