41,492 research outputs found
Input Matrix Construction and Approximation Using a Graphic Approach
Given a state transition matrix (STM), we reinvestigate the problem of
constructing the sparest input matrix with a fixed number of inputs to
guarantee controllability. We give a new and simple graph theoretic
characterization for the sparsity pattern of input matrices to guarantee
controllability for a general STM admitting multiple eigenvalues, and provide a
deterministic procedure with polynomial time complexity to construct real
valued input matrices with arbi- trarily prescribed sparsity pattern satisfying
controllability. Based on this criterion, some novel results on sparsely
controlling a system are obtained. It is proven that the minimal number of
inputs to guarantee controllability equals to the maximum geometric
multiplicity of the STM under the constraint that some states are
actuated-forbidden, extending the results of [28]. The minimal sparsity of
input matrices with a fixed number of inputs is not necessarily equal to the
minimal number of actuated states to ensure controllability. Furthermore, a
graphic sub- modular function is built, leading to a greedy algorithm to
efficiently approximate the minimal actuated states to assure controllability
for general STMs. For the problem of approximating the sparsest input matrices
with a fixed number of inputs, we propose a simple greedy algo- rithm
(non-submodular) and a two-stage algorithm, and demonstrate that the latter
algorithm, inspired from techniques in dynamic coloring, has a provable
approximation guarantee. Finally, we present numerical results to show the
efficiency and effectiveness of our approaches.Comment: to appear in International Journal of Contro
Structural Controllability of a Networked Dynamic System with LFT Parameterized Subsystems
This paper studies structural controllability for a networked dynamic system
(NDS), in which each subsystem may have different dynamics, and unknown
parameters may exist both in subsystem dynamics and in subsystem
interconnections. In addition, subsystem parameters are parameterized by a
linear fractional transformation (LFT). It is proven that controllability keeps
to be a generic property for this kind of NDSs. Some necessary and sufficient
conditions are then established respectively for them to be structurally
controllable, to have a fixed uncontrollable mode, and to have a parameter
dependent uncontrollable mode, under the condition that each subsystem
interconnection link can take a weight independently. These conditions are
scalable, and in their verifications, all arithmetic calculations are performed
separately on each subsystem. In addition, these conditions also reveal
influences on NDS controllability from subsystem input-output relations,
subsystem uncontrollable modes and subsystem interconnection topology. Based on
these observations, the problem of selecting the minimal number of subsystem
interconnection links is studied under the requirement of constructing a
structurally controllable NDS. A heuristic method is derived with some provable
approximation bounds and a low computational complexity.Comment: Accepted by IEEE Transactions on Automatic Control as full paper,
scheduled to appear in Volume 64 (2019), Issue 12 (December
An optimal consensus tracking control algorithm for autonomous underwater vehicles with disturbances
The optimal disturbance rejection control problem is considered for consensus
tracking systems affected by external persistent disturbances and noise.
Optimal estimated values of system states are obtained by recursive filtering
for the multiple autonomous underwater vehicles modeled to multi-agent systems
with Kalman filter. Then the feedforward-feedback optimal control law is
deduced by solving the Riccati equations and matrix equations. The existence
and uniqueness condition of feedforward-feedback optimal control law is
proposed and the optimal control law algorithm is carried out. Lastly,
simulations show the result is effectiveness with respect to external
persistent disturbances and noise
C^1-Regularity of planar \infty-harmonic functions - REVISIT
In the seminal paper [Arch. Ration. Mech. Anal. 176 (2005), 351--361],
Savin proved the -regularity of planar -harmonic functions .
