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Venkat Anantharam - One of the best experts on this subject based on the ideXlab platform.

  • one shot variable length secret key agreement approaching mutual information
    IEEE Transactions on Information Theory, 2021
    Co-Authors: Venkat Anantharam
    Abstract:

    This paper studies an information-theoretic one-shot variable-length secret key agreement problem with public discussion. Let $X$ and $Y$ be jointly distributed random variables, each taking values in some Measurable Space. Alice and Bob observe $X$ and $Y$ respectively, can communicate interactively through a public noiseless channel, and want to agree on a key length and a key that is approximately uniformly distributed over all bit sequences with the agreed key length. The public discussion is observed by an eavesdropper, Eve. The key should be approximately independent of the public discussion, conditional on the key length. We show that the optimal expected key length is close to the mutual information $I(X;Y)$ within a logarithmic gap. Moreover, an upper bound and a lower bound on the optimal expected key length can be written down in terms of $I(X;Y)$ only. This means that the optimal one-shot performance is always within a small gap of the optimal asymptotic performance regardless of the distribution of the pair $(X,Y)$ . This one-shot result may find applications in situations where the components of an i.i.d. pair source $(X^{n},Y^{n})$ are observed sequentially and the key is output bit by bit, or in situations where the random source is not an i.i.d. or ergodic process.

  • one shot variable length secret key agreement approaching mutual information
    arXiv: Information Theory, 2018
    Co-Authors: Venkat Anantharam
    Abstract:

    This paper studies an information-theoretic one-shot variable-length secret key agreement problem with public discussion. Let $X$ and $Y$ be jointly distributed random variables, each taking values in some Measurable Space. Alice and Bob observe $X$ and $Y$ respectively, can communicate interactively through a public noiseless channel, and want to agree on a key length and a key that is approximately uniformly distributed over all bit sequences with the agreed key length. The public discussion is observed by an eavesdropper, Eve. The key should be approximately independent of the public discussion, conditional on the key length. We show that the optimal expected key length is close to the mutual information $I(X;Y)$ within a logarithmic gap. Moreover, an upper bound and a lower bound on the optimal expected key length can be written down in terms of $I(X;Y)$ only. This means that the optimal one-shot performance is always within a small gap of the optimal asymptotic performance regardless of the distribution of the pair $(X,Y)$. This one-shot result may find applications in situations where the components of an i.i.d. pair source $(X^{n},Y^{n})$ are observed sequentially and the key is output bit by bit with small delay, or in situations where the random source is not an i.i.d. or ergodic process.

  • a variational characterization of renyi divergences
    International Symposium on Information Theory, 2017
    Co-Authors: Venkat Anantharam
    Abstract:

    We present a variational characterization of the Renyi divergences between any two probability distributions on an arbitrary Measurable Space, in terms of relative entropies. This yields as a corollary a recently developed variational formula, due to Atar, Chowdhary and Dupuis, for exponential integrals of bounded Measurable functions in terms of Renyi divergences. We then develop a similar variational characterization of the Renyi divergence rates between two stationary finite state Markov chains in terms of relative entropy rates. This leads to an analog of the variational formula of Atar, Chowdhary and Dupuis in the framework of stationary finite state Markov chains.

Ledoux James - One of the best experts on this subject based on the ideXlab platform.

  • State-discretization of V -geometrically ergodic Markov chains and convergence to the stationary distribution
    Springer Verlag, 2020
    Co-Authors: Hervé Loïc, Ledoux James
    Abstract:

    International audienceLet $(X_n)_{n \in\mathbb{N}}$ be a $V$-geometrically ergodic Markov chain on a Measurable Space $\mathbb{X}$ with invariant probability distribution $\pi$. In this paper, we propose a discretization scheme providing a computable sequence $(\widehat\pi_k)_{k\ge 1}$ of probability measures which approximates $\pi$ as $k$ growths to infinity. The probability measure $\widehat\pi_k$ is computed from the invariant probability distribution of a finite Markov chain. The convergence rate in total variation of $(\widehat\pi_k)_{k\ge 1}$ to $\pi$ is given. As a result, the specific case of first order autoregressive processes with linear and non-linear errors is studied. Finally, illustrations of the procedure for such autoregressive processes are provided, in particular when no explicit formula for $\pi$ is known

