# Trimester Seminar

Venue: HIM lecture hall, Poppelsdorfer Allee 45
Organizers: Massimo Fornasier, Mauro Maggioni, Holger Rauhut, Thomas Strohmer

## Tuesday, January 5

14:30 - 15:30 Akram Aldroubi (Vanderbilt University, Nashville): Dynamical sampling, cyclical sets, cyclical frames and the spectral theory

Abstract: Let f be a signal at time t=0 of a dynamical process controlled by an operator A that produces the signals Af, A2f, ... at times t=1,2,... . Let M be a measurement operator applied to the series f, Af, A2f, ... at times t=0,1,2,... . The problem is to recover f from the measurements Y = {Mf, MAf, MA2f, ..., MALf}. This is the so called Dynamical Sampling Problem. A prototypical example is when f ∈ l2(ℤ), X a proper subset of ℤ and Y={f(X), Af(X), A2f(X), ..., ALf(X)}. The problem is to find conditions on A, X, L, that are sufficient for the recovery of f. This problem has connection to many areas of mathematics including frames, Banach algebras, and the recently solved Kadison-Singer/Feichtinger conjecture. We will discuss the problem, its applications, and some of the recent results obtained with in collaboration Carlos Cabrelli, Ursula Molter, Armenak Petrosyan, and Sui Tang.

15:30 - 16:00 Round of Introductions

## Thursday, January 7

14:30 - 15:15 Mark Iwen (Michigan State University): Fast phase retrieval from local correlation measurements

Abstract: We develop a fast phase retrieval method which can utilize a large class of local phase- less correlation-based measurements in order to recover a given signal x in Cd (up to an unknown global phase) in near-linear O(d log4 d)-time. Accompanying theoretical analysis proves that the proposed algorithm is guaranteed to deterministically recover all signals x satisfying a natural flat-ness (i.e., non-sparsity) condition for a particular choice of deterministic correlation-based measurements. A randomized version of these same measurements are then shown to provide nonuniform probabilistic recovery guarantees for arbitrary signals x in Cd. Numerical experiments demonstrate the method’s speed, accuracy, and robustness in practice – all code is made publicly available. Finally, we develop an extension of the proposed method to the sparse phase retrieval problem; specifically, we demonstrate a sublinear-time compressive phase retrieval algorithm which is guaranteed to recover a given s-sparse vector x in Cd with high probability in just O(s log5 s · log d)-time using only O(s log4 s · log d) magnitude measurements. In doing so we prove the existence of compressive phase retrieval algorithms with near-optimal linear-in-sparsity runtime complexities.

This talk will discuss joint work with various subsets of:  Aditya Viswanathan (MSU), Wang Yang (HKUST), Rayan Saab (UCSD), and Brian Preskitt (UCSD).

15:15 - 16:00 Rayan Saab (University of California, San Diego): Near-optimal quantization and encoding under various measurement models

Abstract: In the era of digital computing, data acquisition consists of a series of steps. First one samples a signal of interest (modeled as a vector) by taking its inner products with appropriate measurement vectors. This sampling or measurement process is typically followed by quantization, or digitization. Here, the inner products are replaced by elements from a finite set, which in the extreme case may contain only two elements. In turn, quantization is often followed by encoding, or compression, to efficiently represent the quantized data. In this talk we discuss quantization and encoding schemes for a variety of measurement processes, along with their reconstruction algorithms and a theoretical analysis of the associated reconstruction error. We show results for classically oversampled band-limited functions, oversampled linear measurements of finite dimensional signals, and compressed sensing measurements of sparse and compressible signals. For example, in the compressed sensing setup our encoding and reconstruction methods are practical, work in the extreme case of 1-bit quantization, and are robust to noise as well as model mismatch. Moreover, they yield near-optimal approximation accuracy as a function of the bit-rate.

