15,388 research outputs found
Piecewise smooth systems near a co-dimension 2 discontinuity manifold: can one say what should happen?
We consider a piecewise smooth system in the neighborhood of a co-dimension 2
discontinuity manifold . Within the class of Filippov solutions, if
is attractive, one should expect solution trajectories to slide on
. It is well known, however, that the classical Filippov
convexification methodology is ambiguous on . The situation is further
complicated by the possibility that, regardless of how sliding on is
taking place, during sliding motion a trajectory encounters so-called generic
first order exit points, where ceases to be attractive.
In this work, we attempt to understand what behavior one should expect of a
solution trajectory near when is attractive, what to expect
when ceases to be attractive (at least, at generic exit points), and
finally we also contrast and compare the behavior of some regularizations
proposed in the literature.
Through analysis and experiments we will confirm some known facts, and
provide some important insight: (i) when is attractive, a solution
trajectory indeed does remain near , viz. sliding on is an
appropriate idealization (of course, in general, one cannot predict which
sliding vector field should be selected); (ii) when loses attractivity
(at first order exit conditions), a typical solution trajectory leaves a
neighborhood of ; (iii) there is no obvious way to regularize the
system so that the regularized trajectory will remain near as long as
is attractive, and so that it will be leaving (a neighborhood of)
when looses attractivity.
We reach the above conclusions by considering exclusively the given piecewise
smooth system, without superimposing any assumption on what kind of dynamics
near (or sliding motion on ) should have been taking place.Comment: 19 figure
Asymptotic Analysis of MAP Estimation via the Replica Method and Applications to Compressed Sensing
The replica method is a non-rigorous but well-known technique from
statistical physics used in the asymptotic analysis of large, random, nonlinear
problems. This paper applies the replica method, under the assumption of
replica symmetry, to study estimators that are maximum a posteriori (MAP) under
a postulated prior distribution. It is shown that with random linear
measurements and Gaussian noise, the replica-symmetric prediction of the
asymptotic behavior of the postulated MAP estimate of an n-dimensional vector
"decouples" as n scalar postulated MAP estimators. The result is based on
applying a hardening argument to the replica analysis of postulated posterior
mean estimators of Tanaka and of Guo and Verdu.
The replica-symmetric postulated MAP analysis can be readily applied to many
estimators used in compressed sensing, including basis pursuit, lasso, linear
estimation with thresholding, and zero norm-regularized estimation. In the case
of lasso estimation the scalar estimator reduces to a soft-thresholding
operator, and for zero norm-regularized estimation it reduces to a
hard-threshold. Among other benefits, the replica method provides a
computationally-tractable method for precisely predicting various performance
metrics including mean-squared error and sparsity pattern recovery probability.Comment: 22 pages; added details on the replica symmetry assumptio
Which graphical models are difficult to learn?
We consider the problem of learning the structure of Ising models (pairwise
binary Markov random fields) from i.i.d. samples. While several methods have
been proposed to accomplish this task, their relative merits and limitations
remain somewhat obscure. By analyzing a number of concrete examples, we show
that low-complexity algorithms systematically fail when the Markov random field
develops long-range correlations. More precisely, this phenomenon appears to be
related to the Ising model phase transition (although it does not coincide with
it)
Existence and Blow-Up Behavior for Solutions of the Generalized Jang Equation
The generalized Jang equation was introduced in an attempt to prove the
Penrose inequality in the setting of general initial data for the Einstein
equations. In this paper we give an extensive study of this equation, proving
existence, regularity, and blow-up results. In particular, precise asymptotics
for the blow-up behavior are given, and it is shown that blow-up solutions are
not unique.Comment: 33 pages; final versio
- …