3,010 research outputs found
From Random Lines to Metric Spaces
Consider an improper Poisson line process, marked by positive speeds so as to
satisfy a scale-invariance property (actually, scale-equivariance). The line
process can be characterized by its intensity measure, which belongs to a
one-parameter family if scale and Euclidean invariance are required. This paper
investigates a proposal by Aldous, namely that the line process could be used
to produce a scale-invariant random spatial network (SIRSN) by means of
connecting up points using paths which follow segments from the line process at
the stipulated speeds. It is shown that this does indeed produce a
scale-invariant network, under suitable conditions on the parameter; indeed
that this produces a parameter-dependent random geodesic metric for
d-dimensional space (), where geodesics are given by minimum-time
paths. Moreover in the planar case it is shown that the resulting geodesic
metric space has an almost-everywhere-unique-geodesic property, that geodesics
are locally of finite mean length, and that if an independent Poisson point
process is connected up by such geodesics then the resulting network places
finite length in each compact region. It is an open question whether the result
is a SIRSN (in Aldous' sense; so placing finite mean length in each compact
region), but it may be called a pre-SIRSN.Comment: Version 1: 46 pages, 10 figures Version 2: 47 pages, 10 figures
(various typos and stylistic amendments, added dedication to Burkholder,
added references concerning Lipschitz property and Sobolev space
Actor-Critic Reinforcement Learning for Control with Stability Guarantee
Reinforcement Learning (RL) and its integration with deep learning have
achieved impressive performance in various robotic control tasks, ranging from
motion planning and navigation to end-to-end visual manipulation. However,
stability is not guaranteed in model-free RL by solely using data. From a
control-theoretic perspective, stability is the most important property for any
control system, since it is closely related to safety, robustness, and
reliability of robotic systems. In this paper, we propose an actor-critic RL
framework for control which can guarantee closed-loop stability by employing
the classic Lyapunov's method in control theory. First of all, a data-based
stability theorem is proposed for stochastic nonlinear systems modeled by
Markov decision process. Then we show that the stability condition could be
exploited as the critic in the actor-critic RL to learn a controller/policy. At
last, the effectiveness of our approach is evaluated on several well-known
3-dimensional robot control tasks and a synthetic biology gene network tracking
task in three different popular physics simulation platforms. As an empirical
evaluation on the advantage of stability, we show that the learned policies can
enable the systems to recover to the equilibrium or way-points when interfered
by uncertainties such as system parametric variations and external disturbances
to a certain extent.Comment: IEEE RA-L + IROS 202
Optimization with Sparsity-Inducing Penalties
Sparse estimation methods are aimed at using or obtaining parsimonious
representations of data or models. They were first dedicated to linear variable
selection but numerous extensions have now emerged such as structured sparsity
or kernel selection. It turns out that many of the related estimation problems
can be cast as convex optimization problems by regularizing the empirical risk
with appropriate non-smooth norms. The goal of this paper is to present from a
general perspective optimization tools and techniques dedicated to such
sparsity-inducing penalties. We cover proximal methods, block-coordinate
descent, reweighted -penalized techniques, working-set and homotopy
methods, as well as non-convex formulations and extensions, and provide an
extensive set of experiments to compare various algorithms from a computational
point of view
Variational approximation of functionals defined on 1-dimensional connected sets: the planar case
In this paper we consider variational problems involving 1-dimensional
connected sets in the Euclidean plane, such as the classical Steiner tree
problem and the irrigation (Gilbert-Steiner) problem. We relate them to optimal
partition problems and provide a variational approximation through
Modica-Mortola type energies proving a -convergence result. We also
introduce a suitable convex relaxation and develop the corresponding numerical
implementations. The proposed methods are quite general and the results we
obtain can be extended to -dimensional Euclidean space or to more general
manifold ambients, as shown in the companion paper [11].Comment: 30 pages, 5 figure
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