The Experts below are selected from a list of 360 Experts worldwide ranked by ideXlab platform
Kannan Ramchandran - One of the best experts on this subject based on the ideXlab platform.
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information theoretic limits on sparse signal recovery dense versus sparse measurement matrices
IEEE Transactions on Information Theory, 2010Co-Authors: Wei Wang, Martin J Wainwright, Kannan RamchandranAbstract:We study the information-theoretic limits of exactly recovering the support set of a sparse signal, using noisy projections defined by various classes of measurement matrices. Our analysis is high-dimensional in nature, in which the number of observations n, the ambient signal dimension p, and the signal sparsity k are all allowed to tend to infinity in a general manner. This paper makes two novel contributions. First, we provide sharper necessary conditions for exact support recovery using general (including non-Gaussian) dense measurement matrices. Combined with previously known sufficient conditions, this result yields sharp characterizations of when the optimal decoder can recover a signal for various scalings of the signal sparsity k and sample size n, including the Important Special Case of linear sparsity (k = ?(p)) using a linear scaling of observations (n = ?(p)). Our second contribution is to prove necessary conditions on the number of observations n required for asymptotically reliable recovery using a class of ?-sparsified measurement matrices, where the measurement sparsity parameter ?(n, p, k) ? (0,1] corresponds to the fraction of nonzero entries per row. Our analysis allows general scaling of the quadruplet (n, p, k, ?) , and reveals three different regimes, corresponding to whether measurement sparsity has no asymptotic effect, a minor effect, or a dramatic effect on the information-theoretic limits of the subset recovery problem.
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information theoretic limits on sparse signal recovery dense versus sparse measurement matrices
arXiv: Statistics Theory, 2008Co-Authors: Wei Wang, Martin J Wainwright, Kannan RamchandranAbstract:We study the information-theoretic limits of exactly recovering the support of a sparse signal using noisy projections defined by various classes of measurement matrices. Our analysis is high-dimensional in nature, in which the number of observations $n$, the ambient signal dimension $p$, and the signal sparsity $k$ are all allowed to tend to infinity in a general manner. This paper makes two novel contributions. First, we provide sharper necessary conditions for exact support recovery using general (non-Gaussian) dense measurement matrices. Combined with previously known sufficient conditions, this result yields sharp characterizations of when the optimal decoder can recover a signal for various scalings of the sparsity $k$ and sample size $n$, including the Important Special Case of linear sparsity ($k = \Theta(p)$) using a linear scaling of observations ($n = \Theta(p)$). Our second contribution is to prove necessary conditions on the number of observations $n$ required for asymptotically reliable recovery using a class of $\gamma$-sparsified measurement matrices, where the measurement sparsity $\gamma(n, p, k) \in (0,1]$ corresponds to the fraction of non-zero entries per row. Our analysis allows general scaling of the quadruplet $(n, p, k, \gamma)$, and reveals three different regimes, corresponding to whether measurement sparsity has no effect, a minor effect, or a dramatic effect on the information-theoretic limits of the subset recovery problem.
Gregory A Fiete - One of the best experts on this subject based on the ideXlab platform.
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analog of hamilton jacobi theory for the time evolution operator
Physical Review A, 2019Co-Authors: Michael Vogl, Pontus Laurell, Aaron Barr, Gregory A FieteAbstract:In this paper we develop an analogue of Hamilton-Jacobi theory for the time-evolution operator of a quantum many-particle system. The theory offers a useful approach to develop approximations to the time-evolution operator, and also provides a unified framework and starting point for many well-known approximations to the time-evolution operator. In the Important Special Case of periodically driven systems at stroboscopic times, we find relatively simple equations for the coupling constants of the Floquet Hamiltonian, where a straightforward truncation of the couplings leads to a powerful class of approximations. Using our theory, we construct a flow chart that illustrates the connection between various common approximations, which also highlights some missing connections and associated approximation schemes. These missing connections turn out to imply an analytically accessible approximation that is the "inverse" of a rotating frame approximation and thus has a range of validity complementary to it. We numerically test the various methods on the one-dimensional Ising model to confirm the ranges of validity that one would expect from the approximations used. The theory provides a map of the relations between the growing number of approximations for the time-evolution operator. We describe these relations in a table showing the limitations and advantages of many common approximations, as well as the new approximations introduced in this paper.
