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## Information geometric similarity measurement for near-random stochastic processes

Fonte: Universidade Federal do Rio Grande do Sul
Publicador: Universidade Federal do Rio Grande do Sul

Tipo: Artigo de Revista Científica
Formato: application/pdf

ENG

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#Gamma models#Matemática#Information geometry#Multisymbol sequences#Random#Search#Stochastic process

We outline the information-theoretic differential geometry of gamma distributions, which contain exponential distributions as a special case, and log-gamma distributions. Our arguments support the opinion that these distributions have a natural role in representing departures from randomness, uniformity, and Gaussian behavior in stochastic processes. We show also how the information geometry provides a surprisingly tractable Riemannian manifold and product spaces thereof, on which may be represented the evolution of a stochastic process, or the comparison of different processes, by means of well-founded maximum likelihood parameter estimation. Our model incorporates possible correlations among parameters. We discuss applications and provide some illustrations from a recent study of amino acid self-clustering in protein sequences; we provide also some results from simulations for multisymbol sequences.

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## Information Geometry for Landmark Shape Analysis: Unifying Shape Representation and Deformation

Fonte: PubMed
Publicador: PubMed

Tipo: Artigo de Revista Científica

Publicado em /02/2009
EN

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Shape matching plays a prominent role in the comparison of similar structures. We present a unifying framework for shape matching that uses mixture models to couple both the shape representation and deformation. The theoretical foundation is drawn from information geometry wherein information matrices are used to establish intrinsic distances between parametric densities. When a parameterized probability density function is used to represent a landmark-based shape, the modes of deformation are automatically established through the information matrix of the density. We first show that given two shapes parameterized by Gaussian mixture models (GMMs), the well-known Fisher information matrix of the mixture model is also a Riemannian metric (actually, the Fisher-Rao Riemannian metric) and can therefore be used for computing shape geodesics. The Fisher-Rao metric has the advantage of being an intrinsic metric and invariant to reparameterization. The geodesic—computed using this metric—establishes an intrinsic deformation between the shapes, thus unifying both shape representation and deformation. A fundamental drawback of the Fisher-Rao metric is that it is not available in closed form for the GMM. Consequently, shape comparisons are computationally very expensive. To address this...

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## Geometria da informação : métrica de Fisher; Information geometry : Fisher's metric

Fonte: Biblioteca Digital da Unicamp
Publicador: Biblioteca Digital da Unicamp

Tipo: Dissertação de Mestrado
Formato: application/pdf

Publicado em 23/08/2013
PT

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#Geometria da informação#Matriz de informação de Fisher#Distância de Rao#Geometria diferencial#Estatística matemática#Information geometry#Fisher information metric#Rao distance#Differential geometry#Mathematical statistics

A Geometria da Informação é uma área da matemática que utiliza ferramentas geométricas no estudo de modelos estatísticos. Em 1945, Rao introduziu uma métrica Riemanniana no espaço das distribuições de probabilidade usando a matriz de informação, dada por Ronald Fisher em 1921. Com a métrica associada a essa matriz, define-se uma distância entre duas distribuições de probabilidade (distância de Rao), geodésicas, curvaturas e outras propriedades do espaço. Desde então muitos autores veem estudando esse assunto, que está naturalmente ligado a diversas aplicações como, por exemplo, inferência estatística, processos estocásticos, teoria da informação e distorção de imagens. Neste trabalho damos uma breve introdução à geometria diferencial e Riemanniana e fazemos uma coletânea de alguns resultados obtidos na área de Geometria da Informação. Mostramos a distância de Rao entre algumas distribuições de probabilidade e damos uma atenção especial ao estudo da distância no espaço formado por distribuições Normais Multivariadas. Neste espaço, como ainda não é conhecida uma fórmula fechada para a distância e nem para a curva geodésica, damos ênfase ao cálculo de limitantes para a distância de Rao. Conseguimos melhorar...

