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Documents  94A12 | enregistrements trouvés : 103

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Research schools

We consider the convergence of the iterative projected gradient (IPG) algorithm for arbitrary (typically nonconvex) sets and when both the gradient and projection oracles are only computed approximately. We consider different notions of approximation of which we show that the Progressive Fixed Precision (PFP) and (1+epsilon) optimal oracles can achieve the same accuracy as for the exact IPG algorithm. We also show that the former scheme is also able to maintain the (linear) rate of convergence of the exact algorithm, under the same embedding assumption, while the latter requires a stronger embedding condition, moderate compression ratios and typically exhibits slower convergence. We apply our results to accelerate solving a class of data driven compressed sensing problems, where we replace iterative exhaustive searches over large datasets by fast approximate nearest neighbour search strategies based on the cover tree data structure. Finally, if there is time we will give examples of this theory applied in practice for rapid enhanced solutions to an emerging MRI protocol called magnetic resonance fingerprinting for quantitative MRI. We consider the convergence of the iterative projected gradient (IPG) algorithm for arbitrary (typically nonconvex) sets and when both the gradient and projection oracles are only computed approximately. We consider different notions of approximation of which we show that the Progressive Fixed Precision (PFP) and (1+epsilon) optimal oracles can achieve the same accuracy as for the exact IPG algorithm. We also show that the former scheme is also ...

65C60 ; 62D05 ; 94A12

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Research talks;Analysis and its Applications;Mathematical Physics

Retrieving an arbitrary signal from the magnitudes of its inner products with the elements of a frame is not possible in infinite dimensions. Under certain conditions, signals can be retrieved satisfactorily however.

42C15 ; 46C05 ; 94A12 ; 94A15 ; 94A20

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- viii; 369 p.
ISBN 978-3-0348-0584-1

Operator theory: advances and applications , 0231

Localisation : Collection 1er étage

opérateur pseudo-différentiel # équation différentielle # équation aux dérivées partielles # groupe topologique

22A10 ; 32A40 ; 32A45 ; 35A17 ; 35A22 ; 35B05 ; 35B40 ; 35B60 ; 35J70 ; 35K05 ; 35K65 ; 35L05 ; 35L40 ; 35S05 ; 35S15 ; 35S30 ; 43A77 ; 46F15 ; 47B10 ; 47B35 ; 47B37 ; 47G10 ; 47G30 ; 47L15 ; 58J35 ; 58J40 ; 58J50 ; 65R10 ; 94A12 ; 22C05 ; 30E25 ; 35G05 ; 35H10 ; 35J05 ; 42B10 ; 42B35 ; 47A10 ; 47A53 ; 47F05 ; 58J20 ; 65M60 ; 65T10

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- vi; 305 p.
ISBN 978-3-0348-0048-8

Operator theory: advances and applications , 0213

Localisation : Collection 1er étage

EDP # opérateurs pseudo-différentiels

22A10 ; 32A40 ; 32A45 ; 35A17 ; 35A22 ; 35B05 ; 35B40 ; 35B60 ; 35J70 ; 35K05 ; 35K65 ; 35L05 ; 35L40 ; 35S05 ; 35S15 ; 35S30 ; 43A77 ; 46F15 ; 47B10 ; 47B35 ; 47B37 ; 47G10 ; 47G30 ; 47L15 ; 58J35 ; 58J40 ; 58J50 ; 65R10 ; 94A12 ; 22C05 ; 30E25 ; 35G05 ; 35H10 ; 35J05 ; 42B10 ; 42B35 ; 47A10 ; 47A53 ; 47F05 ; 58J20 ; 65M60 ; 65T10

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- 291 p.
ISBN 978-0-8218-4144-0

Contemporary mathematics , 0451

Localisation : Collection 1er étage

analyse de Fourier # théorie des opérateurs # théorie analytique des nombres # groupe de Coxeter # opérateur pseudodifférentiel # approximation # dévellopement de Fourier généralisé # ondelette # opérateurs pour l'analyse numérique # théorie des opérateurs non classiques # traitement du signal