Joint work (in part) with: Ingrid Daubechies, Mark Iwen, Rongrong Wang, Ozgur Yilmaz

## Friday, January 8

10:30 - 11:15 Hans Feichtinger (University of Vienna): Fourier Analysis via the Banach Gelfand Triple

Abstract: Normally the Fourier transform is defined for L1-functions or tempered distributions. For LCA (locally compact Abelian) groups one has to resort to the even more complicated Schwartz-Bruhat space. The talk will indicate how the Banach Gelfand triple viewpoint, using a Banach space of test functions, namely the Segal algebra S0(G), the Hilbert space L2(G) and the dual space SO'(G) can be used to properly define the Fourier transform in a generality suitable for most engineering applications, but also for the purpose of abstract harmonic analysis. The talk is meant to provide a first overview on the definition and basic properties of this Banach Gelfand triple and how it can be used to prove typical results (e.g. the Shannon sampling theorem for band-limited signals).

## Monday, January 18

14:30 - 15:30 Ivan Oseledets (Skolkovo Institute of Science and Technology): Tensor-trains: theory, algorithms, applications

Abstract: In this talk I will describe basic properties of the Tensor-Train (TT) format, which is one of the possible generalization of the matrix singular value decomposition (SVD) to tensors. I will compare it to other approaches (canonical and Tucker decompositions). I will also describe algorithmic challenges and promising application areas. One of the most interesting problems is the optimization over manifolds with multiple low-rank constraints. Several optimization tools have been developed, including nuclear norm optimization, Riemannian optimization, and also more recently AMEN and DMRG approach. What is similar and what is different between those approaches (and the best is still to be found).

15:30 - 16:00 Round of Introductions

## Wednesday, January 20

11:00 - 12:00 Session on open/interesting problems

15:00 - 15:30 Mads Sielemann Jakobsen (Technical University of Denmark): On the duality theorem for Gabor frames

Abstract: The duality principle of Gabor theory states that a Gabor system is a Gabor frame if and only if the adjoint Gabor system is a Riesz sequence. The result goes back to three groups of authors: Dauchechies-Landau-Landau, Jannsen and Ron-Shen. The theorem was presented by Karlheinz Gröchenig last week during the winter school. I will present a proof based on Jannsens ideas. Unlike the proofs by Daubechie-Landau-Landau and Ron-Shen the proof by Jannsen generalizes without much trouble to non-separable lattices.

15:30 - 16:00 Antonio Cicone (Università degli Studi dell'Aquila): A new approach to the decomposition and analysis of nonlinear and nonstationary signals: the Adaptive Local Iterative Filtering method

Abstract: The analysis and decomposition of nonstationary and nonlinear signals in the quest for the identification of hidden quasiperiodicities and trends is of high theoretical and applied interest nowadays. Linear techniques like Fourier and Wavelet Transform, historically used in signal processing, cannot capture properly nonlinear and non stationary phenomena. For this reason in the last few years new nonlinear methods have been developed like the groundbreaking Empirical Mode Decomposition algorithm, aka Hilbert-Huang Transform, and the Iterative Filtering technique. In this seminar I will give an overview of this kind of methods and I will introduce a new algorithm, the Adaptive Local Iterative Filtering, as well as a new definition for Instantaneous frequency that are all together an all-round toolbox for a completely local signal analysis. Some convergence results will be showed as well as examples of applications of these techniques to both artificial and real life signals to give a foretaste of their potential and robustness.

## Thursday, January 21

15:15 - 16:00 Jeff Hogan (University of Newcastle, Australia): Bandpass prolates, prolate shift frames and sampling

Abstract: The classical prolate spheroidal wavefunctions (prolates) arise when solving the Helmholtz equation by separation of variables in prolate spheroidal coordinates. In a beautiful series of papers published in the Bell Labs Technical Journal in the 1960's, they were rediscovered by Landau, Slepian and Pollak in connection with the spectral concentration problem. After years spent out of the limelight while wavelets drew the focus of mathematicians, physicists and electrical engineers, the popularity of the prolates has recently surged through their appearance in certain communication technologies. In this talk we outline some developments in the sampling theory of bandlimited signals that employ the prolates, the construction of bandpass prolate functions, and the frame properties of shifted prolates. This is joint work with Joe Lakey (New Mexico State University).