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## Information geometry, dynamics and discrete quantum mechanics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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We consider a system with a discrete configuration space. We show that the
geometrical structures associated with such a system provide the tools
necessary for a reconstruction of discrete quantum mechanics once dynamics is
brought into the picture. We do this in three steps. Our starting point is
information geometry, the natural geometry of the space of probability
distributions. Dynamics requires additional structure. To evolve the
probabilities $P^k$, we introduce coordinates $S^k$ canonically conjugate to
the $P^k$ and a symplectic structure. We then seek to extend the metric
structure of information geometry, to define a geometry over the full space of
the $P^k$ and $S^k$. Consistency between the metric tensor and the symplectic
form forces us to introduce a K\"ahler geometry. The construction has notable
features. A complex structure is obtained in a natural way. The canonical
coordinates of the K\"ahler space are precisely the wave functions of quantum
mechanics. The full group of unitary transformations is obtained. Finally, one
may associate a Hilbert space with the K\"ahler space, which leads to the
standard version of quantum theory. We also show that the metric that we derive
here using purely geometrical arguments is precisely the one that leads to
Wootters' expression for the statistical distance for quantum systems.; Comment: 12 pages. Presented at MaxEnt 2012...

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## Information geometry and sufficient statistics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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Information geometry provides a geometric approach to families of statistical
models. The key geometric structures are the Fisher quadratic form and the
Amari-Chentsov tensor. In statistics, the notion of sufficient statistic
expresses the criterion for passing from one model to another without loss of
information. This leads to the question how the geometric structures behave
under such sufficient statistics. While this is well studied in the finite
sample size case, in the infinite case, we encounter technical problems
concerning the appropriate topologies. Here, we introduce notions of
parametrized measure models and tensor fields on them that exhibit the right
behavior under statistical transformations. Within this framework, we can then
handle the topological issues and show that the Fisher metric and the
Amari-Chentsov tensor on statistical models in the class of symmetric 2-tensor
fields and 3-tensor fields can be uniquely (up to a constant) characterized by
their invariance under sufficient statistics, thereby achieving a full
generalization of the original result of Chentsov to infinite sample sizes.
More generally, we decompose Markov morphisms between statistical models in
terms of statistics. In particular, a monotonicity result for the Fisher
information naturally follows.; Comment: 37 p...

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## Transversely Hessian foliations and information geometry

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 29/03/2015

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A family of probability distributions parametrized by an open domain
$\Lambda$ in $R^n$ defines the Fisher information matrix on this domain which
is positive semi-definite. In information geometry the standard assumption has
been that the Fisher information matrix tensor is positive definite defining in
this way a Riemannian metric on $\Lambda$. If we replace the "positive
definite" assumption by the existence of a suitable torsion-free connection, a
foliation with a transversely Hessian structure appears naturally. In the paper
we develop the study of transversely Hessian foliations in view of applications
in information geometry.

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## Information Geometry and Statistical Manifold

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 08/10/2014

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We review basic notions in the field of information geometry such as Fisher
metric on statistical manifold, $\alpha$-connection and corresponding curvature
following Amari's work . We show application of information geometry to
asymptotic statistical inference.

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## The Information Geometry of Mirror Descent

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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Information geometry applies concepts in differential geometry to probability
and statistics and is especially useful for parameter estimation in exponential
families where parameters are known to lie on a Riemannian manifold.
Connections between the geometric properties of the induced manifold and
statistical properties of the estimation problem are well-established. However
developing first-order methods that scale to larger problems has been less of a
focus in the information geometry community. The best known algorithm that
incorporates manifold structure is the second-order natural gradient descent
algorithm introduced by Amari. On the other hand, stochastic approximation
methods have led to the development of first-order methods for optimizing noisy
objective functions. A recent generalization of the Robbins-Monro algorithm
known as mirror descent, developed by Nemirovski and Yudin is a first order
method that induces non-Euclidean geometries. However current analysis of
mirror descent does not precisely characterize the induced non-Euclidean
geometry nor does it consider performance in terms of statistical relative
efficiency. In this paper, we prove that mirror descent induced by Bregman
divergences is equivalent to the natural gradient descent algorithm on the dual
Riemannian manifold. Using this equivalence...

