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Gianluigi Pillonetto - One of the best experts on this subject based on the ideXlab platform.

  • The connection between Bayesian estimation of a Gaussian Random Field and RKHS
    IEEE Transactions on Neural Networks and Learning Systems, 2015
    Co-Authors: Aleksandr Y. Aravkin, Bradley M. Bell, James V Burke, Gianluigi Pillonetto
    Abstract:

    Reconstruction of a function from noisy data is key in machine learning and is often formulated as a regularized optimization problem over an infinite-dimensional reproducing kernel Hilbert space (RKHS). The solution suitably balances adherence to the observed data and the corresponding RKHS norm. When the data fit is measured using a quadratic loss, this estimator has a known statistical interpretation. Given the noisy measurements, the RKHS estimate represents the posterior mean (minimum variance estimate) of a Gaussian Random Field with covariance proportional to the kernel associated with the RKHS. In this brief, we provide a statistical interpretation when more general losses are used, such as absolute value, Vapnik or Huber. Specifically, for any finite set of sampling locations (that includes where the data were collected), the maximum a posteriori estimate for the signal samples is given by the RKHS estimate evaluated at the sampling locations. This connection establishes a firm statistical foundation for several stochastic approaches used to estimate unknown regularization parameters. To illustrate this, we develop a numerical scheme that implements a Bayesian estimator with an absolute value loss. This estimator is used to learn a function from measurements contaminated by outliers.

  • the connection between bayesian estimation of a Gaussian Random Field and rkhs
    arXiv: Machine Learning, 2013
    Co-Authors: Aleksandr Y. Aravkin, Bradley M. Bell, James V Burke, Gianluigi Pillonetto
    Abstract:

    Reconstruction of a function from noisy data is often formulated as a regularized optimization problem over an infinite-dimensional reproducing kernel Hilbert space (RKHS). The solution describes the observed data and has a small RKHS norm. When the data fit is measured using a quadratic loss, this estimator has a known statistical interpretation. Given the noisy measurements, the RKHS estimate represents the posterior mean (minimum variance estimate) of a Gaussian Random Field with covariance proportional to the kernel associated with the RKHS. In this paper, we provide a statistical interpretation when more general losses are used, such as absolute value, Vapnik or Huber. Specifically, for any finite set of sampling locations (including where the data were collected), the MAP estimate for the signal samples is given by the RKHS estimate evaluated at these locations.

Aleksandr Y. Aravkin - One of the best experts on this subject based on the ideXlab platform.

  • The connection between Bayesian estimation of a Gaussian Random Field and RKHS
    IEEE Transactions on Neural Networks and Learning Systems, 2015
    Co-Authors: Aleksandr Y. Aravkin, Bradley M. Bell, James V Burke, Gianluigi Pillonetto
    Abstract:

    Reconstruction of a function from noisy data is key in machine learning and is often formulated as a regularized optimization problem over an infinite-dimensional reproducing kernel Hilbert space (RKHS). The solution suitably balances adherence to the observed data and the corresponding RKHS norm. When the data fit is measured using a quadratic loss, this estimator has a known statistical interpretation. Given the noisy measurements, the RKHS estimate represents the posterior mean (minimum variance estimate) of a Gaussian Random Field with covariance proportional to the kernel associated with the RKHS. In this brief, we provide a statistical interpretation when more general losses are used, such as absolute value, Vapnik or Huber. Specifically, for any finite set of sampling locations (that includes where the data were collected), the maximum a posteriori estimate for the signal samples is given by the RKHS estimate evaluated at the sampling locations. This connection establishes a firm statistical foundation for several stochastic approaches used to estimate unknown regularization parameters. To illustrate this, we develop a numerical scheme that implements a Bayesian estimator with an absolute value loss. This estimator is used to learn a function from measurements contaminated by outliers.

  • the connection between bayesian estimation of a Gaussian Random Field and rkhs
    arXiv: Machine Learning, 2013
    Co-Authors: Aleksandr Y. Aravkin, Bradley M. Bell, James V Burke, Gianluigi Pillonetto
    Abstract:

    Reconstruction of a function from noisy data is often formulated as a regularized optimization problem over an infinite-dimensional reproducing kernel Hilbert space (RKHS). The solution describes the observed data and has a small RKHS norm. When the data fit is measured using a quadratic loss, this estimator has a known statistical interpretation. Given the noisy measurements, the RKHS estimate represents the posterior mean (minimum variance estimate) of a Gaussian Random Field with covariance proportional to the kernel associated with the RKHS. In this paper, we provide a statistical interpretation when more general losses are used, such as absolute value, Vapnik or Huber. Specifically, for any finite set of sampling locations (including where the data were collected), the MAP estimate for the signal samples is given by the RKHS estimate evaluated at these locations.

