1,362 research outputs found
Laplace's rule of succession in information geometry
Laplace's "add-one" rule of succession modifies the observed frequencies in a
sequence of heads and tails by adding one to the observed counts. This improves
prediction by avoiding zero probabilities and corresponds to a uniform Bayesian
prior on the parameter. The canonical Jeffreys prior corresponds to the
"add-one-half" rule. We prove that, for exponential families of distributions,
such Bayesian predictors can be approximated by taking the average of the
maximum likelihood predictor and the \emph{sequential normalized maximum
likelihood} predictor from information theory. Thus in this case it is possible
to approximate Bayesian predictors without the cost of integrating or sampling
in parameter space
Parametric Fokker-Planck equation
We derive the Fokker-Planck equation on the parametric space. It is the
Wasserstein gradient flow of relative entropy on the statistical manifold. We
pull back the PDE to a finite dimensional ODE on parameter space. Some
analytical example and numerical examples are presented
Hamilton-Jacobi Theory and Information Geometry
Recently, a method to dynamically define a divergence function for a
given statistical manifold by means of the
Hamilton-Jacobi theory associated with a suitable Lagrangian function
on has been proposed. Here we will review this
construction and lay the basis for an inverse problem where we assume the
divergence function to be known and we look for a Lagrangian function
for which is a complete solution of the associated
Hamilton-Jacobi theory. To apply these ideas to quantum systems, we have to
replace probability distributions with probability amplitudes.Comment: 8 page
Magnetohydrostatic solar prominences in near-potential coronal magnetic fields
We present numerical magnetohydrostatic solutions describing the
gravitationally stratified, bulk equilibrium of cool, dense prominence plasma
embedded in a near-potential coronal field. These solutions are calculated
using the FINESSE magnetohydrodynamics equilibrium solver and describe the
morphologies of magnetic field distributions in and around prominences and the
cool prominence plasma that these fields support. The equilibrium condition for
this class of problem is usually different in distinct subdomains, separated by
free boundaries, across which solutions are matched by suitable continuity or
jump conditions describing force balance. We employ our precise finite element
elliptic solver to calculate solutions not accessible by previous analytical
techniques with temperature or entropy prescribed as free functions of the
magnetic flux function, including a range of values of the polytropic index,
temperature variations mainly across magnetic field lines and photospheric
field profiles sheared close to the polarity inversion line. Out of the many
examples computed here, perhaps the most noteworthy is one which reproduces
precisely the three-part structure often encountered in observations: a cool
dense prominence within a cavity/flux rope embedded in a hot corona. The
stability properties of these new equilibria, which may be relevant to solar
eruptions, can be determined in the form of a full resistive MHD spectrum using
a companion hyperbolic stability solver.Comment: To appear in ApJ August 200
Statistical physics of independent component analysis
Statistical physics is used to investigate independent component analysis
with polynomial contrast functions. While the replica method fails, an adapted
cavity approach yields valid results. The learning curves, obtained in a
suitable thermodynamic limit, display a first order phase transition from poor
to perfect generalization.Comment: 7 pages, 1 figure, to appear in Europhys. Lett
Riemannian Walk for Incremental Learning: Understanding Forgetting and Intransigence
Incremental learning (IL) has received a lot of attention recently, however,
the literature lacks a precise problem definition, proper evaluation settings,
and metrics tailored specifically for the IL problem. One of the main
objectives of this work is to fill these gaps so as to provide a common ground
for better understanding of IL. The main challenge for an IL algorithm is to
update the classifier whilst preserving existing knowledge. We observe that, in
addition to forgetting, a known issue while preserving knowledge, IL also
suffers from a problem we call intransigence, inability of a model to update
its knowledge. We introduce two metrics to quantify forgetting and
intransigence that allow us to understand, analyse, and gain better insights
into the behaviour of IL algorithms. We present RWalk, a generalization of
EWC++ (our efficient version of EWC [Kirkpatrick2016EWC]) and Path Integral
[Zenke2017Continual] with a theoretically grounded KL-divergence based
perspective. We provide a thorough analysis of various IL algorithms on MNIST
and CIFAR-100 datasets. In these experiments, RWalk obtains superior results in
terms of accuracy, and also provides a better trade-off between forgetting and
intransigence
Quantum state decorrelation
We address the general problem of removing correlations from quantum states
while preserving local quantum information as much as possible. We provide a
complete solution in the case of two qubits, by evaluating the minimum amount
of noise that is necessary to decorrelate covariant sets of bipartite states.
We show that two harmonic oscillators in arbitrary Gaussian state can be
decorrelated by a Gaussian covariant map. Finally, for finite-dimensional
Hilbert spaces, we prove that states obtained from most cloning channels (e.g.,
universal and phase-covariant cloning) can be decorrelated only at the expense
of a complete erasure of information about the copied state. More generally, in
finite dimension, cloning without correlations is impossible for continuous
sets of states. On the contrary, for continuos variables cloning, a slight
modification of the customary set-up for cloning coherent states allows one to
obtain clones without correlations.Comment: 11 pages, 2 figures, RevTex
SIMS chemical analysis of extended impacts on the leading and trailing edges of LDEF experiment AO187-2
Numerous 'extended impacts' found in both leading and trailing edge capture cells were successfully analyzed for the chemical composition of projectile residues by secondary ion mass spectrometry (SIMS). Most data were obtained from the trailing edge cells where 45 of 58 impacts were classified as 'probably natural' and the remainder as 'possibly man-made debris.' This is in striking contrast to leading edge cells where 9 of 11 impacts so far measured are definitely classified as orbital debris. Although all the leading edge cells had lost their plastic entrance foils during flight, the rate of foil failure was similar to that of the trailing edge cells, 10 percent of which were recovered intact. Ultraviolet embrittlement is suspected as the major cause of failure on both leading and trailing edges. The major impediment to the accurate determination of projectile chemistry is the fractionation of volatile and refractory elements in the hypervelocity impact and redeposition processes. This effect had been noted in a simulation experiment but is more pronounced in the LDEF capture cells, probably due to the higher average velocities of the space impacts. Surface contamination of the pure Ge surfaces with a substance rich in Si, but also containing Mg and Al, provides an additional problem for the accurate determination of impactor chemistry. The effect is variable, being much larger on surfaces that were exposed to space than in those cells that remained intact. Future work will concentrate on the analyses of more leading edge impacts and the development of new SIMS techniques for the measurement of elemental abundances in extended impacts
Functional Optimisation of Online Algorithms in Multilayer Neural Networks
We study the online dynamics of learning in fully connected soft committee
machines in the student-teacher scenario. The locally optimal modulation
function, which determines the learning algorithm, is obtained from a
variational argument in such a manner as to maximise the average generalisation
error decay per example. Simulations results for the resulting algorithm are
presented for a few cases. The symmetric phase plateaux are found to be vastly
reduced in comparison to those found when online backpropagation algorithms are
used. A discussion of the implementation of these ideas as practical algorithms
is given
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