The Experts below are selected from a list of 234747 Experts worldwide ranked by ideXlab platform
Rocco A Servedio - One of the best experts on this subject based on the ideXlab platform.
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a robust khintchine inequality and algorithms for computing optimal constants in Fourier Analysis and high dimensional geometry
International Colloquium on Automata Languages and Programming, 2013Co-Authors: Ilias Diakonikolas, Rocco A ServedioAbstract:This paper makes two contributions towards determining some well-studied optimal constants in Fourier Analysis of Boolean functions and high-dimensional geometry. 1 It has been known since 1994 [GL94] that every linear threshold function has squared Fourier mass at least 1/2 on its degree-0 and degree-1 coefficients. Denote the minimum such Fourier mass by W≤1[LTF], where the minimum is taken over all n-variable linear threshold functions and all n≥0. Benjamini, Kalai and Schramm [BKS99] have conjectured that the true value of W≤1[LTF] is 2/π. We make progress on this conjecture by proving that W≤1[LTF]≥1/2+c for some absolute constant c>0. The key ingredient in our proof is a "robust" version of the well-known Khintchine inequality in functional Analysis, which we believe may be of independent interest. 2 We give an algorithm with the following property: given any η>0, the algorithm runs in time 2poly(1/η) and determines the value of W≤1[LTF] up to an additive error of ±η. We give a similar 2poly(1/η)-time algorithm to determine Tomaszewski's constant to within an additive error of ±η; this is the minimum (over all origin-centered hyperplanes H) fraction of points in {−1,1}n that lie within Euclidean distance 1 of H. Tomaszewski's constant is conjectured to be 1/2; lower bounds on it have been given by Holzman and Kleitman [HK92] and independently by Ben-Tal, Nemirovski and Roos [BTNR02]. Our algorithms combine tools from anti-concentration of sums of independent random variables, Fourier Analysis, and Hermite Analysis of linear threshold functions.
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a robust khintchine inequality and algorithms for computing optimal constants in Fourier Analysis and high dimensional geometry
arXiv: Computational Complexity, 2012Co-Authors: Ilias Diakonikolas, Rocco A ServedioAbstract:This paper makes two contributions towards determining some well-studied optimal constants in Fourier Analysis \newa{of Boolean functions} and high-dimensional geometry. \begin{enumerate} \item It has been known since 1994 \cite{GL:94} that every linear threshold function has squared Fourier mass at least 1/2 on its degree-0 and degree-1 coefficients. Denote the minimum such Fourier mass by $\w^{\leq 1}[\ltf]$, where the minimum is taken over all $n$-variable linear threshold functions and all $n \ge 0$. Benjamini, Kalai and Schramm \cite{BKS:99} have conjectured that the true value of $\w^{\leq 1}[\ltf]$ is $2/\pi$. We make progress on this conjecture by proving that $\w^{\leq 1}[\ltf] \geq 1/2 + c$ for some absolute constant $c>0$. The key ingredient in our proof is a "robust" version of the well-known Khintchine inequality in functional Analysis, which we believe may be of independent interest. \item We give an algorithm with the following property: given any $\eta > 0$, the algorithm runs in time $2^{\poly(1/\eta)}$ and determines the value of $\w^{\leq 1}[\ltf]$ up to an additive error of $\pm\eta$. We give a similar $2^{{\poly(1/\eta)}}$-time algorithm to determine \emph{Tomaszewski's constant} to within an additive error of $\pm \eta$; this is the minimum (over all origin-centered hyperplanes $H$) fraction of points in $\{-1,1\}^n$ that lie within Euclidean distance 1 of $H$. Tomaszewski's constant is conjectured to be 1/2; lower bounds on it have been given by Holzman and Kleitman \cite{HK92} and independently by Ben-Tal, Nemirovski and Roos \cite{BNR02}. Our algorithms combine tools from anti-concentration of sums of independent random variables, Fourier Analysis, and Hermite Analysis of linear threshold functions. \end{enumerate}
Kenji Yamanishi - One of the best experts on this subject based on the ideXlab platform.