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## Information Geometry and Evolutionary Game Theory

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 09/11/2009

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#Computer Science - Information Theory#Computer Science - Computer Science and Game Theory#Mathematics - Dynamical Systems#Nonlinear Sciences - Adaptation and Self-Organizing Systems

The Shahshahani geometry of evolutionary game theory is realized as the
information geometry of the simplex, deriving from the Fisher information
metric of the manifold of categorical probability distributions. Some essential
concepts in evolutionary game theory are realized information-theoretically.
Results are extended to the Lotka-Volterra equation and to multiple population
systems.; Comment: Added references

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## Extension of information geometry for modelling non-statistical systems

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 05/01/2015

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In this dissertation, an abstract formalism extending information geometry is
introduced. This framework encompasses a broad range of modelling problems,
including possible applications in machine learning and in the information
theoretical foundations of quantum theory. Its purely geometrical foundations
make no use of probability theory and very little assumptions about the data or
the models are made. Starting only from a divergence function, a Riemannian
geometrical structure consisting of a metric tensor and an affine connection is
constructed and its properties are investigated. Also the relation to
information geometry and in particular the geometry of exponential families of
probability distributions is elucidated. It turns out this geometrical
framework offers a straightforward way to determine whether or not a
parametrised family of distributions can be written in exponential form. Apart
from the main theoretical chapter, the dissertation also contains a chapter of
examples illustrating the application of the formalism and its geometric
properties, a brief introduction to differential geometry and a historical
overview of the development of information geometry.; Comment: PhD thesis, University of Antwerp, Advisors: Prof. dr. Jan Naudts and
Prof. dr. Jacques Tempere...

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## Notes on information geometry and evolutionary processes

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 20/08/2004

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#Nonlinear Sciences - Adaptation and Self-Organizing Systems#Computer Science - Neural and Evolutionary Computing

In order to analyze and extract different structural properties of
distributions, one can introduce different coordinate systems over the manifold
of distributions. In Evolutionary Computation, the Walsh bases and the Building
Block Bases are often used to describe populations, which simplifies the
analysis of evolutionary operators applying on populations. Quite independent
from these approaches, information geometry has been developed as a geometric
way to analyze different order dependencies between random variables (e.g.,
neural activations or genes).
In these notes I briefly review the essentials of various coordinate bases
and of information geometry. The goal is to give an overview and make the
approaches comparable. Besides introducing meaningful coordinate bases,
information geometry also offers an explicit way to distinguish different order
interactions and it offers a geometric view on the manifold and thereby also on
operators that apply on the manifold. For instance, uniform crossover can be
interpreted as an orthogonal projection of a population along an m-geodesic,
monotonously reducing the theta-coordinates that describe interactions between
genes.

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## Information geometry in vapour-liquid equilibrium

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 06/09/2008

Relevância na Pesquisa

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Using the square-root map p-->\sqrt{p} a probability density function p can
be represented as a point of the unit sphere S in the Hilbert space of
square-integrable functions. If the density function depends smoothly on a set
of parameters, the image of the map forms a Riemannian submanifold M in S. The
metric on M induced by the ambient spherical geometry of S is the Fisher
information matrix. Statistical properties of the system modelled by a
parametric density function p can then be expressed in terms of information
geometry. An elementary introduction to information geometry is presented,
followed by a precise geometric characterisation of the family of Gaussian
density functions. When the parametric density function describes the
equilibrium state of a physical system, certain physical characteristics can be
identified with geometric features of the associated information manifold M.
Applying this idea, the properties of vapour-liquid phase transitions are
elucidated in geometrical terms. For an ideal gas, phase transitions are absent
and the geometry of M is flat. In this case, the solutions to the geodesic
equations yield the adiabatic equations of state. For a van der Waals gas, the
associated geometry of M is highly nontrivial. The scalar curvature of M
diverges along the spinodal boundary which envelopes the unphysical region in
the phase diagram. The curvature is thus closely related to the stability of
the system.; Comment: A short survey article. 38 Pages...

