Motivated by the idea of perspectivity of rings and modules, we introduce perspective elements of... more Motivated by the idea of perspectivity of rings and modules, we introduce perspective elements of a ring. We notably prove that perspective elements form a proper subset of special clean elements, and that perspectivity of elements is a left-right symmetric property. An equational characterization of perspective elements is also given and examples are provided.
Sous-espaces hilbertiens, sous-dualités et applications (Hilbertian subspaces, subdualities and a... more Sous-espaces hilbertiens, sous-dualités et applications (Hilbertian subspaces, subdualities and applications) Présentée et soutenue publiquement par
Proc of Statistica l Lear ning: Theor y and Applica・tions. Par is, 2002
Page 1. Wavelet Kernels and RKHS Alain Rakotomamonjy, Xavier Mary and Stéphane Canu alain.rakotom... more Page 1. Wavelet Kernels and RKHS Alain Rakotomamonjy, Xavier Mary and Stéphane Canu alain.rakotomamonjy@insa-rouen.fr asi.insa-rouen.fr/˜arakotom . INSA Rouen -Département ASI Laboratoire PSI Wavelet Kernels and RKHS – p.1/17 Page 2. Motivations Justify wavelet networks (Zhang, 1992) as a particular case of Regularization Networks (Girosi, 1995) Enlarge choice of hypothesis space where one looks for the solution of a learning problem by including wavelet span Develop algorithms that adapt the regularization to the scale of data. ...
We study generalized inverses on semigroups by means of Green's relations. We first define th... more We study generalized inverses on semigroups by means of Green's relations. We first define the notion of inverse along an element and study its properties. Then we show that the classical generalized inverses (group inverse, Drazin inverse and Moore-Penrose inverse) belong to this class.
In this article, recent results about point processes are used in sampling theory. Precisely, we ... more In this article, recent results about point processes are used in sampling theory. Precisely, we define and study a new class of sampling designs: determinantal sampling designs. The law of such designs is known, and there exists a simple selection algorithm. We compute exactly the variance of linear estimators constructed upon these designs by using the first and second order inclusion probabilities. Moreover, we obtain asymptotic and finite sample theorems. We construct explicitly fixed size determinantal sampling designs with given first order inclusion probabilities. We also address the search of optimal determinantal sampling designs.
We prove that special clean decompositions of a given element of a ring are in one-to-one corresp... more We prove that special clean decompositions of a given element of a ring are in one-to-one correspondence with the set of solutions of a simple equation in a corner ring. We then derive ?constructive? proofs that in many rings, regular elements are special clean by solving this equation in specific cases. Other applications, such as uniqueness of decompositions, are given. Many examples of special clean decompositions of 2-2 matrices found by this methodology are also presented.
Abstract We introduce and study a new class of modules and rings we call -perspective, which can ... more Abstract We introduce and study a new class of modules and rings we call -perspective, which can either be described in terms of perspective direct summands, associate idempotents, or generalized inverses. When n is small ( ), we recover existing class of modules and rings: (endo)abelian, strongly IC, and perspective ones. And 3/2-perspective rings are characterized by all their regular elements being special clean. Standard constructions are also discussed and examples are provided.
In this short note, we propose a concrete analogue of the space L(H) for local operator spaces, t... more In this short note, we propose a concrete analogue of the space L(H) for local operator spaces, the multinormed C ∗-algebra Y L(Hα).
We study unitary representations of groups in Kreĭn spaces, irreducibility criteria and integral ... more We study unitary representations of groups in Kreĭn spaces, irreducibility criteria and integral decompositions. Our main tool is the theory of Kreĭn subspaces and their (reproducing) kernels and a variant of Choquet’s theorem. Keywords representation
This paper review the functional aspects of the statistical learning theory. It regards mainly th... more This paper review the functional aspects of the statistical learning theory. It regards mainly the nature of the hypothesis set when no prior information is available but the data. In this framework our starting point is a discussion arguing three principles about the hypothesis set: it is a vectorial space, it is a set of pointwise defined functions and the evaluation functional on this set is a continuous mapping. Based on these principles an original theory is developed generalizing the notion of reproduction kernel Hilbert space to non hilbertian sets. Then it is shown that the hypothesis set of any learning machine has to be a generalized reproducing set. Therefore, thanks to a general "representer theorem", the solution of the learning problem is still a linear combination of some kernel. Furthermore, a way to design these kernels is given. To illustrates this framework some examples of such reproducing sets and kernels are given.
We study extensions of Cline’s formula and Jacobson lemma for one-sided, commuting and bicommutin... more We study extensions of Cline’s formula and Jacobson lemma for one-sided, commuting and bicommuting weak inverses, in semigroups and general rings. In particular, we provide various isomorphisms between the (one-sided, commuting and bicommuting) weak inverses of [Formula: see text] and those of [Formula: see text] (or [Formula: see text] and [Formula: see text] when [Formula: see text] and [Formula: see text] satisfy additional identities).
Applied Stochastic Models in Business and Industry, 2005
Abstract This paper introduces a method to construct a reproducing wavelet kernel Hilbert spaces ... more Abstract This paper introduces a method to construct a reproducing wavelet kernel Hilbert spaces for non-parametric regression estimation when the sampling points are not equally spaced. Another objective is to make high-dimensional wavelet estimation problems tractable. It then provides a theoretical foundation to build reproducing kernel from operators and a practical technique to obtain reproducing kernel Hilbert spaces spanned by a set of wavelets. A multiscale approximation technique that aims at taking advantage of the ...