## Monday, January 25

11:00 - 12:00 Hans G. Feichtinger (University of Vienna): Gabor Analysis and the numerical realization using MATLAB

Abstract: In this MATLAB-based presentation the author will explain how one can understand and illustrate the foundations of Gabor analysis with the help of MATLAB. From the point of view of Abstract Harmonic Analysis one works with functions over finite Abelian groups, hence linear operators can be represented by matrices and optimal estimates can be computed explicitely using matrix analysis methods. The fact that the algebraic structures are the same as in the continuous case, but with the advantage that all sums are finite, allows to destill (and check numerically) various identities in a concrete setting. MATLAB (or OCTAVE, or the LTFAT toolbox) also allow to carry out experiments in order to come up with conjectures or first ideas about possible claims in the continuous case.

Various talks on the subject can be found at
http://www.univie.ac.at/nuhag-php/nuhag_talks/
author = Feichtinger, title = Gabor

14:30 - 15:30 Philipp Grohs (ETH Zurich): Some mathematical properties of deep convolutional networks

15:30 - 16:00 Meeting of all trimester program participants

## Wednesday, January 27

15:15 - 16:00 Ville Turunen (Aalto University): Born–Jordan time-frequency analysis

Abstract: Born-Jordan quantization originates from Heisenberg's matrix mechanics, leading to sharp time-frequency localization of signals of finite energy. The related Born-Jordan distribution provides an attractive alternative to the spectrograms. We characterize the Born-Jordan time-frequency distribution within Cohen's class. Each Cohen's class time-frequency distribution gives rise to a quantization (symbol-to-operator mapping) of pseudo-differential operators, and vice versa. We study properties of the operators especially in the Born-Jordan quantization, and we show how this can be applied in acoustic signal processing, quantum mechanics and medical sciences.

## Thursday, January 28

15:15 - 16:00 Roy Lederman (Princeton University): The Truncated Laplace Transform (and why we can handle a condition number of 101000 - in this case)

Abstract: The Laplace Transform is frequently encountered in mathematics, physics, engineering and other ﬁelds. However, the spectral properties of the Laplace Transform tend to complicate its numerical treatment; therefore, the closely related "Truncated" Laplace Transforms are often used in applications.

I will talk about algorithms for computing the Singular Value Decomposition (SVD) of the truncated Laplace transform. I will focus on some "everyday" numerical problems, and how we bypass them in this case using the remarkable properties of the truncated Laplace transform.

## Wednesday, February 3

14:00 - 14:30 Michael Kech (TU München): Explicit Frames for Phase Retrieval via PhaseLift

Abstract: Phase retrieval is the task of reconstructing a signal x ∈ ℂn up to a global phase from intensity measurements, i.e. from measurements of the form |<x,vi>|2, i=1,...,m. In 2011, Candès, Strohmer and Voroninski showed that when choosing the vectors vi at random, the signal x can be recovered exactly up to a global phase by solving a semidefinite program. I will present an explicit construction of intensity measurements that allow for phase retrieval of every signal via the same semidefinite program, requiring a close to optimal number of measurements.

14:30 - 15:00 Alessandro Buccini (Università degli Studi dell'Insubria): On nonstationary preconditioned iterative regularization methods for image deblurring

Abstract

15:00 - 15:30 Ilaria Giulini (INRIA Saclay): Spectral Clustering & Reproducing kernels

Abstract: We present a new algorithm for spectral clustering applied to an i.i.d. sample of points in a Hilbert space. This algorithm can be described as a change of representation in a reproducing kernel Hilbert space followed by a (greedy) classification. We describe the clustering effect in terms of Markov chains with exponential transitions and we discuss the stability of the algorithm through the study of Gram operators in Hilbert spaces.

