33,851 research outputs found
Bayesian Detection of Changepoints in Finite-State Markov Chains for Multiple Sequences
We consider the analysis of sets of categorical sequences consisting of
piecewise homogeneous Markov segments. The sequences are assumed to be governed
by a common underlying process with segments occurring in the same order for
each sequence. Segments are defined by a set of unobserved changepoints where
the positions and number of changepoints can vary from sequence to sequence. We
propose a Bayesian framework for analyzing such data, placing priors on the
locations of the changepoints and on the transition matrices and using Markov
chain Monte Carlo (MCMC) techniques to obtain posterior samples given the data.
Experimental results using simulated data illustrates how the methodology can
be used for inference of posterior distributions for parameters and
changepoints, as well as the ability to handle considerable variability in the
locations of the changepoints across different sequences. We also investigate
the application of the approach to sequential data from two applications
involving monsoonal rainfall patterns and branching patterns in trees
A randomized kinodynamic planner for closed-chain robotic systems
Kinodynamic RRT planners are effective tools for finding feasible trajectories in many classes of robotic systems. However, they are hard to apply to systems with closed-kinematic chains, like parallel robots, cooperating arms manipulating an object, or legged robots keeping their feet in contact with the environ- ment. The state space of such systems is an implicitly-defined manifold, which complicates the design of the sampling and steering procedures, and leads to trajectories that drift away from the manifold when standard integration methods are used. To address these issues, this report presents a kinodynamic RRT planner that constructs an atlas of the state space incrementally, and uses this atlas to both generate ran- dom states, and to dynamically steer the system towards such states. The steering method is based on computing linear quadratic regulators from the atlas charts, which greatly increases the planner efficiency in comparison to the standard method that simulates random actions. The atlas also allows the integration of the equations of motion as a differential equation on the state space manifold, which eliminates any drift from such manifold and thus results in accurate trajectories. To the best of our knowledge, this is the first kinodynamic planner that explicitly takes closed kinematic chains into account. We illustrate the performance of the approach in significantly complex tasks, including planar and spatial robots that have to lift or throw a load at a given velocity using torque-limited actuators.Peer ReviewedPreprin
Bounds on the maximum multiplicity of some common geometric graphs
We obtain new lower and upper bounds for the maximum multiplicity of some
weighted and, respectively, non-weighted common geometric graphs drawn on n
points in the plane in general position (with no three points collinear):
perfect matchings, spanning trees, spanning cycles (tours), and triangulations.
(i) We present a new lower bound construction for the maximum number of
triangulations a set of n points in general position can have. In particular,
we show that a generalized double chain formed by two almost convex chains
admits {\Omega}(8.65^n) different triangulations. This improves the bound
{\Omega}(8.48^n) achieved by the double zig-zag chain configuration studied by
Aichholzer et al.
(ii) We present a new lower bound of {\Omega}(12.00^n) for the number of
non-crossing spanning trees of the double chain composed of two convex chains.
The previous bound, {\Omega}(10.42^n), stood unchanged for more than 10 years.
(iii) Using a recent upper bound of 30^n for the number of triangulations,
due to Sharir and Sheffer, we show that n points in the plane in general
position admit at most O(68.62^n) non-crossing spanning cycles.
(iv) We derive lower bounds for the number of maximum and minimum weighted
geometric graphs (matchings, spanning trees, and tours). We show that the
number of shortest non-crossing tours can be exponential in n. Likewise, we
show that both the number of longest non-crossing tours and the number of
longest non-crossing perfect matchings can be exponential in n. Moreover, we
show that there are sets of n points in convex position with an exponential
number of longest non-crossing spanning trees. For points in convex position we
obtain tight bounds for the number of longest and shortest tours. We give a
combinatorial characterization of the longest tours, which leads to an O(nlog
n) time algorithm for computing them
The hexagon in the mirror: the three-point function in the SoV representation
We derive an integral expression for the leading-order type I-I-I three-point
functions in the -sector of super Yang-Mills
theory, for which no determinant formula is known. To this end, we first map
the problem to the partition function of the six vertex model with a hexagonal
boundary. The advantage of the six-vertex model expression is that it reveals
an extra symmetry of the problem, which is the invariance under 90
rotation. On the spin-chain side, this corresponds to the exchange of the
quantum space and the auxiliary space and is reminiscent of the mirror
transformation employed in the worldsheet S-matrix approaches. After the
rotation, we then apply Sklyanin's separation of variables (SoV) and obtain a
multiple-integral expression of the three-point function. The resulting
integrand is expressed in terms of the so-called Baxter polynomials, which is
closely related to the quantum spectral curve approach. Along the way, we also
derive several new results about the SoV, such as the explicit construction of
the basis with twisted boundary conditions and the overlap between the orginal
SoV state and the SoV states on the subchains.Comment: 37 pages, 10 figure
Total variation on a tree
We consider the problem of minimizing the continuous valued total variation
subject to different unary terms on trees and propose fast direct algorithms
based on dynamic programming to solve these problems. We treat both the convex
and the non-convex case and derive worst case complexities that are equal or
better than existing methods. We show applications to total variation based 2D
image processing and computer vision problems based on a Lagrangian
decomposition approach. The resulting algorithms are very efficient, offer a
high degree of parallelism and come along with memory requirements which are
only in the order of the number of image pixels.Comment: accepted to SIAM Journal on Imaging Sciences (SIIMS
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