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exact calculation of normalized maximum likelihood code length using Fourier Analysis
International Symposium on Information Theory, 2018Co-Authors: Atsushi Suzuki, Kenji YamanishiAbstract:The normalized maximum likelihood code length has been widely used in model selection, and its favorable properties, such as its consistency and the upper bound of its statistical risk, have been demonstrated. This paper proposes a novel methodology for calculating the normalized maximum likelihood code length on the basis of Fourier Analysis. Our methodology provides an efficient non-asymptotic calculation formula for exponential family models and an asymptotic calculation formula for general parametric models with a weaker assumption compared to that in previous work. 2018 International Symposium on Information Theory. A full version of this paper is accessible at https://arxiv.org/abs/1801.03705 [21]
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exact calculation of normalized maximum likelihood code length using Fourier Analysis
arXiv: Statistics Theory, 2018Co-Authors: Atsushi Suzuki, Kenji YamanishiAbstract:The normalized maximum likelihood code length has been widely used in model selection, and its favorable properties, such as its consistency and the upper bound of its statistical risk, have been demonstrated. This paper proposes a novel methodology for calculating the normalized maximum likelihood code length on the basis of Fourier Analysis. Our methodology provides an efficient non-asymptotic calculation formula for exponential family models and an asymptotic calculation formula for general parametric models with a weaker assumption compared to that in previous work.
Ilias Diakonikolas - One of the best experts on this subject based on the ideXlab platform.
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a robust khintchine inequality and algorithms for computing optimal constants in Fourier Analysis and high dimensional geometry
International Colloquium on Automata Languages and Programming, 2013Co-Authors: Ilias Diakonikolas, Rocco A ServedioAbstract:This paper makes two contributions towards determining some well-studied optimal constants in Fourier Analysis of Boolean functions and high-dimensional geometry. 1 It has been known since 1994 [GL94] that every linear threshold function has squared Fourier mass at least 1/2 on its degree-0 and degree-1 coefficients. Denote the minimum such Fourier mass by W≤1[LTF], where the minimum is taken over all n-variable linear threshold functions and all n≥0. Benjamini, Kalai and Schramm [BKS99] have conjectured that the true value of W≤1[LTF] is 2/π. We make progress on this conjecture by proving that W≤1[LTF]≥1/2+c for some absolute constant c>0. The key ingredient in our proof is a "robust" version of the well-known Khintchine inequality in functional Analysis, which we believe may be of independent interest. 2 We give an algorithm with the following property: given any η>0, the algorithm runs in time 2poly(1/η) and determines the value of W≤1[LTF] up to an additive error of ±η. We give a similar 2poly(1/η)-time algorithm to determine Tomaszewski's constant to within an additive error of ±η; this is the minimum (over all origin-centered hyperplanes H) fraction of points in {−1,1}n that lie within Euclidean distance 1 of H. Tomaszewski's constant is conjectured to be 1/2; lower bounds on it have been given by Holzman and Kleitman [HK92] and independently by Ben-Tal, Nemirovski and Roos [BTNR02]. Our algorithms combine tools from anti-concentration of sums of independent random variables, Fourier Analysis, and Hermite Analysis of linear threshold functions.
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a robust khintchine inequality and algorithms for computing optimal constants in Fourier Analysis and high dimensional geometry
arXiv: Computational Complexity, 2012Co-Authors: Ilias Diakonikolas, Rocco A ServedioAbstract:This paper makes two contributions towards determining some well-studied optimal constants in Fourier Analysis \newa{of Boolean functions} and high-dimensional geometry. \begin{enumerate} \item It has been known since 1994 \cite{GL:94} that every linear threshold function has squared Fourier mass at least 1/2 on its degree-0 and degree-1 coefficients. Denote the minimum such Fourier mass by $\w^{\leq 1}[\ltf]$, where the minimum is taken over all $n$-variable linear threshold functions and all $n \ge 0$. Benjamini, Kalai and Schramm \cite{BKS:99} have conjectured that the true value of $\w^{\leq 1}[\ltf]$ is $2/\pi$. We make progress on this conjecture by proving that $\w^{\leq 1}[\ltf] \geq 1/2 + c$ for some absolute constant $c>0$. The key ingredient in our proof is a "robust" version of the well-known Khintchine inequality in functional Analysis, which we believe may be of independent interest. \item We give an algorithm with the following property: given any $\eta > 0$, the algorithm runs in time $2^{\poly(1/\eta)}$ and determines the value of $\w^{\leq 1}[\ltf]$ up to an additive error of $\pm\eta$. We give a similar $2^{{\poly(1/\eta)}}$-time algorithm to determine \emph{Tomaszewski's constant} to within an additive error of $\pm \eta$; this is the minimum (over all origin-centered hyperplanes $H$) fraction of points in $\{-1,1\}^n$ that lie within Euclidean distance 1 of $H$. Tomaszewski's constant is conjectured to be 1/2; lower bounds on it have been given by Holzman and Kleitman \cite{HK92} and independently by Ben-Tal, Nemirovski and Roos \cite{BNR02}. Our algorithms combine tools from anti-concentration of sums of independent random variables, Fourier Analysis, and Hermite Analysis of linear threshold functions. \end{enumerate}
Olaf Ronneberger - One of the best experts on this subject based on the ideXlab platform.