Wei Wang - One of the best experts on this subject based on the ideXlab platform.
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tikhonov regularization with l 0 term complementing a convex penalty l 1 convergence under sparsity constraints
arXiv: Numerical Analysis, 2018Co-Authors: Wei Wang, Bernd Hofmann, Jin ChengAbstract:Measuring the error by an l^1-norm, we analyze under sparsity assumptions an l^0-regularization approach, where the penalty in the Tikhonov functional is complemented by a general stabilizing convex functional. In this context, ill-posed operator equations Ax = y with an injective and bounded linear operator A mapping between l^2 and a Banach space Y are regularized. For sparse solutions, error estimates as well as linear and sublinear convergence rates are derived based on a variational inequality approach, where the regularization parameter can be chosen either a priori in an appropriate way or a posteriori by the sequential discrepancy principle. To further illustrate the balance between the l^0-term and the complementing convex penalty, the Important Special Case of the l^2-norm square penalty is investigated showing explicit dependence between both terms. Finally, some numerical experiments verify and illustrate the sparsity promoting properties of corresponding regularized solutions.
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information theoretic limits on sparse signal recovery dense versus sparse measurement matrices
IEEE Transactions on Information Theory, 2010Co-Authors: Wei Wang, Martin J Wainwright, Kannan RamchandranAbstract:We study the information-theoretic limits of exactly recovering the support set of a sparse signal, using noisy projections defined by various classes of measurement matrices. Our analysis is high-dimensional in nature, in which the number of observations n, the ambient signal dimension p, and the signal sparsity k are all allowed to tend to infinity in a general manner. This paper makes two novel contributions. First, we provide sharper necessary conditions for exact support recovery using general (including non-Gaussian) dense measurement matrices. Combined with previously known sufficient conditions, this result yields sharp characterizations of when the optimal decoder can recover a signal for various scalings of the signal sparsity k and sample size n, including the Important Special Case of linear sparsity (k = ?(p)) using a linear scaling of observations (n = ?(p)). Our second contribution is to prove necessary conditions on the number of observations n required for asymptotically reliable recovery using a class of ?-sparsified measurement matrices, where the measurement sparsity parameter ?(n, p, k) ? (0,1] corresponds to the fraction of nonzero entries per row. Our analysis allows general scaling of the quadruplet (n, p, k, ?) , and reveals three different regimes, corresponding to whether measurement sparsity has no asymptotic effect, a minor effect, or a dramatic effect on the information-theoretic limits of the subset recovery problem.
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information theoretic limits on sparse signal recovery dense versus sparse measurement matrices
arXiv: Statistics Theory, 2008Co-Authors: Wei Wang, Martin J Wainwright, Kannan RamchandranAbstract:We study the information-theoretic limits of exactly recovering the support of a sparse signal using noisy projections defined by various classes of measurement matrices. Our analysis is high-dimensional in nature, in which the number of observations $n$, the ambient signal dimension $p$, and the signal sparsity $k$ are all allowed to tend to infinity in a general manner. This paper makes two novel contributions. First, we provide sharper necessary conditions for exact support recovery using general (non-Gaussian) dense measurement matrices. Combined with previously known sufficient conditions, this result yields sharp characterizations of when the optimal decoder can recover a signal for various scalings of the sparsity $k$ and sample size $n$, including the Important Special Case of linear sparsity ($k = \Theta(p)$) using a linear scaling of observations ($n = \Theta(p)$). Our second contribution is to prove necessary conditions on the number of observations $n$ required for asymptotically reliable recovery using a class of $\gamma$-sparsified measurement matrices, where the measurement sparsity $\gamma(n, p, k) \in (0,1]$ corresponds to the fraction of non-zero entries per row. Our analysis allows general scaling of the quadruplet $(n, p, k, \gamma)$, and reveals three different regimes, corresponding to whether measurement sparsity has no effect, a minor effect, or a dramatic effect on the information-theoretic limits of the subset recovery problem.