15:30 - 16:00 Thang Huynh (New York University): Quantization of Phaseless Measurements

Abstract: In this talk, I will discuss how the distributed noise-shaping method of Chou and Gunturk can be extended to the quantization problem of phaseless measurements and will show that a suitably modified recovery algorithm based on PhaseLift guarantees near-optimal error performance. Joint work with Gunturk and Jeong.

## Wednesday, February 10

15:15 - 16:00 Peter Jung (TU Berlin): A Szegö-formula for the Capacity of Doubly-Dispersive Gaussian Channels

Abstract: In this talk I will consider time-continuous doubly-dispersive communication channels with additive Gaussian noise. An asymptotic formula for the channel capacity (supremum of achievable data rates) is established for the case where the (Kohn-Nirenberg) symbol of the channel operator fulfills certain integrability, smoothness and oscillation conditions.

More precisely, the Holsinger-Gallager model for translating the time-continuous channel for a sequence of increasing time-intervals to a series of equivalent sets of discrete, parallel channels is applied. I will quantify conditions when this procedure converges.

The key to this is result is a new Szegö formula for certain pseudo-differential operators with real-valued symbol. The limit holds if the symbol is (with respect to time) contained in Besov spaces Bs∞,1 for smoothness parameters s > 1 requiring certain oscillatory behavior in time. Finally, the formula justifies the so called "water-filling principle" in time and frequency as general technique independent of a sampling scheme.

## Thursday, February 11

11:00 - 12:00 Emily King (Universität Bremen): Shearlets and Morphological Component Analysis

## Thursday, February 25

11:00 - 12:00 Holger Boche (TU München): Mathematics of Signal Design for Communication Systems and Szemerédi’s and Green-Tao’s Theorems

Abstract: Orthogonal transmission schemes constitute the foundations of both our present -, and future communication standards. One of the major drawback of orthogonal transmission schemes is their high dynamical behaviour, which can be measured by the so called Peak-to-Average power value { the ratio between the peak value (i.e. L-norm) and the average power (i.e. L2-norm) of a signal. This undesired behaviour of orthogonal schemes has remarkable negative impacts to the performance, the energy efficiency, and the maintain cost of the transmission systems. In this talk, we give some discussions concerning to the problem of reduction of the high dynamics of an orthogonal transmission scheme. We show that this problem is connected with some mathematical fields, such as functional analysis (Hahn-Banach Theorem and Baire Category), additive combinatorics (Szemeredi Theorem, Green-Tao Theorem on arithmetic progressions in the primes, sparse Szemeredi type Theorems, by Conlon and Gowers, and the famous Erdos problem on arithmetic progressions), and both trigonometric - and non-trigonometric harmonic analysis.

15:00 - 16:00 Luis Daniel Abreu (Austrian Academy of Sciences): Multitaper Spectral Estimation and Phase Retrieval from Coded Diffraction Patterns

Abstract: After a tutorial-level presentation of spectral estimation, we will point out the similarities and differences between the multi-taper approach to the problem and the one of  phase retrieval from power spectra of masked signals (known in statistics as taped periodograms and in image analysis as coded diffraction patterns).

The most important result to be presented is a bound for the spectral leackage in David Thomson´s multi-taper method. This confirms his conjecture, for which numerical evidence was offered in the 1982 Proc. IEEE paper where the method was presented. By bounding the spectral leackage of the estimator, we were able to explicitly quantify the bias-variance tradeoff in the method.

A short discussion about extension of existing methods of phase retrieval and multi-taper spectral estimation to time-frequency analysis will be also presented.