Yimin Xiao - One of the best experts on this subject based on the ideXlab platform.

  • A packing dimension theorem for Gaussian Random Fields
    Statistics & Probability Letters, 2020
    Co-Authors: Yimin Xiao
    Abstract:

    Let be a Gaussian Random Field with values in defined by where X1,...,Xd are independent copies of a centered Gaussian Random Field X0. Under certain general conditions, Xiao [Xiao, Y., 2007. Strong local nondeterminism and the sample path properties of Gaussian Random Fields. In: Lai, Tze Leung, Shao, Qiman, Qian, Lianfen (Eds.), Asymptotic Theory in Probability and Statistics with Applications. Higher Education Press, Beijing, pp. 136-176] defined an upper index [alpha]* and a lower index [alpha]* for X0 and showed that the Hausdorff dimensions of the range X([0,1]N) and graph are determined by the upper index [alpha]*. In this paper, we prove that the packing dimensions of X([0,1]N) and are determined by the lower index [alpha]* of X0. Namely, and This verifies a conjecture of Xiao in the above-cited reference. Our method is based on the potential-theoretic approach to packing dimension due to Falconer and Howroyd [Falconer, K.J., Howroyd, J.D., 1997. Packing dimensions for projections and dimension profiles. Math. Proc. Cambridge Philos. Soc. 121, 269-286].

  • Packing dimensions of the images of Gaussian Random Fields
    Statistics & Probability Letters, 2015
    Co-Authors: Yali Du, Dongsheng Wu, Junjie Miao, Yimin Xiao
    Abstract:

    Let X={X(t):t∈RN} be a Gaussian Random Field with values in Rd and let E⊆RN be a Borel set. We determine the packing dimension of the image set X(E) in terms of the packing dimension profiles in the canonical metric ρ of X, which are extensions of the packing dimension profiles of Falconer and Howroyd (1997) and the box-counting dimension profiles of Howroyd (2001).

  • tail estimation of the spectral density for a stationary Gaussian Random Field
    Journal of Multivariate Analysis, 2013
    Co-Authors: Weiying Wu, Yimin Xiao
    Abstract:

    Consider a stationary Gaussian Random Field on R^d with spectral density f(@l) that satisfies f(@l)~c|@l|^-^@q as |@l|->~. The parameters c and @q control the tail behavior of the spectral density. c is related to a microergodic parameter and @q is related to a fractal index. For data observed on a grid, we propose estimators of c and @q by minimizing an objective function, which can be viewed as a weighted local Whittle likelihood, study their properties under the fixed-domain asymptotics and provide simulation results.

  • Spectral conditions for strong local nondeterminism and exact Hausdorff measure of ranges of Gaussian Random Fields
    arXiv: Probability, 2011
    Co-Authors: Nana Luan, Yimin Xiao
    Abstract:

    Let $X= \{X(t), t \in \R^N\}$ be a Gaussian Random Field with values in $\R^d$ defined by \[ X(t) = \big(X_1(t),..., X_d(t)\big),\qquad t \in \R^N, \] where $X_1, ..., X_d$ are independent copies of a real-valued, centered, anisotropic Gaussian Random Field $X_0$ which has stationary increments and the property of strong local nondeterminism. In this paper we determine the exact Hausdorff measure function for the range $X([0, 1]^N)$. We also provide a sufficient condition for a Gaussian Random Field with stationary increments to be strongly locally nondeterministic. This condition is given in terms of the spectral measures of the Gaussian Random Fields which may contain either an absolutely continuous or discrete part. This result strengthens and extends significantly the related theorems of Berman (1973, 1988), Pitt (1978) and Xiao (2007, 2009), and will have wider applicability beyond the scope of the present paper.

  • Packing dimension results for anisotropic Gaussian Random Fields
    Communications on Stochastic Analysis, 2011
    Co-Authors: Anne Estrade, Dongsheng Wu, Yimin Xiao
    Abstract:

    Let $X=\{X(t), t \in \R^N\}$ be a Gaussian Random Field with values in $\R^d$ defined by $$X(t) = \big(X_1(t), \ldots, X_d(t)\big), \qquad \forall \ t \in \R^N, $$ where $X_1, \ldots, X_d$ are independent copies of a centered real-valued Gaussian Random Field $X_0$. We consider the case when $X_0$ is anisotropic and study the packing dimension of the range $X(E)$, where $E\subseteq \R^N$ is a Borel set. For this purpose we extend the original notion of packing dimension profile due to Falconer and Howroyd (1997) to the anisotropic metric space $(\R^N, \rho)$, where $\rho(s, t) = \sum_{j=1}^N |s_j - t_j|^{H_j}$ and $(H_1, \ldots, H_N) \in (0, 1)^N$ is a given vector. The extended notion of packing dimension profile is of independent interest.

