I am working in signal processing, information theory, communicationtheory and applied math. My research profile is the interface betweencommunication engineering, data science and its mathematicaltreatment.
2017 42nd International Conference on Infrared, Millimeter, and Terahertz Waves (IRMMW-THz), 2017
We present a single detector THz imaging system with millimeter spatial resolution that operates ... more We present a single detector THz imaging system with millimeter spatial resolution that operates without any mechanical movements of the detector and the THz optics. The imaging system, an optically controllable 0.35 THz single-pixel camera, is used in transmission mode to image binary targets. The typical size of the targets is several centimeters diameter and the acquisition time for such targets is less than 2 minutes. The software reconstruction of the acquired signals yields images with more than 2500 pixels, respective < 1 mm spatial resolution. As an outlook first results obtained with the camera's radar-reflection mode will be presented.
Block-sparse regularization is already well known in active thermal imaging and is used for multi... more Block-sparse regularization is already well known in active thermal imaging and is used for multiple-measurement-based inverse problems. The main bottleneck of this method is the choice of regularization parameters which differs for each experiment. We show the benefits of using a learned block iterative shrinkage thresholding algorithm (LBISTA) that is able to learn the choice of regularization parameters, without the need to manually select them. In addition, LBISTA enables the determination of a suitable weight matrix to solve the underlying inverse problem. Therefore, in this paper we present LBISTA and compare it with state-of-the-art block iterative shrinkage thresholding using synthetically generated and experimental test data from active thermography for defect reconstruction. Our results show that the use of the learned block-sparse optimization approach provides smaller normalized mean square errors for a small fixed number of iterations. Thus, this allows us to improve th...
Countless signal processing applications include the reconstruction of signals from few indirect ... more Countless signal processing applications include the reconstruction of signals from few indirect linear measurements. The design of effective measurement operators is typically constrained by the underlying hardware and physics, posing a challenging and often even discrete optimization task. While the potential of gradient-based learning via the unrolling of iterative recovery algorithms has been demonstrated, it has remained unclear how to leverage this technique when the set of admissible measurement operators is structured and discrete. We tackle this problem by combining unrolled optimization with Gumbel reparametrizations, which enable the computation of low-variance gradient estimates of categorical random variables. Our approach is formalized by GLODISMO (Gradient-based Learning of DIscrete Structured Measurement Operators). This novel method is easy-to-implement, computationally efficient, and extendable due to its compatibility with automatic differentiation. We empirically...
This work deals with the problem of distributed data acquisition under non-linear communication c... more This work deals with the problem of distributed data acquisition under non-linear communication constraints. More specifically, we consider a model setup where M distributed nodes take individual measurements of an unknown structured source vector x_0 ∈R^n, communicating their readings simultaneously to a central receiver. Since this procedure involves collisions and is usually imperfect, the receiver measures a superposition of non-linearly distorted signals. In a first step, we will show that an s-sparse vector x_0 can be successfully recovered from O(s ·(2n/s)) of such superimposed measurements, using a traditional Lasso estimator that does not rely on any knowledge about the non-linear corruptions. This direct method however fails to work for several "uncalibrated" system configurations. These blind reconstruction tasks can be easily handled with the ℓ^1,2-Group Lasso, but coming along with an increased sampling rate of O(s·{M, (2n/s) }) observations. In fact, the purp...
The success of the compressed sensing paradigm has shown that a substantial reduction in sampling... more The success of the compressed sensing paradigm has shown that a substantial reduction in sampling and storage complexity can be achieved in certain linear and non-adaptive estimation problems. It is therefore an advisable strategy for noncoherent information retrieval in, for example, sporadic blind and semi-blind communication and sampling problems. But, the conventional model is not practical here since the compressible signals have to be estimated from samples taken solely on the output of an un-calibrated system which is unknown during measurement but often compressible. Conventionally, one has either to operate at suboptimal sampling rates or the recovery performance substantially suffers from the dominance of model mismatch. In this work we discuss such type of estimation problems and we focus on bilinear inverse problems. We link this problem to the recovery of low-rank and sparse matrices and establish stable low-dimensional embeddings of the uncalibrated receive signals whe...