## Monday, February 29

15:00 - 16:00 Roy Ledermann (Princeton University): Common Variable Learning Using Alternating Diffusion and Deep Siamese Networks

Abstract: One of the challenges in signal processing is to distinguish between different sources of variability. In this work, we consider the manifold learning perspective and deep networks perspective on using diversity for separating sources of variability. From the manifold learning standpoint, we introduce a method based on alternating products of diffusion operators, which extracts the common source of variability from multimodal recordings. From the deep networks perspective, we discuss an approach based on Siamese Networks.

## Tuesday, March 1

15:00 - 16:00 Christine De Mol (Université Libre de Bruxelles): Inverse and blind imaging with positivity

Abstract: We describe blind inversion algorithms for the case where the unknown image, the matrix to invert and the data are positive, in which case the problem amounts to nonnegative matrix factorization (NMF). The cost function to be minimized contains a data fidelity term (least squares residual or Kullback-Leibler divergence) as well as various regularizing penalties. The proposed algorithms consist of alternating multiplicative updates preserving the positivity at each iteration. We will discuss their convergence properties and present results of numerical simulations for blind deconvolution and hyperspectral imagery.

## Friday, March 4

11:00 - 11:30 Sandra Keiper (TU Berlin): Discrete-Valued Sparse Signals

Abstract

11:30 - 12:00 Yin Xian (Duke University): DCTNet and PCANet for acoustic signal feature extraction

Abstract: We introduce the use of DCTNet, an efficient alternative to PCANet, for acoustic signal classification. When the eigenfunctions of the local sample covariance matrix (PCA) are well approximated by the Discrete Cosine Transform (DCT) functions, and are used as filterbanks for convolution and feature extraction, each layer of PCANet and DCTNet is essentially a time-frequency representation. We relate DCTNet to spectral feature representation methods, such as the the short time Fourier transform (STFT), spectrogram and linear frequency spectral coefficients (LFSC). Experimental results on whale vocalization data show that DCTNet improves classification rate, demonstrating DCTNet's applicability to signal processing problems such as underwater acoustics.

General introduction on ocean acoustic environment, whale vocalizations characteristics, intrinsic structure and parameters of whale vocalizations will also be included.

## Monday, March 7

14:45 - 15:30 Claire Boyer (Institut de Mathématiques de Toulouse): Structured compressed sensing

Abstract: The talk will be divided into 2 parts. First, we will theoretically justify the applicability of Compressed Sensing (CS) in real-life applications. To do so, I will introduce new CS theorems compatible with physical acquisition constraints. These new results do not only encompass structure in the acquisition but also structured sparsity of the signal of interest. Then, we will present a new way to generate subsampling schemes that can be implemented on real sensors and that give good reconstruction results. This work relies on measure projection and will be illustrated in the case of MRI.

15:30 - 16:00 Round of Introductions

## Thursday, March 10

15:00 - 16:00 Bruno Torresani (Aix-Marseille Université): Analysis and Estimation of deformed stationary signals

Abstract: We consider random processes generated by deformations of stationary processes. In the 1D situation, examples of such deformations include modulation and time warping. Under regularity assumptions on the deformation, the deformation can be approximated by a translation in a suitable representation domain. This permits the estimation of the deformation and the statistical characteristics of the underlying stationary process. We will discuss identifiability issues, and approximation results, present the estimation algorithm and illustrate with non-stationary sound analysis examples.

Joint work with Harold Omer (I2M Marseille).

## Monday, March 21

15:15 - 16:00 Markus Hansen (Technische Universität München): Inferring interaction rules from observations of evolutive systems

Abstracts: The description of social dynamics and self-organizing systems via systems of ODEs has attracted a lot of atention in recent years, in particular the seminal work of Cucker and Smale. We wish to study a related inverse problem: Form observations of such dynamical systems we wish to reconstruct the underlying interaction rules. In particular, for dynamical systems which are obtained from a gradient descent of some energy functional depending only on the mutual distances, i.e. systems of the form

$\dot x_i=\sum_{j\neq i}a(|x_i-x_j|)\frac{x_i-x_j}{|x_i-x_j|}$

we wish to infer the interaction kernel function a. In this talk we will present a variational approach to this problem, and (if the time allows) also sketch a second approach for an adaptive algorithm.