11S80 ; 20F55 ; 35S99 ; 41A30 ; 42C15 ; 42C40 ; 47N40 ; 47S10 ; 94A12

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- 414 p.
ISBN 978-0-8218-4676-8

Fields institute communications , 0052

Localisation : Collection 1er étage

EDP # opérateur pseudo-différentiel # analyse de Fourier # opérateur intégral # analyse globale # analyse de variétés # méthode numérique pour l'analyse de Fourier # ondes # théorie du signal

35-06 ; 35S05 ; 42-06 ; 47G10 ; 47G30 ; 58-06 ; 58J40 ; 65T60 ; 94A12

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- 162 p.
ISBN 978-0-8176-4358-4

Progress in mathematics , 0238

Localisation : Collection 1er étage

analyse harmonique # traitement du signal # complexité calculatoire # problème de Cauchy #idéal arithmétique # idéal analytique # EDP # opérateur d'Hermite # groupe d'Heisenberg # fonction subharmonique # structure discrète # variété hyperbolique # condition de Phragmen-Lindelöf # déconvolation locale # projection orthogonale # espace hyperbolique # ondelette

05C05 ; 05C50 ; 31C20 ; 32A26 ; 32A50 ; 35C15 ; 35N05 ; 35R30 ; 42A85 ; 42B10 ; 42B35 ; 43A85 ; 44A12 ; 46F12 ; 65R30 ; 92C55 ; 94A12

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- 156 p.
ISBN 978-3-7643-6790-9

Operator theory: advances and applications , 0132

Localisation : Collection 1er étage

théorie des opérateurs # ondelette # transformation d'ondelette # opérateur de localisation # opérateur intégrale # représentation de groupe de dimension infinie # groupe localement compact # groupe de Hausdorff # groupe de Weyl-Heisenberg # théorie spectrale # inégalité de Schatten-Von Neumann # opérateur de Daubechies # opérateur pseudo-differentiel # transformation de Weyl

47-02 ; 47G10 ; 47G30 ; 22A10 ; 42C40 ; 81S40 ; 94A12

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ISBN 978-0-8186-6275-1

Localisation : Colloque 1er étage (JERU)

algorithmique parallèle # architecture parallèle # calcul parallèle # codage # intelligence artificielle # reconnaissance des formes # théorie d

60-06 ; 68Q10 ; 94A12 ; 94A24 ; 94Axx

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Banach center publications , 0041

Localisation : Salle des périodiques 1er étage

champ d

46C20 ; 83B05 ; 83C25 ; 83C35 ; 83CXX ; 85A45 ; 94A12 ; 94A13

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ISBN 978-0-86720-225-0

Localisation : Colloque 1er étage (LOWE)

analyse de signal # analyse numérique # mécanique quantique # ondelette

42-06 ; 60T10 ; 93E14 ; 94A05 ; 94A12

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- 175 p.
ISBN 978-3-211-82143-5

C.I.S.M. courses and lectures , 0309

Localisation : Colloque 1er étage (UDIN)

circuits # communication # information et communication # théorie du signal

94-06 ; 94-XX ; 94A12 ; 94Axx

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- 376 p.
ISBN 978-3-540-56320-4

Lecture notes in computer science , 0653

Localisation : Collection 1er étage

automate # automatisation # contrôle # image # informatique # ingineering # intelligence artificielle # logiciel # logique symbolique # programmation # reconnaissance # robotique # science cognitive # sciences de l'ingénieur # signal # software # système cognitif # théorie du signal # vision

68D99 ; 68Q40 ; 68Q68 ; 68Txx ; 94A12

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Localisation : Colloque 1er étage (PARI)

algorithme # codage # signal # theorie de l'information # theorie du signal

94A12 ; 94Bxx

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ISBN 978-0-19-853613-0

The institute of mathematics and its applications conference series , 0012

Localisation : Colloque 1er étage (BATH)

algorithme numerique # graphisme # image informatique # informati que graphique # processus stochastique # signal # theorie distribution invariante signal

