150 research outputs found
Combinatorial and Asymptotical Results on the Neighborhood Grid
In 2009, Joselli et al introduced the Neighborhood Grid data structure for
fast computation of neighborhood estimates in point clouds. Even though the
data structure has been used in several applications and shown to be
practically relevant, it is theoretically not yet well understood. The purpose
of this paper is to present a polynomial-time algorithm to build the data
structure. Furthermore, it is investigated whether the presented algorithm is
optimal. This investigations leads to several combinatorial questions for which
partial results are given. Finally, we present several limits and experiments
regarding the quality of the obtained neighborhood relation.Comment: 33 pages, 18 Figure
A Fibonacci control system with application to hyper-redundant manipulators
We study a robot snake model based on a discrete linear control system
involving Fibonacci sequence and closely related to the theory of expansions in
non-integer bases. The present paper includes an investigation of the reachable
workspace, a more general analysis of the control system underlying the model,
its reachability and local controllability properties and the relation with
expansions in non-integer bases and with iterated function systems
(Almost) tight bounds for randomized and quantum Local Search on hypercubes and grids
The Local Search problem, which finds a local minimum of a black-box function
on a given graph, is of both practical and theoretical importance to many areas
in computer science and natural sciences. In this paper, we show that for the
Boolean hypercube \B^n, the randomized query complexity of Local Search is
and the quantum query complexity is
. We also show that for the constant dimensional grid
, the randomized query complexity is for and the quantum query complexity is for . New
lower bounds for lower dimensional grids are also given. These improve the
previous results by Aaronson [STOC'04], and Santha and Szegedy [STOC'04].
Finally we show for a new upper bound of on the quantum query complexity, which implies that Local Search on
grids exhibits different properties at low dimensions.Comment: 18 pages, 1 figure. v2: introduction rewritten, references added. v3:
a line for grant added. v4: upper bound section rewritte
Hidden Gibbs random fields model selection using Block Likelihood Information Criterion
Performing model selection between Gibbs random fields is a very challenging
task. Indeed, due to the Markovian dependence structure, the normalizing
constant of the fields cannot be computed using standard analytical or
numerical methods. Furthermore, such unobserved fields cannot be integrated out
and the likelihood evaluztion is a doubly intractable problem. This forms a
central issue to pick the model that best fits an observed data. We introduce a
new approximate version of the Bayesian Information Criterion. We partition the
lattice into continuous rectangular blocks and we approximate the probability
measure of the hidden Gibbs field by the product of some Gibbs distributions
over the blocks. On that basis, we estimate the likelihood and derive the Block
Likelihood Information Criterion (BLIC) that answers model choice questions
such as the selection of the dependency structure or the number of latent
states. We study the performances of BLIC for those questions. In addition, we
present a comparison with ABC algorithms to point out that the novel criterion
offers a better trade-off between time efficiency and reliable results
Covering Points by Disjoint Boxes with Outliers
For a set of n points in the plane, we consider the axis--aligned (p,k)-Box
Covering problem: Find p axis-aligned, pairwise-disjoint boxes that together
contain n-k points. In this paper, we consider the boxes to be either squares
or rectangles, and we want to minimize the area of the largest box. For general
p we show that the problem is NP-hard for both squares and rectangles. For a
small, fixed number p, we give algorithms that find the solution in the
following running times:
For squares we have O(n+k log k) time for p=1, and O(n log n+k^p log^p k time
for p = 2,3. For rectangles we get O(n + k^3) for p = 1 and O(n log n+k^{2+p}
log^{p-1} k) time for p = 2,3.
In all cases, our algorithms use O(n) space.Comment: updated version: - changed problem from 'cover exactly n-k points' to
'cover at least n-k points' to avoid having non-feasible solutions. Results
are unchanged. - added Proof to Lemma 11, clarified some sections - corrected
typos and small errors - updated affiliations of two author
Metaheuristics for NP-hard combinatorial optimization problems
Ph.DDOCTOR OF PHILOSOPH
Optimization with Discrete Simultaneous Perturbation Stochastic Approximation Using Noisy Loss Function Measurements
Discrete stochastic optimization considers the problem of minimizing (or
maximizing) loss functions defined on discrete sets, where only noisy
measurements of the loss functions are available. The discrete stochastic
optimization problem is widely applicable in practice, and many algorithms have
been considered to solve this kind of optimization problem. Motivated by the
efficient algorithm of simultaneous perturbation stochastic approximation
(SPSA) for continuous stochastic optimization problems, we introduce the middle
point discrete simultaneous perturbation stochastic approximation (DSPSA)
algorithm for the stochastic optimization of a loss function defined on a
p-dimensional grid of points in Euclidean space. We show that the sequence
generated by DSPSA converges to the optimal point under some conditions.
Consistent with other stochastic approximation methods, DSPSA formally
accommodates noisy measurements of the loss function. We also show the rate of
convergence analysis of DSPSA by solving an upper bound of the mean squared
error of the generated sequence. In order to compare the performance of DSPSA
with the other algorithms such as the stochastic ruler algorithm (SR) and the
stochastic comparison algorithm (SC), we set up a bridge between DSPSA and the
other two algorithms by comparing the probability in a big-O sense of not
achieving the optimal solution. We show the theoretical and numerical
comparison results of DSPSA, SR, and SC. In addition, we consider an
application of DSPSA towards developing optimal public health strategies for
containing the spread of influenza given limited societal resources
Approximation of length minimization problems among compact connected sets
In this paper we provide an approximation \`a la Ambrosio-Tortorelli of some
classical minimization problems involving the length of an unknown
one-dimensional set, with an additional connectedness constraint, in dimension
two. We introduce a term of new type relying on a weighted geodesic distance
that forces the minimizers to be connected at the limit. We apply this approach
to approximate the so-called Steiner Problem, but also the average distance
problem, and finally a problem relying on the p-compliance energy. The proof of
convergence of the approximating functional, which is stated in terms of
Gamma-convergence relies on technical tools from geometric measure theory, as
for instance a uniform lower bound for a sort of average directional Minkowski
content of a family of compact connected sets
- …