## Tuesday, March 22

15:15 - 16:00 Wenjing Liao (Duke University): Learning multiscale adaptive approximations of data near low-dimensional sets

Abstract: Many data sets in image analysis and signal processing are in a high-dimensional space but exhibit a low-dimensional structure. We are interested in building efficient representations of these data for compression and inference. I will discuss a multiscale geometric method to build a dictionary which provides sparse representations for data lying on a low-dimensional manifold embedded in a high-dimensional space. Our method is based on a multiscale partition of data and constructing empirical piecewise linear approximations, which are adaptive even when the low dimensional set has geometric regularity that varies at different locations and scales. Indeed our algorithm will automatically learn the distribution of the data and choose the right partition to use. The finite-sample performance guarantee of our adaptive method is proved for a large model class.

## Wednesday, March 23

15:15 - 16:00 Yutong Chen (Princeton University): Non-unique games over compact groups and orientation estimation in cryo-EM

Abstract: Let G be a compact group and let fij ∈ L2(G). We define the Non-Unique Games (NUG) problem as finding g1, …, gn ∈ G to minimize ∑i,j fij (gigj-1). We devise a relaxation of the NUG problem to a semidefinite program (SDP) by taking the Fourier transform of fij over G, which can then be solved efficiently. The NUG framework can be seen as a generalization of the little Grothendieck problem over the orthogonal group and the Unique Games problem and includes many practically relevant problems, such as the maximum likelihood estimator to registering bandlimited functions over the unit sphere in d-dimensions and orientation estimation in cryo-Electron Microscopy.

This is joint work with Afonso Bandeira (MIT) and Amit Singer (Princeton).

## Tuesday, March 29

15:00 - 16:00 Alain Pajor (University of Paris-Est Marne la Vallée): On the covariance matrix

Abstract: We will survey recent results on the empirical covariance matrix of a sample from a random vector which coordinates are not necessarily independent. We will discuss the quantitative point of view as well as the asymptotic point of view.

## Thursday, March 31

11:00 - 11:45 Dominik Jüstel (Technische Universität München): Diffraction of time-harmonic radiation, radiation design, and reconstruction from intensity measurements

Abstract: In this talk I will present a mathematical model for the diffraction of time/harmonic electromagnetic radiation in the X-ray regime, discuss the design of new waveforms within this framework and ideas how to reconstruct the structure of the illuminated sample from intensity measurements. In particular, I will explain our recent work on twisted X-rays, which are designed to analyze helical structures like carbon nanotubes or filamentous viruses [1,2]. If time allows, the case of structures with more general symmetries will be discussed in an abstract harmonic analysis setting.

This is joint work with Gero Friesecke (Technical University of Munich) and Richard D. James (University of Minnesota).

[1] D. Jüstel, G. Friesecke, R. D. James: Bragg-Von Laue diffraction generalized to twisted X-rays, Acta Cryst. A72, 190-196, 2016.

[2] G. Friesecke, R. D. James, D. Jüstel: Twisted X-rays: incoming waveforms yielding discrete diffraction patterns for helical structures, arXiv:1506.04240.

15:15 - 16:00 Jean-Luc Bouchot (RWTH Aachen): Robust RBF interpolation from cardinal functions

Abstract: In this talk I review some basics of uniform and non-uniform interpolation via multiquadrics and analyze its robustness to noise in the sampling points (both jitter and measurement noise). As a consequence, I will introduce a novel method for computing RBF interpolation based on sampling the Fourier transform of its cardinal function. This is joint work with Keaton Hamm (Vanderbilt University)

## Tuesday, April 5

15:15 - 16:00 Philipp Walk (Caltech): Ambiguities of Discrete Convolutions

Abstract: Blind deconvolution is a bilinear inverse problem and a "Holy Grail" in signal processing. Without any knowledge about the input signals x and y the reconstruction from the convolution output x∗y is an ill-posed problem, since the convolution product has many ambiguities in the input signals. We will characterize in this talk for the one-dimensional convolution x∗y of n-dimensional vectors x and y, all ambiguities by a polynomial factorization.