  • State-discretization of $V$-geometrically ergodic Markov chains and convergence to the stationary distribution
    2019
    Co-Authors: Hervé Loïc, Ledoux James
    Abstract:

    Let $(X_n)_{n \in\mathbb{N}}$ be a $V$-geometrically ergodic Markov chain on a Measurable Space $\mathbb{X}$ with invariant probability distribution $\pi$. In this paper, we propose a discretization scheme providing a computable sequence $(\widehat\pi_k)_{k\ge 1}$ of probability measures which approximates $\pi$ as $k$ growths to infinity. The probability measure $\widehat\pi_k$ is computed from the invariant probability distribution of a finite Markov chain. The convergence rate in total variation of $(\widehat\pi_k)_{k\ge 1}$ to $\pi$ is given. As a result, the specific case of first order autoregressive processes with linear and non-linear errors is studied. Finally, illustrations of the procedure for such autoregressive processes are provided, in particular when no explicit formula for $\pi$ is known.Comment: Submitted 22 november 201

  • Approximating Markov chains and V-geometric ergodicity via weak perturbation theory
    'Elsevier BV', 2014
    Co-Authors: Hervé Loïc, Ledoux James
    Abstract:

    International audienceLet $P$ be a Markov kernel on a Measurable Space $\mathbb{X}$ and let $V:\mathbb{X}\rightarrow [1,+\infty)$. This paper provides explicit connections between the $V$-geometric ergodicity of $P$ and that of finite-rank nonnegative sub-Markov kernels $\widehat{P}_k$ approximating $P$. A special attention is paid to obtain an efficient way to specify the convergence rate for $P$ from that of $\widehat{P}_k$ and conversely. Furthermore, explicit bounds are obtained for the total variation distance between the $P$-invariant probability measure and the $\widehat{P}_k$-invariant positive measure. The proofs are based on the Keller-Liverani perturbation theorem which requires an accurate control of the essential spectral radius of $P$ on usual weighted supremum Spaces. Such computable bounds are derived in terms of standard drift conditions. Our spectral procedure to estimate both the convergence rate and the invariant probability measure of $P$ is applied to truncation of discrete Markov kernels on $\mathbb{X}:=\mathbb{N}$

  • Approximating Markov chains and V-geometric ergodicity via weak perturbation theory
    'Elsevier BV', 2014
    Co-Authors: Hervé Loïc, Ledoux James
    Abstract:

    Let $P$ be a Markov kernel on a Measurable Space $\X$ and let $V:\X\r[1,+\infty)$. This paper provides explicit connections between the $V$-geometric ergodicity of $P$ and that of finite-rank nonnegative sub-Markov kernels $\Pc_k$ approximating $P$. A special attention is paid to obtain an efficient way to specify the convergence rate for $P$ from that of $\Pc_k$ and conversely. Furthermore, explicit bounds are obtained for the total variation distance between the $P$-invariant probability measure and the $\Pc_k$-invariant positive measure. The proofs are based on the Keller-Liverani perturbation theorem which requires an accurate control of the essential spectral radius of $P$ on usual weighted supremum Spaces. Such computable bounds are derived in terms of standard drift conditions. Our spectral procedure to estimate both the convergence rate and the invariant probability measure of $P$ is applied to truncation of discrete Markov kernels on $\X:=\N$

  • Quasi-compactness of Markov kernels on weighted-supremum Spaces and geometrical ergodicity
    2012
    Co-Authors: Guibourg Denis, Hervé Loïc, Ledoux James
    Abstract:

    Let $P$ be a Markov kernel on a Measurable Space $\X$ and let $V:\X\r[1,+\infty)$. We provide various assumptions, based on drift conditions, under which $P$ is quasi-compact on the weighted-supremum Banach Space $(\cB_V,\|\cdot\|_V)$ of all the Measurable functions $f : \X\r\C$ such that $\|f\|_V := \sup_{x\in \X} |f(x)|/V(x) < \infty$. Furthermore we give bounds for the essential spectral radius of $P$. Under additional assumptions, these results allow us to derive the convergence rate of $P$ on $\cB_V$, that is the geometric rate of convergence of the iterates $P^n$ to the stationary distribution in operator norm. Applications to discrete Markov kernels and to iterated function systems are presented.Comment: 45 page

Mathew D Penrose - One of the best experts on this subject based on the ideXlab platform.