Zhidong Zhang - One of the best experts on this subject based on the ideXlab platform.
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Inverse problems for heat equation and space-time fractional diffusion equation with one measurement
Journal of Differential Equations, 2020Co-Authors: Tapio Helin, Matti Lassas, Lauri Ylinen, Zhidong ZhangAbstract:Abstract Given a connected compact Riemannian manifold ( M , g ) without boundary, dim M ≥ 2 , we consider a space–time fractional diffusion equation with an interior source that is supported on an open subset V of the manifold. The time-fractional part of the equation is given by the Caputo derivative of order α ∈ ( 0 , 1 ] , and the space fractional part by ( − Δ g ) β , where β ∈ ( 0 , 1 ] and Δ g is the Laplace–Beltrami operator on the manifold. The Case α = β = 1 , which corresponds to the standard heat equation on the manifold, is an Important Special Case. We construct a specific source such that measuring the evolution of the corresponding solution on V determines the manifold up to a Riemannian isometry.
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Inverse problems for heat equation and space-time fractional diffusion equation with one measurement
arXiv: Analysis of PDEs, 2019Co-Authors: Tapio Helin, Matti Lassas, Lauri Ylinen, Zhidong ZhangAbstract:Given a connected compact Riemannian manifold $(M,g)$ without boundary, $\dim M\ge 2$, we consider a space--time fractional diffusion equation with an interior source that is supported on an open subset $V$ of the manifold. The time-fractional part of the equation is given by the Caputo derivative of order $\alpha\in(0,1]$, and the space fractional part by $(-\Delta_g)^\beta$, where $\beta\in(0,1]$ and $\Delta_g$ is the Laplace--Beltrami operator on the manifold. The Case $\alpha=\beta=1$, which corresponds to the standard heat equation on the manifold, is an Important Special Case. We construct a specific source such that measuring the evolution of the corresponding solution on $V$ determines the manifold up to a Riemannian isometry.
Mark J Van Der Laan - One of the best experts on this subject based on the ideXlab platform.
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an omnibus non parametric test of equality in distribution for unknown functions
Journal of The Royal Statistical Society Series B-statistical Methodology, 2019Co-Authors: Alexander R Luedtke, Marco Carone, Mark J Van Der LaanAbstract:We present a novel family of nonparametric omnibus tests of the hypothesis that two unknown but estimable functions are equal in distribution when applied to the observed data structure. We developed these tests, which represent a generalization of the maximum mean discrepancy tests described in Gretton et al. [2006], using recent developments from the higher-order pathwise differentiability literature. Despite their complex derivation, the associated test statistics can be expressed rather simply as U-statistics. We study the asymptotic behavior of the proposed tests under the null hypothesis and under both fixed and local alternatives. We provide examples to which our tests can be applied and show that they perform well in a simulation study. As an Important Special Case, our proposed tests can be used to determine whether an unknown function, such as the conditional average treatment effect, is equal to zero almost surely.
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an omnibus nonparametric test of equality in distribution for unknown functions
arXiv: Statistics Theory, 2015Co-Authors: Alexander R Luedtke, Marco Carone, Mark J Van Der LaanAbstract:We present a novel family of nonparametric omnibus tests of the hypothesis that two unknown but estimable functions are equal in distribution when applied to the observed data structure. We developed these tests, which represent a generalization of the maximum mean discrepancy tests described in Gretton et al. [2006], using recent developments from the higher-order pathwise differentiability literature. Despite their complex derivation, the associated test statistics can be expressed rather simply as U-statistics. We study the asymptotic behavior of the proposed tests under the null hypothesis and under both fixed and local alternatives. We provide examples to which our tests can be applied and show that they perform well in a simulation study. As an Important Special Case, our proposed tests can be used to determine whether an unknown function, such as the conditional average treatment effect, is equal to zero almost surely.