We will establish in this note a stability result for sparse convolutions on torsion-free additiv... more We will establish in this note a stability result for sparse convolutions on torsion-free additive (discrete) abelian groups. Sparse convolutions on torsion-free groups are free of cancellations and hence admit stability, i.e. injectivity with a universal lower bound α=α(s,f), only depending on the cardinality s and f of the supports of both input sequences. More precisely, we show that α depends only on s and f and not on the ambient dimension. This statement follows from a reduction argument which involves a compression into a small set preserving the additive structure of the supports.
We consider the time-continuous doubly-dispersive channel with additive Gaussian noise and establ... more We consider the time-continuous doubly-dispersive channel with additive Gaussian noise and establish a capacity formula for the case where the channel correlation operator is represented by a symbol which is periodic in time and fulfills some further integrability and smoothness conditions. The key to this result is a new Szeg\"o formula for certain pseudo-differential operators. The formula justifies the water-filling principle along time and frequency in terms of the time--continuous time-varying transfer function (the symbol).
As it becomes increasingly apparent that 4G will not be able to meet the emerging demands of futu... more As it becomes increasingly apparent that 4G will not be able to meet the emerging demands of future mobile communication systems, the question what could make up a 5G system, what are the crucial challenges and what are the key drivers is part of intensive, ongoing discussions. Partly due to the advent of compressive sensing, methods that can optimally exploit sparsity in signals have received tremendous attention in recent years. In this paper we will describe a variety of scenarios in which signal sparsity arises naturally in 5G wireless systems. Signal sparsity and the associated rich collection of tools and algorithms will thus be a viable source for innovation in 5G wireless system design. We will discribe applications of this sparse signal processing paradigm in MIMO random access, cloud radio access networks, compressive channel-source network coding, and embedded security. We will also emphasize important open problem that may arise in 5G system design, for which sparsity wi...
In this article new bounds on weighted p-norms of ambiguity functions and Wigner functions are de... more In this article new bounds on weighted p-norms of ambiguity functions and Wigner functions are derived. Such norms occur frequently in several areas of physics and engineering. In pulse optimization for Weyl--Heisenberg signaling in wide-sense stationary uncorrelated scattering channels for example it is a key step to find the optimal waveforms for a given scattering statistics which is a problem also well known in radar and sonar waveform optimizations. The same situation arises in quantum information processing and optical communication when optimizing pure quantum states for communicating in bosonic quantum channels, i.e. find optimal channel input states maximizing the pure state channel fidelity. Due to the non-convex nature of this problem the optimum and the maximizers itself are in general difficult find, numerically and analytically. Therefore upper bounds on the achievable performance are important which will be provided by this contribution. Based on a result due to E. Li...
In this article we show the relation between the theory of pulse shaping for WSSUS channels and t... more In this article we show the relation between the theory of pulse shaping for WSSUS channels and the notion of approximate eigenstructure for time-varying channels. We consider pulse shaping for a general signaling scheme, called Weyl-Heisenberg signaling, which includes OFDM with cyclic prefix and OFDM/OQAM. The pulse design problem in the view of optimal WSSUS--averaged SINR is an interplay between localization and "orthogonality". The localization problem itself can be expressed in terms of eigenvalues of localization operators and is intimately connected to the concept of approximate eigenstructure of LTV channel operators. In fact, on the L_2-level both are equivalent as we will show. The concept of "orthogonality" in turn can be related to notion of tight frames. The right balance between these two sides is still an open problem. However, several statements on achievable values of certain localization measures and fundamental limits on SINR can already be ma...
We consider the time-continuous doubly--dispersive channel with additive Gaussian noise and estab... more We consider the time-continuous doubly--dispersive channel with additive Gaussian noise and establish a capacity formula for the case where the channel operator is represented by a symbol which is periodic in time and fulfills some further integrability, smoothness and oscillation conditions. More precisely, we apply the well-known Holsinger-Gallager model for translating a time-continuous channel for a sequence of time--intervals of increasing length $\alpha\rightarrow\infty$ to a series of equivalent sets of discrete, parallel channels, known at the transmitter. We quantify conditions when this procedure converges. Finally, under periodicity assumptions this result can indeed be justified as the channel capacity in the sense Shannon. The key to this is result is a new Szeg\"o formula for certain pseudo--differential operators with real-valued symbol. The Szeg\"o limit holds if the symbol belongs to the homogeneous Besov space $\dot{B}^1_{\infty,1}$ with respect to its ti...
