54 research outputs found
On Time-optimal Trajectories for a Car-like Robot with One Trailer
In addition to the theoretical value of challenging optimal control problmes,
recent progress in autonomous vehicles mandates further research in optimal
motion planning for wheeled vehicles. Since current numerical optimal control
techniques suffer from either the curse of dimens ionality, e.g. the
Hamilton-Jacobi-Bellman equation, or the curse of complexity, e.g.
pseudospectral optimal control and max-plus methods, analytical
characterization of geodesics for wheeled vehicles becomes important not only
from a theoretical point of view but also from a prac tical one. Such an
analytical characterization provides a fast motion planning algorithm that can
be used in robust feedback loops. In this work, we use the Pontryagin Maximum
Principle to characterize extremal trajectories, i.e. candidate geodesics, for
a car-like robot with one trailer. We use time as the distance function. In
spite of partial progress, this problem has remained open in the past two
decades. Besides straight motion and turn with maximum allowed curvature, we
identify planar elastica as the third piece of motion that occurs along our
extr emals. We give a detailed characterization of such curves, a special case
of which, called \emph{merging curve}, connects maximum curvature turns to
straight line segments. The structure of extremals in our case is revealed
through analytical integration of the system and adjoint equations
Exact Learning of RNA Energy Parameters From Structure
We consider the problem of exact learning of parameters of a linear RNA
energy model from secondary structure data. A necessary and sufficient
condition for learnability of parameters is derived, which is based on
computing the convex hull of union of translated Newton polytopes of input
sequences. The set of learned energy parameters is characterized as the convex
cone generated by the normal vectors to those facets of the resulting polytope
that are incident to the origin. In practice, the sufficient condition may not
be satisfied by the entire training data set; hence, computing a maximal subset
of training data for which the sufficient condition is satisfied is often
desired. We show that problem is NP-hard in general for an arbitrary
dimensional feature space. Using a randomized greedy algorithm, we select a
subset of RNA STRAND v2.0 database that satisfies the sufficient condition for
separate A-U, C-G, G-U base pair counting model. The set of learned energy
parameters includes experimentally measured energies of A-U, C-G, and G-U
pairs; hence, our parameter set is in agreement with the Turner parameters
The RNA Newton Polytope and Learnability of Energy Parameters
Despite nearly two scores of research on RNA secondary structure and RNA-RNA
interaction prediction, the accuracy of the state-of-the-art algorithms are
still far from satisfactory. Researchers have proposed increasingly complex
energy models and improved parameter estimation methods in anticipation of
endowing their methods with enough power to solve the problem. The output has
disappointingly been only modest improvements, not matching the expectations.
Even recent massively featured machine learning approaches were not able to
break the barrier. In this paper, we introduce the notion of learnability of
the parameters of an energy model as a measure of its inherent capability. We
say that the parameters of an energy model are learnable iff there exists at
least one set of such parameters that renders every known RNA structure to date
the minimum free energy structure. We derive a necessary condition for the
learnability and give a dynamic programming algorithm to assess it. Our
algorithm computes the convex hull of the feature vectors of all feasible
structures in the ensemble of a given input sequence. Interestingly, that
convex hull coincides with the Newton polytope of the partition function as a
polynomial in energy parameters. We demonstrated the application of our theory
to a simple energy model consisting of a weighted count of A-U and C-G base
pairs. Our results show that this simple energy model satisfies the necessary
condition for less than one third of the input unpseudoknotted
sequence-structure pairs chosen from the RNA STRAND v2.0 database. For another
one third, the necessary condition is barely violated, which suggests that
augmenting this simple energy model with more features such as the Turner loops
may solve the problem. The necessary condition is severely violated for 8%,
which provides a small set of hard cases that require further investigation
An Efficient Algorithm for Upper Bound on the Partition Function of Nucleic Acids
It has been shown that minimum free energy structure for RNAs and RNA-RNA
interaction is often incorrect due to inaccuracies in the energy parameters and
inherent limitations of the energy model. In contrast, ensemble based
quantities such as melting temperature and equilibrium concentrations can be
more reliably predicted. Even structure prediction by sampling from the
ensemble and clustering those structures by Sfold [7] has proven to be more
reliable than minimum free energy structure prediction. The main obstacle for
ensemble based approaches is the computational complexity of the partition
function and base pairing probabilities. For instance, the space complexity of
the partition function for RNA-RNA interaction is and the time
complexity is which are prohibitively large [4,12]. Our goal in this
paper is to give a fast algorithm, based on sparse folding, to calculate an
upper bound on the partition function. Our work is based on the recent
algorithm of Hazan and Jaakkola [10]. The space complexity of our algorithm is
the same as that of sparse folding algorithms, and the time complexity of our
algorithm is for single RNA and for RNA-RNA
interaction in practice, in which is the running time of sparse folding
and () is a sequence dependent parameter
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