436,340 research outputs found
Stochastic Gradient Hamiltonian Monte Carlo
Hamiltonian Monte Carlo (HMC) sampling methods provide a mechanism for
defining distant proposals with high acceptance probabilities in a
Metropolis-Hastings framework, enabling more efficient exploration of the state
space than standard random-walk proposals. The popularity of such methods has
grown significantly in recent years. However, a limitation of HMC methods is
the required gradient computation for simulation of the Hamiltonian dynamical
system-such computation is infeasible in problems involving a large sample size
or streaming data. Instead, we must rely on a noisy gradient estimate computed
from a subset of the data. In this paper, we explore the properties of such a
stochastic gradient HMC approach. Surprisingly, the natural implementation of
the stochastic approximation can be arbitrarily bad. To address this problem we
introduce a variant that uses second-order Langevin dynamics with a friction
term that counteracts the effects of the noisy gradient, maintaining the
desired target distribution as the invariant distribution. Results on simulated
data validate our theory. We also provide an application of our methods to a
classification task using neural networks and to online Bayesian matrix
factorization.Comment: ICML 2014 versio
Hamiltonian structure of Hamiltonian chaos
From a kinematical point of view, the geometrical information of hamiltonian
chaos is given by the (un)stable directions, while the dynamical information is
given by the Lyapunov exponents. The finite time Lyapunov exponents are of
particular importance in physics. The spatial variations of the finite time
Lyapunov exponent and its associated (un)stable direction are related. Both of
them are found to be determined by a new hamiltonian of same number of degrees
of freedom as the original one. This new hamiltonian defines a flow field with
characteristically chaotic trajectories. The direction and the magnitude of the
phase flow field give the (un)stable direction and the finite time Lyapunov
exponent of the original hamiltonian. Our analysis was based on a
degree of freedom hamiltonian system
Real Hamiltonian forms of Hamiltonian systems
We introduce the notion of a real form of a Hamiltonian dynamical system in
analogy with the notion of real forms for simple Lie algebras. This is done by
restricting the complexified initial dynamical system to the fixed point set of
a given involution. The resulting subspace is isomorphic (but not
symplectomorphic) to the initial phase space. Thus to each real Hamiltonian
system we are able to associate another nonequivalent (real) ones. A crucial
role in this construction is played by the assumed analyticity and the
invariance of the Hamiltonian under the involution. We show that if the initial
system is Liouville integrable, then its complexification and its real forms
will be integrable again and this provides a method of finding new integrable
systems starting from known ones. We demonstrate our construction by finding
real forms of dynamics for the Toda chain and a family of Calogero--Moser
models. For these models we also show that the involution of the complexified
phase space induces a Cartan-like involution of their Lax representations.Comment: 15 pages, No figures, EPJ-style (svjour.cls
Input-output decoupling of Hamiltonian systems: The linear case
In this note we give necessary and sufficient conditions for a linear Hamiltonian system to be input-output decouplable by Hamiltonian feedback, i.e. feedback that preserves the Hamiltonian structure. In a second paper we treat the same problem for nonlinear Hamiltonian systems
Automatic Hermiticity
We study a diagonalizable Hamiltonian that is not at first hermitian.
Requirement that a measurement shall not change one Hamiltonian eigenstate into
another one with a different eigenvalue imposes that an inner product must be
defined so as to make the Hamiltonian normal with regard to it. After a long
time development with the non-hermitian Hamiltonian, only a subspace of
possible states will effectively survive. On this subspace the effect of the
anti-hermitian part of the Hamiltonian is suppressed, and the Hamiltonian
becomes hermitian. Thus hermiticity emerges automatically, and we have no
reason to maintain that at the fundamental level the Hamiltonian should be
hermitian. If the Hamiltonian is given in a local form, a conserved probability
current density can be constructed with two kinds of wave functions. We also
point out a possible misestimation of a past state by extrapolating back in
time with the hermitian Hamiltonian. It is a seeming past state, not a true
one.Comment: 8 pages, references added, typos etc. corrected, the final version to
appear in Prog.Theor.Phy
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