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## K\"ahlerian information geometry for signal processing

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Mathematics - Differential Geometry#Computer Science - Information Theory#Computer Science - Systems and Control#Mathematics - Statistics Theory

We prove the correspondence between the information geometry of a signal
filter and a K\"ahler manifold. The information geometry of a minimum-phase
linear system with a finite complex cepstrum norm is a K\"ahler manifold. The
square of the complex cepstrum norm of the signal filter corresponds to the
K\"ahler potential. The Hermitian structure of the K\"ahler manifold is
explicitly emergent if and only if the impulse response function of the highest
degree in $z$ is constant in model parameters. The K\"ahlerian information
geometry takes advantage of more efficient calculation steps for the metric
tensor and the Ricci tensor. Moreover, $\alpha$-generalization on the geometric
tensors is linear in $\alpha$. It is also robust to find Bayesian predictive
priors, such as superharmonic priors, because Laplace-Beltrami operators on
K\"ahler manifolds are in much simpler forms than those of the non-K\"ahler
manifolds. Several time series models are studied in the K\"ahlerian
information geometry.; Comment: 24 pages, published version

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## Critical phenomena and information geometry in black hole physics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 13/01/2010

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We discuss the use of information geometry in black hole physics and present
the outcomes. The type of information geometry we utilize in this approach is
the thermodynamic (Ruppeiner) geometry defined on the state space of a given
thermodynamic system in equilibrium. The Ruppeiner geometry can be used to
analyze stability and critical phenomena in black hole physics with results
consistent with those from the Poincare stability analysis for black holes and
black rings. Furthermore other physical phenomena are well encoded in the
Ruppeiner metric such as the sign of specific heat and the extremality of the
solutions. The black hole families we discuss in particular in this manuscript
are the Myers-Perry black holes.; Comment: Contribution to ERE2009, 5 pages

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## From information to quanta: A derivation of the geometric formulation of quantum theory from information geometry

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 02/12/2013

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It is shown that the geometry of quantum theory can be derived from
geometrical structure that may be considered more fundamental. The basic
elements of this reconstruction of quantum theory are the natural metric on the
space of probabilities (information geometry), the description of dynamics
using a Hamiltonian formalism (symplectic geometry), and requirements of
consistency (K\"{a}hler geometry). The theory that results is standard quantum
mechanics, but in a geometrical formulation that includes also a particular
case of a family of nonlinear gauge transformations introduced by Doebner and
Goldin. The analysis is carried out for the case of discrete quantum mechanics.
The work presented here relies heavily on, and extends, previous work done in
collaboration with M. J. W. Hall.; Comment: 18 pages. Presented at Symmetries in Science XVI, Bregenz, Austria,
July 21-26, 2013

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## Sketching, Embedding, and Dimensionality Reduction for Information Spaces

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 17/03/2015

Relevância na Pesquisa

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#Computer Science - Data Structures and Algorithms#Computer Science - Computational Geometry#Computer Science - Information Theory

Information distances like the Hellinger distance and the Jensen-Shannon
divergence have deep roots in information theory and machine learning. They are
used extensively in data analysis especially when the objects being compared
are high dimensional empirical probability distributions built from data.
However, we lack common tools needed to actually use information distances in
applications efficiently and at scale with any kind of provable guarantees. We
can't sketch these distances easily, or embed them in better behaved spaces, or
even reduce the dimensionality of the space while maintaining the probability
structure of the data.
In this paper, we build these tools for information distances---both for the
Hellinger distance and Jensen--Shannon divergence, as well as related measures,
like the $\chi^2$ divergence. We first show that they can be sketched
efficiently (i.e. up to multiplicative error in sublinear space) in the
aggregate streaming model. This result is exponentially stronger than known
upper bounds for sketching these distances in the strict turnstile streaming
model. Second, we show a finite dimensionality embedding result for the
Jensen-Shannon and $\chi^2$ divergences that preserves pair wise distances.
Finally we prove a dimensionality reduction result for the Hellinger...