Motivated by the idea of perspectivity of rings and modules, we introduce perspective elements of... more Motivated by the idea of perspectivity of rings and modules, we introduce perspective elements of a ring. We notably prove that perspective elements form a proper subset of special clean elements, and that perspectivity of elements is a left-right symmetric property. An equational characterization of perspective elements is also given and examples are provided.
Sous-espaces hilbertiens, sous-dualités et applications (Hilbertian subspaces, subdualities and a... more Sous-espaces hilbertiens, sous-dualités et applications (Hilbertian subspaces, subdualities and applications) Présentée et soutenue publiquement par
Proc of Statistica l Lear ning: Theor y and Applica・tions. Par is, 2002
Page 1. Wavelet Kernels and RKHS Alain Rakotomamonjy, Xavier Mary and Stéphane Canu alain.rakotom... more Page 1. Wavelet Kernels and RKHS Alain Rakotomamonjy, Xavier Mary and Stéphane Canu alain.rakotomamonjy@insa-rouen.fr asi.insa-rouen.fr/˜arakotom . INSA Rouen -Département ASI Laboratoire PSI Wavelet Kernels and RKHS – p.1/17 Page 2. Motivations Justify wavelet networks (Zhang, 1992) as a particular case of Regularization Networks (Girosi, 1995) Enlarge choice of hypothesis space where one looks for the solution of a learning problem by including wavelet span Develop algorithms that adapt the regularization to the scale of data. ...
We study generalized inverses on semigroups by means of Green's relations. We first define th... more We study generalized inverses on semigroups by means of Green's relations. We first define the notion of inverse along an element and study its properties. Then we show that the classical generalized inverses (group inverse, Drazin inverse and Moore-Penrose inverse) belong to this class.
In this article, recent results about point processes are used in sampling theory. Precisely, we ... more In this article, recent results about point processes are used in sampling theory. Precisely, we define and study a new class of sampling designs: determinantal sampling designs. The law of such designs is known, and there exists a simple selection algorithm. We compute exactly the variance of linear estimators constructed upon these designs by using the first and second order inclusion probabilities. Moreover, we obtain asymptotic and finite sample theorems. We construct explicitly fixed size determinantal sampling designs with given first order inclusion probabilities. We also address the search of optimal determinantal sampling designs.
We prove that special clean decompositions of a given element of a ring are in one-to-one corresp... more We prove that special clean decompositions of a given element of a ring are in one-to-one correspondence with the set of solutions of a simple equation in a corner ring. We then derive ?constructive? proofs that in many rings, regular elements are special clean by solving this equation in specific cases. Other applications, such as uniqueness of decompositions, are given. Many examples of special clean decompositions of 2-2 matrices found by this methodology are also presented.
Abstract We introduce and study a new class of modules and rings we call -perspective, which can ... more Abstract We introduce and study a new class of modules and rings we call -perspective, which can either be described in terms of perspective direct summands, associate idempotents, or generalized inverses. When n is small ( ), we recover existing class of modules and rings: (endo)abelian, strongly IC, and perspective ones. And 3/2-perspective rings are characterized by all their regular elements being special clean. Standard constructions are also discussed and examples are provided.
In this short note, we propose a concrete analogue of the space L(H) for local operator spaces, t... more In this short note, we propose a concrete analogue of the space L(H) for local operator spaces, the multinormed C ∗-algebra Y L(Hα).
We study unitary representations of groups in Kreĭn spaces, irreducibility criteria and integral ... more We study unitary representations of groups in Kreĭn spaces, irreducibility criteria and integral decompositions. Our main tool is the theory of Kreĭn subspaces and their (reproducing) kernels and a variant of Choquet’s theorem. Keywords representation
This paper review the functional aspects of the statistical learning theory. It regards mainly th... more This paper review the functional aspects of the statistical learning theory. It regards mainly the nature of the hypothesis set when no prior information is available but the data. In this framework our starting point is a discussion arguing three principles about the hypothesis set: it is a vectorial space, it is a set of pointwise defined functions and the evaluation functional on this set is a continuous mapping. Based on these principles an original theory is developed generalizing the notion of reproduction kernel Hilbert space to non hilbertian sets. Then it is shown that the hypothesis set of any learning machine has to be a generalized reproducing set. Therefore, thanks to a general "representer theorem", the solution of the learning problem is still a linear combination of some kernel. Furthermore, a way to design these kernels is given. To illustrates this framework some examples of such reproducing sets and kernels are given.
We study extensions of Cline’s formula and Jacobson lemma for one-sided, commuting and bicommutin... more We study extensions of Cline’s formula and Jacobson lemma for one-sided, commuting and bicommuting weak inverses, in semigroups and general rings. In particular, we provide various isomorphisms between the (one-sided, commuting and bicommuting) weak inverses of [Formula: see text] and those of [Formula: see text] (or [Formula: see text] and [Formula: see text] when [Formula: see text] and [Formula: see text] satisfy additional identities).
Applied Stochastic Models in Business and Industry, 2005
Abstract This paper introduces a method to construct a reproducing wavelet kernel Hilbert spaces ... more Abstract This paper introduces a method to construct a reproducing wavelet kernel Hilbert spaces for non-parametric regression estimation when the sampling points are not equally spaced. Another objective is to make high-dimensional wavelet estimation problems tractable. It then provides a theoretical foundation to build reproducing kernel from operators and a practical technique to obtain reproducing kernel Hilbert spaces spanned by a set of wavelets. A multiscale approximation technique that aims at taking advantage of the ...
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