# HATA seminar

## Upcoming talks

## Past talks

**Felix Voigtländer (Katholische Universität Eichstätt-Ingolstadt)**: *Approximation theoretic properties of deep ReLU neural networks*.

Abstract:
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 rectified linear unit (ReLU) 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 hypersurfaces.
Given such a function, and a desired approximation accuracy, we construct a neural network which achieves the desired approximation accuracy, where the error is measured in L^p. 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.

(*lunch seminar, 10.10.2018 at 12:05, DTU, Building 324/Room 141*)

**Qaiser Jahan**: *Characterization of low-pass filters on local fields of positive characteristic*.

Abstract: In this talk, we will start from basic definition of local fields after that we give necessary and sufficient conditions on a function to be a low-pass filter on a local field K of positive characteristic associated to the scaling function for multiresolution analysis of L^2(K). We also give the sketch of the proof of main result.

(*HATA seminar, 1.11.2016 at 13:00, DTU, Building 324/Room 141*)

**Mads S. Jakobsen**: *Gabor frames on locally compact abelian groups and related topics*.

Abstract: This thesis consists of four papers. The first one introduces generalized translation invariant systems and considers their frame properties, the second and third paper give new results on the theory of Gabor frames, and the fourth is a review paper with proofs and new results on the Feichtinger algebra.

The generalized translation invariant (GTI) systems provide, for the first time, a framework which can describe frame properties of both discrete and continuous systems. The results yield the well-known characterizations of dual frame pairs and Parseval frames of Gabor-, wavelet-, curvelet- and
shearlet-type and for (generalized) shift-invariant systems and their continuous formulations.

This thesis advances the theory of both separable and non-separable, discrete, semi-continuous and continuous Gabor systems. In particular, the well established structure theory for separable lattice Gabor frames is extended and generalized significantly to Gabor systems with time-frequency shifts along closed subgroups in the time-frequency plane. This includes density results, the Walnut representation, the Wexler-Raz biorthogonality relations, the Bessel duality and the duality principle between Gabor frames and Gabor Riesz bases.

The theory of GTI systems and Gabor frames in this thesis is developed and presented in the setting of locally compact abelian groups, however, even in the euclidean setting the results given here improve the existing theory.

Finally, the thesis contains a review paper with proofs of all the major results on the Banach space of functions known as the Feichtinger algebra. This includes many of its different characterizations and treatment of its many equivalent norms, its minimality among all time-frequency shift invariant Banach spaces and aspects of its dual space, operators on the space and the kernel theorem for the Feichtinger algebra. The work also includes new findings such as a characterization among all Banach spaces, a forgotten theorem by Reiter on Banach space isomorphisms of the Feichtinger algebra, and new useful inequalities.

(*PhD defense, 28.10.2016 at 14:00, DTU, Building 303A/Aud 41*)

**Eugenio Hernandez**: *Systems generated by unitary representations of discrete groups*.

Abstract: Unitary representations of discrete groups acting on a Hilbert space \mathbb H (such as translations, modulations and dilations) play an important role in Multiresolution Analysis and approximation theory. For a group \Gamma and a unitary representation \Pi : \Gamma \to \cal U (\mathbb H) we study reproducing properties of the orbit of \psi \in \mathbb H under \Pi, that is O_{\Pi, \Gamma}(\psi)=\{\Pi(\gamma)\psi: \gamma \in \Gamma\}. Previous results known for the case of \Gamma been abelian are extended to the non abelian case. The results are obtained using left regular representations and the von Neumann algebra of the group \Gamma.

(*HATA seminar, 27.10.2016 at 13:00, DTU, Building 324/Room 1xx*)

**Hans G. Feichtinger**: *An Alternative Approach to Convolution and the Fourier Transform*.

Abstract: When one looks at the Fourier transform in the mathematical literature the description starts usually with Fourier Series for periodic functions or right away with the Fourier transform as an integral transform. In either case the transform requires to use integrals, and of course the Lebesgue integral appears to provide the natural domain, namely the space L^1 of integrable functions. Similar arguments apply to the convolution integral. Combining the two concepts one can then derive the all-important convolution theorem, Fourier inversion and Plancherel's theorem,
showing that the ``complicated convolution* is turned into easy pointwise multiplication. But why should we be interested in convolution? Is it a natural product for integrable functions? And which functions do ``have a Fourier transform*?
Aside from heuristic manipulations, leading to the forward and inverse Fourier transform the above results are certainly important to (electrical) engineers, when they deal with translation invariant systems, which are usually described by black boxes. They correspond to convolution operators with the so-called impulse response, which is the output of the system to a ``Dirac delta-function'', and can be described alternatively by their transfer function.
We will describe a mathematically correct approach to convolution and Fourier transform which is based on simple functional analytic principles and encompasses the two aspects of basic Fourier analysis in a way (hopefully well) understandable for both sides (mathematicians and engineers). Furthermore, many aspects of this approach can by supported by a set of simple MATLAB experiments, which connects the material with basic concepts from linear algebra and polynomials with complex coefficients.