60G35 ; 62M20 ; 68U10 ; 94-06 ; 94A12

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Localisation : Colloque 1er étage (LIBL)

augmentation d'entropie en mécanique quantique # # canal ergodique # canal stationnaire # détection de signal connu # entropie du tchèque # fonction de décision statistique # formule de Palm # processus aléatoire # théorie de l'information # théorème de Shannon

60-06 ; 60Gxx ; 94A12 ; 94A40 ; 94Axx

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Research schools

Can modern signal processing be used to overcome the diffraction limit? The classical diffraction limit states that the resolution of a linear imaging system is fundamentally limited by one half of the wavelength of light. This implies that conventional light microscopes cannot distinguish two objects placed within a distance closer than 0.5 × 400 = 200nm (blue) or 0.5 × 700 = 350nm (red). This significantly impedes biomedical discovery by restricting our ability to observe biological structure and processes smaller than 100nm. Recent progress in sparsity-driven signal processing has created a powerful paradigm for increasing both the resolution and overall quality of imaging by promoting model-based image acquisition and reconstruction. This has led to multiple influential results demonstrating super-resolution in practical imaging systems. To date, however, the vast majority of work in signal processing has neglected the fundamental nonlinearity of the object-light interaction and its potential to lead to resolution enhancement. As a result, modern theory heavily focuses on linear measurement models that are truly effective only when object-light interactions are weak. Without a solid signal processing foundation for understanding such nonlinear interactions, we undervalue their impact on information transfer in the image formation. This ultimately limits our capability to image a large class of objects, such as biological tissue, that generally are in large-volumes and interact strongly and nonlinearly with light.
The goal of this talk is to present the recent progress in model-based imaging under multiple scattering. We will discuss several key applications including optical diffraction tomography, Fourier Ptychography, and large-scale Holographic microscopy. We will show that all these application can benefit from models, such as the Rytov approximation and beam propagation method, that take light scattering into account. We will discuss the integration of such models into the state-of-the-art optimization algorithms such as FISTA and ADMM. Finally, we will describe the most recent work that uses learned-priors for improving the quality of image reconstruction under multiple scattering.
Can modern signal processing be used to overcome the diffraction limit? The classical diffraction limit states that the resolution of a linear imaging system is fundamentally limited by one half of the wavelength of light. This implies that conventional light microscopes cannot distinguish two objects placed within a distance closer than 0.5 × 400 = 200nm (blue) or 0.5 × 700 = 350nm (red). This significantly impedes biomedical discovery by ...

94A12 ; 94A08 ; 65T50 ; 65N21 ; 65K10 ; 62H35

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Research schools

Many problems in computational science require the approximation of a high-dimensional function from limited amounts of data. For instance, a common task in Uncertainty Quantification (UQ) involves building a surrogate model for a parametrized computational model. Complex physical systems involve computational models with many parameters, resulting in multivariate functions of many variables. Although the amount of data may be large, the curse of dimensionality essentially prohibits collecting or processing enough data to reconstruct such a function using classical approximation techniques. Over the last five years, spurred by its successful application in signal and image processing, compressed sensing has begun to emerge as potential tool for surrogate model construction UQ. In this talk, I will give an overview of application of compressed sensing to high-dimensional approximation. I will demonstrate how the appropriate implementation of compressed sensing overcomes the curse of dimensionality (up to a log factor). This is based on weighted l1 regularizers, and structured sparsity in so-called lower sets. If time, I will also discuss several variations and extensions relevant to UQ applications, many of which have links to the standard compressed sensing theory. These include dealing with corrupted data, the effect of model error, functions defined on irregular domains and incorporating additional information such as gradient data. I will also highlight several challenges and open problems. Many problems in computational science require the approximation of a high-dimensional function from limited amounts of data. For instance, a common task in Uncertainty Quantification (UQ) involves building a surrogate model for a parametrized computational model. Complex physical systems involve computational models with many parameters, resulting in multivariate functions of many variables. Although the amount of data may be large, the curse ...

