Workshop Donau-Isar-Inn          WDI² - Approximation Theory and Applications             in honor of Prof. Dr. Rupert Lasser

Program

• Date: Friday, July 20, 2018, 1 PM to 6 PM.
• Location: Zentrum Mathematik der TUM, Boltzmannstr. 3, 85747 Garching, Room MI 00.06.011 (Lecture Hall MI HS 3).

(preliminary) Schedule

1pm	Welcome Address
1:05pm	Frames and dynamical samplingFrames and dynamical sampling The talk will give a short survey on frame theory in Hilbert spaces, followed by a more detailed discussion of the recent recearch topic "dynamical sampling." Formulated in purely mathematical terms, the key question is when and how a frame a frame can be represented via iterations of a certain bounded operator, acting on a fixed vector in the underlying Hilbert space. The talk presents joint work with Marzieh Hasannasab. ​ Ole Christensen  DTU - Technical University of Denmark
1:50pm	Break
2:00pm	Approximation theoretic properties of deep ReLU neural networks Approximation theoretic properties of deep ReLU neural networks Studying the approximation theoretic properties of neural networks with smooth activation function is a classical topic. The networks that are used in practice, however, most often use the non-smooth Re-L-U activation function. Despite the recent incredible performance of such networks in many classification tasks, a solid theoretical explanation of this success story is still missing. In this talk, we will present recent results concerning the approximation theoretic properties of deep ReLU neural networks which help to explain some of the characteristics of such networks; in particular we will see that deeper networks can approximate certain classification functions much more efficiently than shallow networks, which is not the case for most smooth activation functions. We emphasize though that these approximation theoretic properties do not explain why simple algorithms like stochastic gradient descent work so well in practice, or why deep neural networks tend to generalize so well; we purely focus on the expressive power of such networks. As a model class for classifier functions we consider the class of (possibly discontinuous) piecewise smooth functions for which the different "smooth regions" are separated by smooth hyper surfaces. Given such a function, and a desired approximation accuracy, we construct a neural network which achieves the desired approximation accuracy, where the erroris measured in L^2. We give precise bounds on the required size (in terms of the number of weights) and depth of the network, depending on the approximation accuracy, on the smoothness parameters of the given function, and on the dimension of its domain of definition. Finally, we show that this size of the networks is optimal, and that networks of smaller depth would need significantly more weights than the deep networks that we construct, in order to achieve the desired approximation accuracy.​ Felix Voigtländer  Catholic University of Eichstätt-Ingolstadt
2:30pm	A Deterministic Sparse FFT for Functions with Structured Fourier Sparsity A Deterministic Sparse FFT for Functions with Structured Fourier Sparsity Abstract here. ​ Sina Bittens  University of Göttingen
3:00pm	Coffee and Cake
3:45pm	Von der Abstrakten zur Anwendungsorientierten Harmonischen AnalyseVon der Abstrakten zur Anwendungsorientierten Harmonischen Analyse In diesem kurzen Beitrag möchte ich ausgehend von einer kurzen historischen Betrachtung des Gebietes der Harmonischen Analyse, über die Tagungen zur Harmonischen Analyse in der abstraktesten Form, bei denen wir einaner erstmals begegnet sind, auf die Entwicklungen der letzten 40 Jahre eingehen. Diese Zeit ist geprägt von einer Reaktivierung und starken Verbreiterung der Forschungsaktivitäten, vor allem aber auch des Anwendungsbezuges sowie der Verwendung von Numerischen Methoden. Die Entwicklung der Wavelets, aber ebenso der Gabor Analysis oder neuerdings der Shearlets, aber genauso Einsichten im Bereich der orthogonalen Polynome und Hypergruppen haben dazu beigetragen. Ich werde an die junge Generation appellieren, weiter Anstrengungen in diese Richtung zu machen, auch um die Bedeutung der Harmonischen Analyse im Gesamtkontext der Angewandten Mathematik hochzuhalten. ​ Hans Georg Feichtinger  University of Vienna
4:05pm	Break
4:10pm	Sampling in shift-invariant spaces with totally positive generatorsSampling in shift-invariant spaces with totally positive generators We will present new results on nonuniform sampling in shift-invariant spaces whose generator is a totally positive function. For a subclass of such generators the sampling theorems can be formulated in analogy to the theorems of Beurling and Landau for bandlimited functions. In contrast to the cardinal series, the reconstruction procedures for sampling in a shift-invariant space with a totally positive generator are local and thus accessible to numerical linear algebra.​ Karlheinz Gröchenig  University of Vienna
4:55pm	Break
5:05pm	Subsampled random convolutions in sparse recoverySubsampled random convolutions in sparse recovery I will discuss recovery of (approximately) sparse vectors from its subsampled convolution with a random vector and, in particular, provide bounds on the number of samples required for successful recovery. Apart from this standard linear compressive sensing setup, we will present new results for one-bit compressed where only the sign of each entry of the subsampled convolution is retained. Finally, we generalize to certain nonlinear functions replacing the sign function. ​ Holger Rauhut  RTWH Aachen
5:50pm	End of Scientific Program
7:00pm	Dinner Location: Scheidegger

Workshop dinner

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