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## A unified framework for information integration based on information geometry

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 15/10/2015

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55.86%

We propose a unified theoretical framework for quantifying spatio-temporal
interactions in a stochastic dynamical system based on information geometry. In
the proposed framework, the degree of interactions is quantified by the
divergence between the actual probability distribution of the system and a
constrained probability distribution where the interactions of interest are
disconnected. This framework provides novel geometric interpretations of
various information theoretic measures of interactions, such as mutual
information, transfer entropy, and stochastic interaction in terms of how
interactions are disconnected. The framework therefore provides an intuitive
understanding of the relationships between the various quantities. By extending
the concept of transfer entropy, we propose a novel measure of integrated
information which measures causal interactions between parts of a system.
Integrated information quantifies the extent to which the whole is more than
the sum of the parts and can be potentially used as a biological measure of the
levels of consciousness.

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## Mapping the Region of Entropic Vectors with Support Enumeration & Information Geometry

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 10/12/2015

Relevância na Pesquisa

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The region of entropic vectors is a convex cone that has been shown to be at
the core of many fundamental limits for problems in multiterminal data
compression, network coding, and multimedia transmission. This cone has been
shown to be non-polyhedral for four or more random variables, however its
boundary remains unknown for four or more discrete random variables. Methods
for specifying probability distributions that are in faces and on the boundary
of the convex cone are derived, then utilized to map optimized inner bounds to
the unknown part of the entropy region. The first method utilizes tools and
algorithms from abstract algebra to efficiently determine those supports for
the joint probability mass functions for four or more random variables that
can, for some appropriate set of non-zero probabilities, yield entropic vectors
in the gap between the best known inner and outer bounds. These supports are
utilized, together with numerical optimization over non-zero probabilities, to
provide inner bounds to the unknown part of the entropy region. Next,
information geometry is utilized to parameterize and study the structure of
probability distributions on these supports yielding entropic vectors in the
faces of entropy and in the unknown part of the entropy region.

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## Loop Calculus for Non-Binary Alphabets using Concepts from Information Geometry

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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The Bethe approximation is a well-known approximation of the partition
function used in statistical physics. Recently, an equality relating the
partition function and its Bethe approximation was obtained for graphical
models with binary variables by Chertkov and Chernyak. In this equality, the
multiplicative error in the Bethe approximation is represented as a weighted
sum over all generalized loops in the graphical model. In this paper, the
equality is generalized to graphical models with non-binary alphabet using
concepts from information geometry.; Comment: 18 pages, 4 figures, submitted to IEEE Trans. Inf. Theory

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## Information geometry and the hydrodynamical formulation of quantum mechanics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 03/04/2012

Relevância na Pesquisa

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Let (M,g) be a compact, connected and oriented Riemannian manifold. We denote
D the space of smooth probability density functions on M.
In this paper, we show that the Frechet manifold D is equipped with a
Riemannian metric g^{D} and an affine connection \nabla^{D} which are infinite
dimensional analogues of the Fisher metric and exponential connection in the
context of information geometry. More precisely, we use Dombrowski's
construction together with the couple (g^{D},\nabla^{D}) to get a
(non-integrable) almost Hermitian structure on D, and we show that the
corresponding fundamental 2-form is a symplectic form from which it is possible
to recover the usual Schrodinger equation for a quantum particle living in M.
These results echo a recent paper of the author where it is stressed that the
Fisher metric and exponential connection are related (via Dombrowski's
construction) to Kahler geometry and quantum mechanics in finite dimension.

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