Here we give a new understanding of it from a capacity viewpoint and drop
several high technique arguments therein. Our argument is essentially based on
a topological lemma of Savin, a flat estimate by Evans and Smart, %
\cite{es11a},
-regularity of and Crandall's flow for infinity
harmonic functions.Comment: 6 page
Adaptive Double-Exploration Tradeoff for Outlier Detection
We study a variant of the thresholding bandit problem (TBP) in the context of
outlier detection, where the objective is to identify the outliers whose
rewards are above a threshold. Distinct from the traditional TBP, the threshold
is defined as a function of the rewards of all the arms, which is motivated by
the criterion for identifying outliers. The learner needs to explore the
rewards of the arms as well as the threshold. We refer to this problem as
"double exploration for outlier detection". We construct an adaptively updated
confidence interval for the threshold, based on the estimated value of the
threshold in the previous rounds. Furthermore, by automatically trading off
exploring the individual arms and exploring the outlier threshold, we provide
an efficient algorithm in terms of the sample complexity. Experimental results
on both synthetic datasets and real-world datasets demonstrate the efficiency
of our algorithm
Collaborative Learning with Limited Interaction: Tight Bounds for Distributed Exploration in Multi-Armed Bandits
Best arm identification (or, pure exploration) in multi-armed bandits is a
fundamental problem in machine learning. In this paper we study the distributed
version of this problem where we have multiple agents, and they want to learn
the best arm collaboratively. We want to quantify the power of collaboration
under limited interaction (or, communication steps), as interaction is
expensive in many settings. We measure the running time of a distributed
algorithm as the speedup over the best centralized algorithm where there is
only one agent. We give almost tight round-speedup tradeoffs for this problem,
along which we develop several new techniques for proving lower bounds on the
number of communication steps under time or confidence constraints.Comment: 33 page
Tight Bounds for Collaborative PAC Learning via Multiplicative Weights
We study the collaborative PAC learning problem recently proposed in Blum et
al.~\cite{BHPQ17}, in which we have players and they want to learn a target
function collaboratively, such that the learned function approximates the
target function well on all players' distributions simultaneously. The quality
of the collaborative learning algorithm is measured by the ratio between the
sample complexity of the algorithm and that of the learning algorithm for a
single distribution (called the overhead). We obtain a collaborative learning
algorithm with overhead , improving the one with overhead in \cite{BHPQ17}. We also show that an overhead is
inevitable when is polynomial bounded by the VC dimension of the hypothesis
class. Finally, our experimental study has demonstrated the superiority of our
algorithm compared with the one in Blum et al. on real-world datasets.Comment: Accepted to NIPS 2018. 14 page
Adaptive Multiple-Arm Identification
We study the problem of selecting arms with the highest expected rewards
in a stochastic -armed bandit game. This problem has a wide range of
applications, e.g., A/B testing, crowdsourcing, simulation optimization. Our
goal is to develop a PAC algorithm, which, with probability at least
, identifies a set of arms with the aggregate regret at most
. The notion of aggregate regret for multiple-arm identification was
first introduced in \cite{Zhou:14} , which is defined as the difference of the
averaged expected rewards between the selected set of arms and the best
arms. In contrast to \cite{Zhou:14} that only provides instance-independent
sample complexity, we introduce a new hardness parameter for characterizing the
difficulty of any given instance. We further develop two algorithms and
establish the corresponding sample complexity in terms of this hardness
parameter. The derived sample complexity can be significantly smaller than
state-of-the-art results for a large class of instances and matches the
instance-independent lower bound upto a factor in the
worst case. We also prove a lower bound result showing that the extra
is necessary for instance-dependent algorithms using the
introduced hardness parameter.Comment: 30 pages, 5 figures, preliminary version to appear in ICML 201
The limit of the m-norms of a class of symmetric matrices and its applications
We consider a special symmetric matrix and obtain a similar formula as the
one obtained by Weyl's criterion. Some applications of the formula are given,
where we give a new way to calculate the integral of on ,
and we claim that one class of matrices are not Hadamard matrices.Comment: 10 pages, 1 figur
Some sharp Sobolev regularity for inhomogeneous -Laplace equation in plane
Suppose and with in . Let be a viscosity
solution to the inhomogeneous -Laplace equation The following are proved in this paper.
(i) For , we have ,
which is (asymptotic) sharp when . Indeed, the function
is a viscosity solution to in . For any ,
whenever
.
(ii) For and , we have
, which is sharp when . Indeed,
.
(iii) For , we have , which is sharp when . Indeed, .
(iv) For , we have -(|Du|^{\alpha})_iu_i= 2\alpha|Du|^{{
\alpha-2}}f \ \mbox{ almost everywhere in $\Omega$}.
Some quantative bounds are also given
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