  • poisson process fock Space representation chaos expansion and covariance inequalities
    Probability Theory and Related Fields, 2011
    Co-Authors: Mathew D Penrose
    Abstract:

    We consider a Poisson process η on an arbitrary Measurable Space with an arbitrary sigma-finite intensity measure. We establish an explicit Fock Space representation of square integrable functions of η. As a consequence we identify explicitly, in terms of iterated difference operators, the integrands in the Wiener–Ito chaos expansion. We apply these results to extend well-known variance inequalities for homogeneous Poisson processes on the line to the general Poisson case. The Poincare inequality is a special case. Further applications are covariance identities for Poisson processes on (strictly) ordered Spaces and Harris–FKG-inequalities for monotone functions of η.

  • poisson process fock Space representation chaos expansion and covariance inequalities
    arXiv: Probability, 2009
    Co-Authors: Mathew D Penrose
    Abstract:

    We consider a Poisson process $\eta$ on an arbitrary Measurable Space with an arbitrary sigma-finite intensity measure. We establish an explicit Fock Space representation of square integrable functions of $\eta$. As a consequence we identify explicitly, in terms of iterated difference operators, the integrands in the Wiener-Ito chaos expansion. We apply these results to extend well-known variance inequalities for homogeneous Poisson processes on the line to the general Poisson case. The Poincare inequality is a special case. Further applications are covariance identities for Poisson processes on (strictly) ordered Spaces and Harris-FKG-inequalities for monotone functions of $\eta$.

Hervé Loïc - One of the best experts on this subject based on the ideXlab platform.

  • State-discretization of V -geometrically ergodic Markov chains and convergence to the stationary distribution
    Springer Verlag, 2020
    Co-Authors: Hervé Loïc, Ledoux James
    Abstract:

    International audienceLet $(X_n)_{n \in\mathbb{N}}$ be a $V$-geometrically ergodic Markov chain on a Measurable Space $\mathbb{X}$ with invariant probability distribution $\pi$. In this paper, we propose a discretization scheme providing a computable sequence $(\widehat\pi_k)_{k\ge 1}$ of probability measures which approximates $\pi$ as $k$ growths to infinity. The probability measure $\widehat\pi_k$ is computed from the invariant probability distribution of a finite Markov chain. The convergence rate in total variation of $(\widehat\pi_k)_{k\ge 1}$ to $\pi$ is given. As a result, the specific case of first order autoregressive processes with linear and non-linear errors is studied. Finally, illustrations of the procedure for such autoregressive processes are provided, in particular when no explicit formula for $\pi$ is known

  • State-discretization of $V$-geometrically ergodic Markov chains and convergence to the stationary distribution
    2019
    Co-Authors: Hervé Loïc, Ledoux James
    Abstract:

    Let $(X_n)_{n \in\mathbb{N}}$ be a $V$-geometrically ergodic Markov chain on a Measurable Space $\mathbb{X}$ with invariant probability distribution $\pi$. In this paper, we propose a discretization scheme providing a computable sequence $(\widehat\pi_k)_{k\ge 1}$ of probability measures which approximates $\pi$ as $k$ growths to infinity. The probability measure $\widehat\pi_k$ is computed from the invariant probability distribution of a finite Markov chain. The convergence rate in total variation of $(\widehat\pi_k)_{k\ge 1}$ to $\pi$ is given. As a result, the specific case of first order autoregressive processes with linear and non-linear errors is studied. Finally, illustrations of the procedure for such autoregressive processes are provided, in particular when no explicit formula for $\pi$ is known.Comment: Submitted 22 november 201

  • Approximating Markov chains and V-geometric ergodicity via weak perturbation theory
    'Elsevier BV', 2014
    Co-Authors: Hervé Loïc, Ledoux James
    Abstract:

