3,922 research outputs found
Sparsity with sign-coherent groups of variables via the cooperative-Lasso
We consider the problems of estimation and selection of parameters endowed
with a known group structure, when the groups are assumed to be sign-coherent,
that is, gathering either nonnegative, nonpositive or null parameters. To
tackle this problem, we propose the cooperative-Lasso penalty. We derive the
optimality conditions defining the cooperative-Lasso estimate for generalized
linear models, and propose an efficient active set algorithm suited to
high-dimensional problems. We study the asymptotic consistency of the estimator
in the linear regression setup and derive its irrepresentable conditions, which
are milder than the ones of the group-Lasso regarding the matching of groups
with the sparsity pattern of the true parameters. We also address the problem
of model selection in linear regression by deriving an approximation of the
degrees of freedom of the cooperative-Lasso estimator. Simulations comparing
the proposed estimator to the group and sparse group-Lasso comply with our
theoretical results, showing consistent improvements in support recovery for
sign-coherent groups. We finally propose two examples illustrating the wide
applicability of the cooperative-Lasso: first to the processing of ordinal
variables, where the penalty acts as a monotonicity prior; second to the
processing of genomic data, where the set of differentially expressed probes is
enriched by incorporating all the probes of the microarray that are related to
the corresponding genes.Comment: Published in at http://dx.doi.org/10.1214/11-AOAS520 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Inferring Multiple Graphical Structures
Gaussian Graphical Models provide a convenient framework for representing
dependencies between variables. Recently, this tool has received a high
interest for the discovery of biological networks. The literature focuses on
the case where a single network is inferred from a set of measurements, but, as
wetlab data is typically scarce, several assays, where the experimental
conditions affect interactions, are usually merged to infer a single network.
In this paper, we propose two approaches for estimating multiple related
graphs, by rendering the closeness assumption into an empirical prior or group
penalties. We provide quantitative results demonstrating the benefits of the
proposed approaches. The methods presented in this paper are embeded in the R
package 'simone' from version 1.0-0 and later
A pressure impulse theory for hemispherical liquid impact problems
Liquid impact problems for hemispherical fluid domain are considered. By
using the concept of pressure impulse we show that the solution of the flow
induced by the impact is reduced to the derivation of Laplace's equation in
spherical coordinates with Dirichlet and Neumann boundary conditions. The
structure of the flow at the impact moment is deduced from the spherical
harmonics representation of the solution. In particular we show that the slip
velocity has a logarithmic singularity at the contact line. The theoretical
predictions are in very good agreement both qualitatively and quantitatively
with the first time step of a numerical simulation with a Navier-Stokes solver
named Gerris.Comment: 11 pages, 14 figures, Accepted for publication in European Journal of
Mechanics - B/Fluid
Tiled QR factorization algorithms
This work revisits existing algorithms for the QR factorization of
rectangular matrices composed of p-by-q tiles, where p >= q. Within this
framework, we study the critical paths and performance of algorithms such as
Sameh and Kuck, Modi and Clarke, Greedy, and those found within PLASMA.
Although neither Modi and Clarke nor Greedy is optimal, both are shown to be
asymptotically optimal for all matrices of size p = q^2 f(q), where f is any
function such that \lim_{+\infty} f= 0. This novel and important complexity
result applies to all matrices where p and q are proportional, p = \lambda q,
with \lambda >= 1, thereby encompassing many important situations in practice
(least squares). We provide an extensive set of experiments that show the
superiority of the new algorithms for tall matrices
Tight Bounds for Consensus Systems Convergence
We analyze the asymptotic convergence of all infinite products of matrices
taken in a given finite set, by looking only at finite or periodic products. It
is known that when the matrices of the set have a common nonincreasing
polyhedral norm, all infinite products converge to zero if and only if all
infinite periodic products with period smaller than a certain value converge to
zero, and bounds exist on that value.
We provide a stronger bound holding for both polyhedral norms and polyhedral
seminorms. In the latter case, the matrix products do not necessarily converge
to 0, but all trajectories of the associated system converge to a common
invariant space. We prove our bound to be tight, in the sense that for any
polyhedral seminorm, there is a set of matrices such that not all infinite
products converge, but every periodic product with period smaller than our
bound does converge.
Our technique is based on an analysis of the combinatorial structure of the
face lattice of the unit ball of the nonincreasing seminorm. The bound we
obtain is equal to half the size of the largest antichain in this lattice.
Explicitly evaluating this quantity may be challenging in some cases. We
therefore link our problem with the Sperner property: the property that, for
some graded posets, -- in this case the face lattice of the unit ball -- the
size of the largest antichain is equal to the size of the largest rank level.
We show that some sets of matrices with invariant polyhedral seminorms lead
to posets that do not have that Sperner property. However, this property holds
for the polyhedron obtained when treating sets of stochastic matrices, and our
bound can then be easily evaluated in that case. In particular, we show that
for the dimension of the space , our bound is smaller than the
previously known bound by a multiplicative factor of
Efficient Algorithms for the Consensus Decision Problem
We address the problem of determining if a discrete time switched consensus
system converges for any switching sequence and that of determining if it
converges for at least one switching sequence. For these two problems, we
provide necessary and sufficient conditions that can be checked in singly
exponential time. As a side result, we prove the existence of a polynomial time
algorithm for the first problem when the system switches between only two
subsystems whose corresponding graphs are undirected, a problem that had been
suggested to be NP-hard by Blondel and Olshevsky.Comment: Small modifications after comments from reviewer
Reachability of Consensus and Synchronizing Automata
We consider the problem of determining the existence of a sequence of
matrices driving a discrete-time consensus system to consensus. We transform
this problem into one of the existence of a product of the transition
(stochastic) matrices that has a positive column. We then generalize some
results from automata theory to sets of stochastic matrices. We obtain as a
main result a polynomial-time algorithm to decide the existence of a sequence
of matrices achieving consensus.Comment: Update after revie
Study of some optimal XFEM type methods
The XFEM method in fracture mechanics is revisited. A first improvement is considered using an enlarged fixed enrichment subdomain aroud the crack tip and a bonding condition for the corresponding degree of freedom. An efficient numerical integration rule is introduced for the nonsmooth enrichment functions. The lack of accuracy due to the transition layer between the enrichment aera and the rest of the domain leads to consider a pointwise matching condition at the boundary of the subdomain. An optimal numerical rate of convergence is then obtained using such a nonconformal method
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