Moreover, we will characterize as a special case the ambiguities in phase retrieval and propose constraints on the input signals which allows a unique reconstruction.

## Thursday, April 7

15:15 - 16:00 Björn Bringmann (Technische Universität München): Solution paths of convex regularizations

Abstract: Energy functionals with quadratic data fidelity terms and l1-regularizations are frequently used in image and signal analysis to recover a signal from linear measurements. To obtain an approximate reconstruction, one often computes minimizers of these energies. In order to get close to the original signal, the regularization parameter t, which determines the trade-off between fitting the measured data and obtaining regularized signals, has to be chosen wisely. Instead of solving the problem just for a fixed t, I will explain how to compute a solution for every nonnegative regularization parameter.

Under a condition on the solution path, called "one-at-a-time", the wellknown homotopy method is able to compute a piecewise linear and continuous solution path by solving a linear system at every kink. To illustrate that this condition is necessary, I will discuss a toy example in which the standard homotopy method fails.

By replacing the linear system with a nonnegative least squares problem, our method works for arbitrary measurement matrices and input data. I will give a full characterization of the set of possible directions and discuss the finite termination property of our algorithm.

If time permits, I will discuss some connections with the spectral framework for convex regularizations as introduced by Gilboa.

Joint work with Daniel Cremers, Felix Krahmer, and Michael Möller.

## Thursday, April 14

14:45 - 15:15 Michael Sandbichler (Universität Innsbruck): Sparsifying transformations and compressed sensing for photoacoustic tomography

Abstract: An important direction of research in tomographic imaging is the reduction of measurement cost, while maintaining (high) image quality. We will discuss a method for using compressed sensing techniques in photoacoustic imaging via sparsifying temporal transformations of the data.

15:15 - 16:00 Richard Küng (Universität zu Köln): The power of non-negative shape constraints: Compressed Sensing without 1-norm regularization

Abstract: Compressed sensing allows for stably reconstructing sparse signals from a sub-linear number of measurements in a computationally tractable way. Most reconstruction algorithms exploit a 1-norm regularization to promote sparsity. Perhaps surprisingly, such a regularization term is often superfluous, when the signals of interest are constrained to have non-negative coefficients. While this behavior is well-understood for perfectly noiseless measurement procedures, much less is known in the noisy setting. We fill this gap by providing novel reconstruction guarantees that assure stability towards noise corruption. These findings have the added benefit of not requiring any a-priori noise bound in the algorithmic reconstruction -- a desirable feature in many applications.

## Friday, April 15

11:00 - 11:45 Illia Karabash (Universität Lübeck): Recovery of periodicities hidden in heavy-tailed noise

based on our joint arxiv preprint arXiv:1512.08732 with Jürgen Prestin

Abstract: We address a parametric joint detection-estimation problem for discrete signals of the form x(t) = Σn=1,…,N αn e-iλnt + εt, t ∈ ℕ, with an additive noise represented by independent centered complex random variables εt. The distributions of εt are assumed to be unknown, but satisfying various sets of conditions. We prove that in the case of a heavy-tailed noise it is possible to construct asymptotically strongly consistent estimators for the unknown parameters of the signal, i.e., the frequencies λn, their number N, and complex amplitudes αn. For example, one of considered classes of noise is the following: εt are independent identically distributed random variables with E(εt) = 0 and E (|εt| ln|εt|) < ∞. The construction of estimators is based on detection of singularities of anti-derivatives for Z-transforms and on a two-level selection procedure for special discretized versions of superlevel sets. The consistency proof relies on the convergence theory for random Fourier series.