    International audienceLet $P$ be a Markov kernel on a Measurable Space $\mathbb{X}$ and let $V:\mathbb{X}\rightarrow [1,+\infty)$. This paper provides explicit connections between the $V$-geometric ergodicity of $P$ and that of finite-rank nonnegative sub-Markov kernels $\widehat{P}_k$ approximating $P$. A special attention is paid to obtain an efficient way to specify the convergence rate for $P$ from that of $\widehat{P}_k$ and conversely. Furthermore, explicit bounds are obtained for the total variation distance between the $P$-invariant probability measure and the $\widehat{P}_k$-invariant positive measure. The proofs are based on the Keller-Liverani perturbation theorem which requires an accurate control of the essential spectral radius of $P$ on usual weighted supremum Spaces. Such computable bounds are derived in terms of standard drift conditions. Our spectral procedure to estimate both the convergence rate and the invariant probability measure of $P$ is applied to truncation of discrete Markov kernels on $\mathbb{X}:=\mathbb{N}$

  • Approximating Markov chains and V-geometric ergodicity via weak perturbation theory
    'Elsevier BV', 2014
    Co-Authors: Hervé Loïc, Ledoux James
    Abstract:

    Let $P$ be a Markov kernel on a Measurable Space $\X$ and let $V:\X\r[1,+\infty)$. This paper provides explicit connections between the $V$-geometric ergodicity of $P$ and that of finite-rank nonnegative sub-Markov kernels $\Pc_k$ approximating $P$. A special attention is paid to obtain an efficient way to specify the convergence rate for $P$ from that of $\Pc_k$ and conversely. Furthermore, explicit bounds are obtained for the total variation distance between the $P$-invariant probability measure and the $\Pc_k$-invariant positive measure. The proofs are based on the Keller-Liverani perturbation theorem which requires an accurate control of the essential spectral radius of $P$ on usual weighted supremum Spaces. Such computable bounds are derived in terms of standard drift conditions. Our spectral procedure to estimate both the convergence rate and the invariant probability measure of $P$ is applied to truncation of discrete Markov kernels on $\X:=\N$

  • Quasi-compactness of Markov kernels on weighted-supremum Spaces and geometrical ergodicity
    2012
    Co-Authors: Guibourg Denis, Hervé Loïc, Ledoux James
    Abstract:

    Let $P$ be a Markov kernel on a Measurable Space $\X$ and let $V:\X\r[1,+\infty)$. We provide various assumptions, based on drift conditions, under which $P$ is quasi-compact on the weighted-supremum Banach Space $(\cB_V,\|\cdot\|_V)$ of all the Measurable functions $f : \X\r\C$ such that $\|f\|_V := \sup_{x\in \X} |f(x)|/V(x) < \infty$. Furthermore we give bounds for the essential spectral radius of $P$. Under additional assumptions, these results allow us to derive the convergence rate of $P$ on $\cB_V$, that is the geometric rate of convergence of the iterates $P^n$ to the stationary distribution in operator norm. Applications to discrete Markov kernels and to iterated function systems are presented.Comment: 45 page

Nigel J Newton - One of the best experts on this subject based on the ideXlab platform.

  • an infinite dimensional statistical manifold modelled on hilbert Space
    Journal of Functional Analysis, 2012
    Co-Authors: Nigel J Newton
    Abstract:

    Abstract We construct an infinite-dimensional Hilbert manifold of probability measures on an abstract Measurable Space. The manifold, M, retains the first- and second-order features of finite-dimensional information geometry: the α-divergences admit first derivatives and mixed second derivatives, enabling the definition of the Fisher metric as a pseudo-Riemannian metric. This is enough for many applications; for example, it justifies certain projections of Markov processes onto finite-dimensional submanifolds in recursive estimation problems. M was constructed with the Fenchel–Legendre transform between Kullback–Leibler divergences, and its role in Bayesian estimation, in mind. This transform retains, on M, the symmetry of the finite-dimensional case. Many of the manifolds of finite-dimensional information geometry are shown to be C ∞ -embedded submanifolds of M. In establishing this, we provide a framework in which many of the formal results of the finite-dimensional subject can be proved with full rigour.