(*HATA seminar, 13.10.2016 at 13:15, DTU, Building 324/Room 141*)

**Shidong Li**: *On Analysis Approaches to Compressed Sensing with Coherent Frames*.

Abstract: Analysis approaches for compressed sensing problems with sparse (coherent) frame representations will be discussed. A notion of the sparsity-inducing dual frame of a non-exact frame and its properties is introduced. The sparse dual frame is a new notion of optimal dual frames。It is motivated in the study of compressed sensing problems where signals are sparse with respect to redundant dictionaries (frames). A sparse-dual-frame based 1_1-analysis approach for compressed sensing (CS) will also be presented. An alternating iterative algorithm is proposed. An error bound ensuring the correct signal recovery is obtained. Empirical studies over generally difficult CS problems demonstrate that the new sparse-dual-based approach provides satisfactory solutions, whereas other existing means do not.

(*HATA seminar, 27.9.2016 at 14:00, DTU, Building 303B/Room 136*)

**Emily King**: *Difference Sets and Grassmannian Packings*.

Abstract:
It is often of interest to find subspaces which are optimally spread apart. For example, if one wants a set of vectors (representing one dimensional subspaces) which have similar properties to orthonormal bases, the vectors should as non-parallel as possible. There are a number of constructions possible using tools from combinatorial design theory: difference sets and their generalizations. In this talk, the connection between algebraic combinatorics and geometric packings will be presented, including brand new constructions of packings.

(*HATA seminar, 20.9.2016 at 13:00, DTU, Building 303B/Room 134*)

**Emil Solsbæk Ottosen**: *Sparse Gabor expansions of music signals*.

Abstract: We investigate sparseness of various Gabor expansions – both stationary and nonstationary – applied to piano music. Also, we show that so-called decomposition spaces can be used as smoothness spaces for certain nonstationary Gabor frames. Both practical examples and theoretical results will be discussed.

(*HATA seminar, 18.8.2016 at 13:00, DTU, Building 303B/Room 136*)

**Ole Christensen**: *On approximately dual frames*.

Abstract: The duality theory for frames is well developed by now. However, often it is not possible to
apply dual frames directly, due to practical constraints. The talk will discuss the more flexible concept
of approximately dual frames and its relation to perturbation theory.

(*HATA seminar, 10.5.2016 at 14:00, DTU, Building 306/Aud. 36*)

**Yavar Khedmati Yengejeh**: *Continuous Generalized Frames in Hilbert Spaces*.

Abstract: Abdollahpour and Faroughi introduced and invistigated
continuous g-frames and Riesz-type continuous g-frames. We
provide some necessary and sufficient conditions under which a
family of bounded operators is a continuous g-frame (Riesz-type
continuous g-frame). Also, we introduce concepts of disjoint,
strongly disjoint, weakly disjoint continuous g-frames in Hilbert
spaces and we get equivalent conditions to these notions.

(*HATA seminar, 19.4.2016 at 14:00, DTU, Building 303B/Room 134*)

**James Murphy** (Duke University): *Single image superresolution through directional representations*.

Abstract: The problem of image superresolution has been significant in the image processing community for many years, and has seen a recent mathematical resurgence. We discuss a superresolution algorithm based on sparse mixing estimators (SME) with shearlet frames. This work introduces the anisotropic frame regime to the state-of-the-art SME algorithm of Mallat and Yu. Synthetic and remotely sensed images will be analyzed, showing the gains from incorporating anisotropy. Joint work with W. Czaja and D. Weinberg.

(*HATA seminar, 9.2.2016 at 14:00, DTU, Building 303B/Room 130*)

**Romanos Malikiosis** (TU Berlin): *Full spark Gabor frames in finite dimensions*.

Abstract: A Gabor frame is the set of all time–frequency translates of a complex function
and is a fundamental tool in utilizing communications channels with wide
applications in time–frequency analysis and signal processing. When the domain
of the function is a finite cyclic group of order N, then the Gabor frame
forms a design on the complex sphere in N dimensions; when the N2 unit vectors
that constitute this Gabor frame are pairwise equiangular then the Gabor
frame forms a spherical 2-design, and in addition, it has minimal coherence, an
ideal property in terms of compressive sensing (whether such an equiangular set
exists is also known as the SIC–POVM existence problem, which is open since
1999).
In this talk, we will deal with the question of existence of a Gabor frame such
that every N vectors form a basis (the discrete analogue of the HRT conjecture);
such a frame is called a full spark Gabor frame. This question was posed by
Lawrence, Pfander and Walnut in 2005 and was answered in the affirmative by
the speaker in 2013 unconditionally. This result has applications in operator
identification, operator sampling, and compressive sensing.