41A05 ; 41A10 ; 65N12 ; 65N15 ; 94A12

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Research talks;Computer Science;Geometry

53B21 ; 53C35 ; 94A08 ; 94A12 ; 94A17

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Research schools;Computer Science;Probability and Statistics

Optimal vector quantization has been originally introduced in Signal processing as a discretization method of random signals, leading to an optimal trade-off between the speed of transmission and the quality of the transmitted signal. In machine learning, similar methods applied to a dataset are the historical core of unsupervised classification methods known as “clustering”. In both case it appears as an optimal way to produce a set of weighted prototypes (or codebook) which makes up a kind of skeleton of a dataset, a signal and more generally, from a mathematical point of view, of a probability distribution.
Quantization has encountered in recent years a renewed interest in various application fields like automatic classification, learning algorithms, optimal stopping and stochastic control, Backward SDEs and more generally numerical probability. In all these various applications, practical implementation of such clustering/quantization methods more or less rely on two procedures (and their countless variants): the Competitive Learning Vector Quantization $(CLV Q)$ which appears as a stochastic gradient descent derived from the so-called distortion potential and the (randomized) Lloyd's procedure (also known as k- means algorithm, nu ees dynamiques) which is but a fixed point search procedure. Batch version of those procedures can also be implemented when dealing with a dataset (or more generally a discrete distribution).
In a more formal form, if is probability distribution on an Euclidean space $\mathbb{R}^d$, the optimal quantization problem at level $N$ boils down to exhibiting an $N$-tuple $(x_{1}^{*}, . . . , x_{N}^{*})$, solution to

argmin$_{(x1,\dotsb,x_N)\epsilon(\mathbb{R}^d)^N} \int_{\mathbb{R}^d 1\le i\le N} \min |x_i-\xi|^2 \mu(d\xi)$

and its distribution i.e. the weights $(\mu(C(x_{i}^{*}))_{1\le i\le N}$ where $(C(x_{i}^{*})$ is a (Borel) partition of $\mathbb{R}^d$ satisfying

$C(x_{i}^{*})\subset \lbrace\xi\epsilon\mathbb{R}^d :|x_{i}^{*} -\xi|\le_{1\le j\le N} \min |x_{j}^{*}-\xi|\rbrace$.

To produce an unsupervised classification (or clustering) of a (large) dataset $(\xi_k)_{1\le k\le n}$, one considers its empirical measure

$\mu=\frac{1}{n}\sum_{k=1}^{n}\delta_{\xi k}$

whereas in numerical probability $\mu = \mathcal{L}(X)$ where $X$ is an $\mathbb{R}^d$-valued simulatable random vector. In both situations, $CLV Q$ and Lloyd's procedures rely on massive sampling of the distribution $\mu$.
As for clustering, the classification into $N$ clusters is produced by the partition of the dataset induced by the Voronoi cells $C(x_{i}^{*}), i = 1, \dotsb, N$ of the optimal quantizer.
In this second case, which is of interest for solving non linear problems like Optimal stopping problems (variational inequalities in terms of PDEs) or Stochastic control problems (HJB equations) in medium dimensions, the idea is to produce a quantization tree optimally fitting the dynamics of (a time discretization) of the underlying structure process.
We will explore (briefly) this vast panorama with a focus on the algorithmic aspects where few theoretical results coexist with many heuristics in a burgeoning literature. We will present few simulations in two dimensions.
Optimal vector quantization has been originally introduced in Signal processing as a discretization method of random signals, leading to an optimal trade-off between the speed of transmission and the quality of the transmitted signal. In machine learning, similar methods applied to a dataset are the historical core of unsupervised classification methods known as “clustering”. In both case it appears as an optimal way to produce a set ...

62L20 ; 93E25 ; 94A12 ; 91G60 ; 65C05

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