Sporadic traffic, e.g. in the Internet of Things (IoT), will dramatically increase in the 5G mark... more Sporadic traffic, e.g. in the Internet of Things (IoT), will dramatically increase in the 5G market. Obviously, such traffic should not be crudely integrated into the bulky synchronization procedures of the current 4G LTE cellular systems. A promising approach is to allow for transmission of control signaling and payload "in one shot" in physical layer random access channel. Hereby, a major challenge is to cope with highly asynchronous access as well as high carrier frequency offsets of low-cost IoT devices. We address this challenge by using a waveform design approach based on bi-orthogonal frequency division multiplexing. More specifically, we introduce a novel mathematical approach to analyze such highly asynchronous random access proving superiority of our proposed waveform design. Numerical experiments, using the configuration of the new access method, confirm the analytical results as well as the performance of the waveform.
We consider the problem of sparse signal recovery from noisy measurements. Many of frequently use... more We consider the problem of sparse signal recovery from noisy measurements. Many of frequently used recovery methods rely on some sort of tuning depending on either noise or signal parameters. If no estimates for either of them are available, the noisy recovery problem is significantly harder. The square root LASSO and the least absolute deviation LASSO are known to be noise-blind, in the sense that the tuning parameter can be chosen independent on the noise and the signal. We generalize those recovery methods to the and give a recovery guarantee once the tuning parameter is above a threshold. Moreover, we analyze the effect of a bad chosen tuning parameter mistuning on a theoretic level and prove the optimality of our recovery guarantee. Further, for Gaussian matrices we give a refined analysis of the threshold of the tuning parameter and proof a new relation of the tuning parameter on the dimensions. Indeed, for a certain amount of measurements the tuning parameter becomes independ...
It is well-established that many iterative sparse reconstruction algorithms can be unrolled to yi... more It is well-established that many iterative sparse reconstruction algorithms can be unrolled to yield a learnable neural network for improved empirical performance. A prime example is learned ISTA (LISTA) where weights, step sizes and thresholds are learned from training data. Recently, Analytic LISTA (ALISTA) has been introduced, combining the strong empirical performance of a fully learned approach like LISTA, while retaining theoretical guarantees of classical compressed sensing algorithms and significantly reducing the number of parameters to learn. However, these parameters are trained to work in expectation, often leading to suboptimal reconstruction of individual targets. In this work we therefore introduce Neurally Augmented ALISTA, in which an LSTM network is used to compute step sizes and thresholds individually for each target vector during reconstruction. This adaptive approach is theoretically motivated by revisiting the recovery guarantees of ALISTA. We show that our ap...
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific ... more HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. A Kashin Approach to the Capacity of the
In this paper, we study the capacity and degree-of-freedom (DoF) scaling for the continuous-time ... more In this paper, we study the capacity and degree-of-freedom (DoF) scaling for the continuous-time amplitude limited AWGN channels in radio frequency (RF) and intensity modulated optical communication (OC) channels. More precisely, we study how the capacity varies in terms of the OFDM block transmission time T, bandwidth W, amplitude A, and the noise spectral density N_0. We first find suitable discrete encoding spaces for both cases, and prove that they are convex sets that have a semi-definite programming (SDP) representation. Using tools from convex geometry, we find lower and upper bounds on the volume of these encoding sets, which we exploit to drive pretty sharp lower and upper bounds on the capacity. We also study a practical Tone-Reservation (TR) encoding algorithm and prove that its performance can be characterized by the statistical width of an appropriate convex set. Recently, it has been observed that in high-dimensional estimation problems under constraints such as those ...
2017 42nd International Conference on Infrared, Millimeter, and Terahertz Waves (IRMMW-THz), 2017
We present a single detector THz imaging system with millimeter spatial resolution that operates ... more We present a single detector THz imaging system with millimeter spatial resolution that operates without any mechanical movements of the detector and the THz optics. The imaging system, an optically controllable 0.35 THz single-pixel camera, is used in transmission mode to image binary targets. The typical size of the targets is several centimeters diameter and the acquisition time for such targets is less than 2 minutes. The software reconstruction of the acquired signals yields images with more than 2500 pixels, respective < 1 mm spatial resolution. As an outlook first results obtained with the camera's radar-reflection mode will be presented.
