The Experts below are selected from a list of 15705 Experts worldwide ranked by ideXlab platform
Gilles Stupfler - One of the best experts on this subject based on the ideXlab platform.
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On the weak convergence of the kernel density estimator in the uniform topology
2016Co-Authors: Gilles StupflerAbstract:The pointwise asymptotic properties of the Parzen-Rosenblatt kernel estimator Fn of a probability density function f on Rd have received great attention, and so have its integrated or uniform errors. It has been pointed out in a couple of recent works that the weak convergence of its centered and rescaled versions in a weighted Lebesgue Lp Space, 1 ≤ p < ∞, considered to be a difficult problem, is in fact essentially uninteresting in the sense that the only possible Borel measurable weak limit is 0 under very mild conditions. This paper examines the weak convergence of such processes in the uniform topology. Specifically, we show that if fn (x) = E(Fn (x)) and (rn) is any nonrandom sequence of positive real numbers such that rn / √n → 0 then, with probability 1, the sample paths of any tight Borel measurable weak limit in an l∞ Space on Rd of the process rn (Fn − fn) must be almost everywhere zero. The particular case when the estimator Fn has continuous sample paths is then considered and simple conditions making it possible to examine the actual existence of a weak limit in this framework are provided.
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On the weak convergence of the kernel density estimator in the uniform topology
Electronic Communications in Probability, 2016Co-Authors: Gilles StupflerAbstract:The pointwise asymptotic properties of the Parzen-Rosenblatt kernel estimator fˆn of a probability density function f on Rd have received great attention, and so have its integrated or uniform errors. It has been pointed out in a couple of recent works that the weak convergence of its centered and rescaled versions in a weighted Lebesgue Lp Space, 1≤p\textless∞, considered to be a difficult problem, is in fact essentially uninteresting in the sense that the only possible Borel measurable weak limit is 0 under very mild conditions. This paper examines the weak convergence of such processes in the uniform topology. Specifically, we show that if fn(x)=E(fˆn(x)) and (rn) is any nonrandom sequence of positive real numbers such that rn/n√→0 then, with probability 1, the sample paths of any tight Borel measurable weak limit in an ℓ∞ Space on Rd of the process rn(fˆn−fn) must be almost everywhere zero. The particular case when the estimator fˆn has continuous sample paths is then considered and simple conditions making it possible to examine the actual existence of a weak limit in this framework are provided.
Wenjun Zeng - One of the best experts on this subject based on the ideXlab platform.
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robust low rank approximation of matrices in Lp Space
2018Co-Authors: Wenjun ZengAbstract:Low-rank approximation plays an important role in many areas of science and engineering such as signal/image processing, machine learning, data mining, imaging, bioinformatics, pattern classification and computer vision because many real-world data exhibit low-rank property. This dissertation devises advanced algorithms for robust low-rank approximation of a single matrix as well as multiple matrices in the presence of outliers, where the conventional dimensionality reduction techniques such as the celebrated principal component analysis (PCA) are not applicable. The proposed methodology is based on minimizing the entry-wise $\ell_p$-norm of the residual including the challenging nonconvex and nonsmooth case of $p<1$. Theoretical analyses are also presented. Extensive practical applications are discussed. Experimental results demonstrate that the superiority of the proposed methods over the state-of-the-art techniques. Two iterative algorithms are designed for low-rank approximation of a single matrix. The first is the iteratively reweighted singular value decomposition (IR-SVD), where the SVD of a reweighted matrix is performed at each iteration. The second converts the nonconvex $\ell_p$-matrix factorization into a series of easily solvable $\ell_p$-norm minimization with vectors being variables. Applications to image demixing, foreground detection in video surveillance, array signal processing, and direction-of-arrival estimation for source localization in impulsive noise are investigated. The low-rank approximation with missing values, i.e., robust matrix completion, is also addressed. Two algorithms are developed for it. The first iteratively solves a set of linear $\ell_p$-regression problems while the second applies the alternating direction method of multipliers (ADMM) in the $\ell_p$-Space. At each iteration of the ADMM, it requires performing a least squares (LS) matrix factorization and calculating the proximity operator of the $p$th power of the $\ell_p$-norm. The LS factorization is efficiently solved using linear LS regression while the proximity operator is obtained by root finding of a scalar nonlinear equation. The two proposed algorithms are scalable to the problem size. Applications to recommender systems, collaborative filtering, and image inpainting are provided. The $\ell_p$-greedy pursuit ($\ell_p$-GP) algorithms are devised for joint robust low-rank approximation of multiple matrices (RLRAMM) with outliers. The $\ell_p$-GP with $0
Imre J. Rudas - One of the best experts on this subject based on the ideXlab platform.
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Pseudo-Lp Space and convergence
Fuzzy Sets and Systems, 2014Co-Authors: Endre Pap, Mirjana Strboja, Imre J. RudasAbstract:In the framework of the pseudo-analysis the classical L^p Space is generalized and there are proved important properties of introduced Space. Three types of convergence of sequences of measurable functions are considered in this Space. The inequalities for integrals based on pseudo-integral have been recently proposed as the Holder, Minkowski and Markov inequalities, which are applied in observing relationship among introduced convergences.
Zeng Wenjun - One of the best experts on this subject based on the ideXlab platform.
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Robust Low-Rank Approximation of Matrices in Lp-Space
2018Co-Authors: Zeng WenjunAbstract:Low-rank approximation plays an important role in many areas of science and engineering such as signal/image processing, machine learning, data mining, imaging, bioinformatics, pattern classification and computer vision because many real-world data exhibit low-rank property. This dissertation devises advanced algorithms for robust low-rank approximation of a single matrix as well as multiple matrices in the presence of outliers, where the conventional dimensionality reduction techniques such as the celebrated principal component analysis (PCA) are not applicable. The proposed methodology is based on minimizing the entry-wise $\ell_p$-norm of the residual including the challenging nonconvex and nonsmooth case of $p
Beloslav Riecan - One of the best experts on this subject based on the ideXlab platform.
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on the Lp Space of observables on product mv algebras
International Journal of Theoretical Physics, 2000Co-Authors: Beloslav RiecanAbstract:A weakly σ-distributive product MV algebra M is considered as a base of aquantum structure model. A state is a morphism from M to the unit interval, andan observable is a morphism from the system of all Borel sets to M. It is provedthat the subSpace Lp of the Space of observables is a complete pseudometricSpace. This result generalizes the previous result; the proof is new.
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on the Lp Space of observables
Fuzzy Sets and Systems, 1999Co-Authors: Beloslav RiecanAbstract:Abstract A Space of fuzzy sets M is considered as a base of a quantum structure model (fuzzy quantum logic, see Riecan and Neubrunn, Integral, Measure and Ordering, Kluwer, Dordrecht, 1997). A state is a morphism from M to the unit interval, an observable is a morphism from Borel sets to M . The subSpace L P of observables is studied and it is proved that L P considered with the corresponding pseudometric is a complete pseudometric Space.