(*HATA seminar, 18.11.2015 at 13:00, DTU, Building 303B/Room 134*)

**Brigitte Forster-Heinlein** (University Passau): *The commutative diagram of signal processing*.

Abstract: We consider variations on the commutative diagram consisting of the Fourier transform, the Sampling Theorem and the Paley-Wiener Theorem. We start from a generalization of the Paley-Wiener theorem and consider entire functions with specific growth properties along half-lines. Our main result shows that the growth exponents are directly related to the shape of the corresponding indicator diagram, e.g., its side lengths. Since many results from sampling theory are derived with the help from a more general function theoretic point of view (the most prominent example for this is the Paley-Wiener Theorem itself), we motivate that a closer examination and understanding of the Bernstein spaces and the corresponding commutative diagrams can—via a a limiting process to the straightline interval [-A,A]-yield new insights into the L^{p}(\mathbb{R})—sampling theory. This is joint work with Gunter Semmler, Technische Universität Bergakademie Freiberg, Germany.

(*HATA seminar, 4.11.2015 at 14:00, DTU, Building 303B/Room 134*)

**Peter Massopust** (TU Munich): *Fractional Cone and Hex Splines*.

Abstract: We introduce an extension of cone splines and box splines to fractional and complex orders. These new families of multivariate splines are defined in the Fourier domain along certain s-dimensional meshes and include as special cases the three-directional box splines and hex splines previously considered by Condat, Van De Ville et al. These cone and hex splines of fractional and complex order generalize the univariate fractional and complex B-splines in a natural way. Explicit time domain representations are derived for these splines on 3-directional meshes. We present some properties of these two multivariate spline families such as recurrence, decay and refinement. Finally it is shown that a bivariate hex spline and its integer lattice translates form a Riesz basis of its linear span. This is joint work with Pat Van Fleet, University of St. Thomas, St. Paul, MN, USA.

(*HATA seminar, 4.11.2015 at 13:00, DTU, Building 303B/Room 134*)

**Marzieh Hasannasab** (Kharazmi University): *Hilbert module frames II*.

(*HATA seminar, 08.06.2015, DTU, Building 303B/Matematicum*)

**Kamilla Haahr Nielsen** (DTU): *The frame set of Gabor systems with B-spline generators*.

(*HATA seminar, 08.06.2015, DTU, Building 303B/Matematicum*)

**Tobias Kloos** (TU Dortmund): *Totally Positive Functions and Exponential B-splines in Gabor Analysis*.

(*HATA seminar, 02-04.02.2015, DTU, Building 303B/Matematicum*)

**Ole Christensen** (DTU): *What I don’t know and would like to know in 2015*.

(*HATA seminar, 03.02.2015, DTU, Building 303B/Matematicum*)

**Jakob Lemvig** (DTU): *Fiberizations and Zak transform methods*.

(*HATA seminar, 03.02.2015, DTU, Building 303B/Matematicum*)

**Diana Stoeva** (Austrian Academy of Sciences): *On the duality principle in Gabor analysis and R-duals*.

(*HATA seminar, 02.02.2015, DTU, Building 303B/Matematicum*)

**Mads S. Jakobsen** (DTU): *The duality principle for Gabor frames*.

(*HATA seminar, 02.02.2015, DTU, Building 303B/Matematicum*)

**Ole Christensen** (DTU): *There are the good guys and the bad guys - Localization of frames*.

(*HATA seminar, 11.12.2014, DTU, Building 303B/Matematicum*)

**Hartmut Führ** (RWTH Aachen): *Wavelet analysis in higher dimensions and the resolution of wavefront sets*.

(*HATA seminar, 25.11.2014, DTU, Building 303B/134*)

**Marzieh Hasannasab** (Kharazmi University): *Hilbert module frames I*.

(*HATA seminar, 29.10.2014, DTU, Building 303B/Matematicum*)

**Peter Massopust** (TU Munich): *B-splines and fractional splines*.

(*HATA seminar, 29.10.2014 at 11:00, DTU, Building 303B/Matematicum*)