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Rotation-Invariant HOG Descriptors Using Fourier Analysis in Polar and Spherical Coordinates
International Journal of Computer Vision, 2014Co-Authors: Kun Liu, Henrik Skibbe, Thorsten Schmidt, Thomas Blein, Klaus Palme, Thomas Brox, Olaf RonnebergerAbstract:The histogram of oriented gradients (HOG) is widely used for image description and proves to be very effective. In many vision problems, rotation-invariant Analysis is necessary or preferred. Popular solutions are mainly based on pose normalization or learning, neglecting some intrinsic properties of rotations. This paper presents a method to build rotation-invariant HOG descriptors using Fourier Analysis in polar/spherical coordinates, which are closely related to the irreducible representation of the 2D/3D rotation groups. This is achieved by considering a gradient histogram as a continuous angular signal which can be well represented by the Fourier basis (2D) or spherical harmonics (3D). As rotation-invariance is established in an analytical way, we can avoid discretization artifacts and create a continuous mapping from the image to the feature space. In the experiments, we first show that our method outperforms the state-of-the-art in a public dataset for a car detection task in aerial images. We further use the Princeton Shape Benchmark and the SHREC 2009 Generic Shape Benchmark to demonstrate the high performance of our method for similarity measures of 3D shapes. Finally, we show an application on microscopic volumetric data.
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rotational invariance based on Fourier Analysis in polar and spherical coordinates
IEEE Transactions on Pattern Analysis and Machine Intelligence, 2009Co-Authors: Qing Wang, Olaf Ronneberger, Hans BurkhardtAbstract:In this paper, polar and spherical Fourier Analysis are defined as the decomposition of a function in terms of eigenfunctions of the Laplacian with the eigenfunctions being separable in the corresponding coordinates. The proposed transforms provide effective decompositions of an image into basic patterns with simple radial and angular structures. The theory is compactly presented with an emphasis on the analogy to the normal Fourier transform. The relation between the polar or spherical Fourier transform and the normal Fourier transform is explored. As examples of applications, rotation-invariant descriptors based on polar and spherical Fourier coefficients are tested on pattern classification problems.
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Fourier Analysis in polar and spherical coordinates
2008Co-Authors: Qing Wang, Olaf Ronneberger, Hans BurkhardtAbstract:In this paper, polar and spherical Fourier Analysis are defined as the decomposition of a function in terms of eigenfunctions of the Laplacian with the eigenfunctions being separable in the corresponding coordinates. Each eigenfunction represents a basic pattern with the wavenumber indicating the scale. The proposed transforms provide an effective radial decomposition in addition to the well-known angular decomposition. The derivation of the basis functions is compactly presented with an emphasis on the analogy to the normal Fourier transform. The relation between the polar or spherical Fourier transform and normal Fourier transform is explored. Possible applications of the proposed transforms are discussed.
Atsushi Suzuki - One of the best experts on this subject based on the ideXlab platform.
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exact calculation of normalized maximum likelihood code length using Fourier Analysis
International Symposium on Information Theory, 2018Co-Authors: Atsushi Suzuki, Kenji YamanishiAbstract:The normalized maximum likelihood code length has been widely used in model selection, and its favorable properties, such as its consistency and the upper bound of its statistical risk, have been demonstrated. This paper proposes a novel methodology for calculating the normalized maximum likelihood code length on the basis of Fourier Analysis. Our methodology provides an efficient non-asymptotic calculation formula for exponential family models and an asymptotic calculation formula for general parametric models with a weaker assumption compared to that in previous work. 2018 International Symposium on Information Theory. A full version of this paper is accessible at https://arxiv.org/abs/1801.03705 [21]
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exact calculation of normalized maximum likelihood code length using Fourier Analysis
arXiv: Statistics Theory, 2018Co-Authors: Atsushi Suzuki, Kenji YamanishiAbstract:The normalized maximum likelihood code length has been widely used in model selection, and its favorable properties, such as its consistency and the upper bound of its statistical risk, have been demonstrated. This paper proposes a novel methodology for calculating the normalized maximum likelihood code length on the basis of Fourier Analysis. Our methodology provides an efficient non-asymptotic calculation formula for exponential family models and an asymptotic calculation formula for general parametric models with a weaker assumption compared to that in previous work.