K J Worsley - One of the best experts on this subject based on the ideXlab platform.

  • Random Field Gaussian
    Encyclopedia of Environmetrics, 2006
    Co-Authors: K J Worsley
    Abstract:

    The Gaussian Random Field Y(t), t ∈ T, is one of the most common models used to describe spatial stochastic processes. In many applications, the domain T is a subset of D-dimensional Euclidean space (usually D = 2 or D = 3), and the function Y(t) is almost surely continuous or smooth in t. The definition is simple: the Gaussian Random Field must be multivariate Gaussian at all finite sets of points, that is, [Y(t1), …, Y(tn)] must be multivariate Gaussian for all n > 0 and all tj ∈ T. Since the multivariate Gaussian is specified uniquely by its mean vector and variance matrix, then the Gaussian Random Field is defined uniquely by its mean function μ(t) = E[Y(t)] and its covariance function C(s, t) = cov[Y(s), Y(t)].

  • testing for a signal with unknown location and scale in a stationary Gaussian Random Field
    Annals of Statistics, 1995
    Co-Authors: David Siegmund, K J Worsley
    Abstract:

    We suppose that our observations can be decomposed into a fixed signal plus Random noise, where the noise is modelled as a particular stationary Gaussian Random Field in N-dimensional Euclidean space. The signal has the form of a known function centered at an unknown location and multiplied by an unknown amplitude, and we are primarily interested in a test to detect such a signal. There are many examples where the signal scale or width is assumed known, and the test is based on maximising a Gaussian Random Field over all locations in a subset of N-dimensional Euclidean space. The novel feature of this work is that the width of the signal is also unknown and the test is based on maximising a Gaussian Random Field in N + 1 dimensions, N dimensions for the location plus one dimension for the width. Two convergent approaches are used to approximate the null distribution: one based on the method of Knowles and Siegmund, which uses a version of Weyl's tube formula for manifolds with boundaries, and the other based on some recent work by Worsley, which uses the Hadwiger characteristic of excursion sets as introduced by Adler. Finally we compare the power of our method with one based on a fixed but perhaps incorrect signal width.

Bradley M. Bell - One of the best experts on this subject based on the ideXlab platform.

  • The connection between Bayesian estimation of a Gaussian Random Field and RKHS
    IEEE Transactions on Neural Networks and Learning Systems, 2015
    Co-Authors: Aleksandr Y. Aravkin, Bradley M. Bell, James V Burke, Gianluigi Pillonetto
    Abstract:

    Reconstruction of a function from noisy data is key in machine learning and is often formulated as a regularized optimization problem over an infinite-dimensional reproducing kernel Hilbert space (RKHS). The solution suitably balances adherence to the observed data and the corresponding RKHS norm. When the data fit is measured using a quadratic loss, this estimator has a known statistical interpretation. Given the noisy measurements, the RKHS estimate represents the posterior mean (minimum variance estimate) of a Gaussian Random Field with covariance proportional to the kernel associated with the RKHS. In this brief, we provide a statistical interpretation when more general losses are used, such as absolute value, Vapnik or Huber. Specifically, for any finite set of sampling locations (that includes where the data were collected), the maximum a posteriori estimate for the signal samples is given by the RKHS estimate evaluated at the sampling locations. This connection establishes a firm statistical foundation for several stochastic approaches used to estimate unknown regularization parameters. To illustrate this, we develop a numerical scheme that implements a Bayesian estimator with an absolute value loss. This estimator is used to learn a function from measurements contaminated by outliers.

  • the connection between bayesian estimation of a Gaussian Random Field and rkhs
    arXiv: Machine Learning, 2013
    Co-Authors: Aleksandr Y. Aravkin, Bradley M. Bell, James V Burke, Gianluigi Pillonetto
    Abstract:

    Reconstruction of a function from noisy data is often formulated as a regularized optimization problem over an infinite-dimensional reproducing kernel Hilbert space (RKHS). The solution describes the observed data and has a small RKHS norm. When the data fit is measured using a quadratic loss, this estimator has a known statistical interpretation. Given the noisy measurements, the RKHS estimate represents the posterior mean (minimum variance estimate) of a Gaussian Random Field with covariance proportional to the kernel associated with the RKHS. In this paper, we provide a statistical interpretation when more general losses are used, such as absolute value, Vapnik or Huber. Specifically, for any finite set of sampling locations (including where the data were collected), the MAP estimate for the signal samples is given by the RKHS estimate evaluated at these locations.