Block-sparse regularization is already well known in active thermal imaging and is used for multi... more Block-sparse regularization is already well known in active thermal imaging and is used for multiple-measurement-based inverse problems. The main bottleneck of this method is the choice of regularization parameters which differs for each experiment. We show the benefits of using a learned block iterative shrinkage thresholding algorithm (LBISTA) that is able to learn the choice of regularization parameters, without the need to manually select them. In addition, LBISTA enables the determination of a suitable weight matrix to solve the underlying inverse problem. Therefore, in this paper we present LBISTA and compare it with state-of-the-art block iterative shrinkage thresholding using synthetically generated and experimental test data from active thermography for defect reconstruction. Our results show that the use of the learned block-sparse optimization approach provides smaller normalized mean square errors for a small fixed number of iterations. Thus, this allows us to improve th...
Countless signal processing applications include the reconstruction of signals from few indirect ... more Countless signal processing applications include the reconstruction of signals from few indirect linear measurements. The design of effective measurement operators is typically constrained by the underlying hardware and physics, posing a challenging and often even discrete optimization task. While the potential of gradient-based learning via the unrolling of iterative recovery algorithms has been demonstrated, it has remained unclear how to leverage this technique when the set of admissible measurement operators is structured and discrete. We tackle this problem by combining unrolled optimization with Gumbel reparametrizations, which enable the computation of low-variance gradient estimates of categorical random variables. Our approach is formalized by GLODISMO (Gradient-based Learning of DIscrete Structured Measurement Operators). This novel method is easy-to-implement, computationally efficient, and extendable due to its compatibility with automatic differentiation. We empirically...
This work deals with the problem of distributed data acquisition under non-linear communication c... more This work deals with the problem of distributed data acquisition under non-linear communication constraints. More specifically, we consider a model setup where M distributed nodes take individual measurements of an unknown structured source vector x_0 ∈R^n, communicating their readings simultaneously to a central receiver. Since this procedure involves collisions and is usually imperfect, the receiver measures a superposition of non-linearly distorted signals. In a first step, we will show that an s-sparse vector x_0 can be successfully recovered from O(s ·(2n/s)) of such superimposed measurements, using a traditional Lasso estimator that does not rely on any knowledge about the non-linear corruptions. This direct method however fails to work for several "uncalibrated" system configurations. These blind reconstruction tasks can be easily handled with the ℓ^1,2-Group Lasso, but coming along with an increased sampling rate of O(s·{M, (2n/s) }) observations. In fact, the purp...
The success of the compressed sensing paradigm has shown that a substantial reduction in sampling... more The success of the compressed sensing paradigm has shown that a substantial reduction in sampling and storage complexity can be achieved in certain linear and non-adaptive estimation problems. It is therefore an advisable strategy for noncoherent information retrieval in, for example, sporadic blind and semi-blind communication and sampling problems. But, the conventional model is not practical here since the compressible signals have to be estimated from samples taken solely on the output of an un-calibrated system which is unknown during measurement but often compressible. Conventionally, one has either to operate at suboptimal sampling rates or the recovery performance substantially suffers from the dominance of model mismatch. In this work we discuss such type of estimation problems and we focus on bilinear inverse problems. We link this problem to the recovery of low-rank and sparse matrices and establish stable low-dimensional embeddings of the uncalibrated receive signals whe...
We will establish in this note a stability result for sparse convolutions on torsion-free additiv... more We will establish in this note a stability result for sparse convolutions on torsion-free additive (discrete) abelian groups. Sparse convolutions on torsion-free groups are free of cancellations and hence admit stability, i.e. injectivity with a universal lower bound α=α(s,f), only depending on the cardinality s and f of the supports of both input sequences. More precisely, we show that α depends only on s and f and not on the ambient dimension. This statement follows from a reduction argument which involves a compression into a small set preserving the additive structure of the supports.
We consider the time-continuous doubly-dispersive channel with additive Gaussian noise and establ... more We consider the time-continuous doubly-dispersive channel with additive Gaussian noise and establish a capacity formula for the case where the channel correlation operator is represented by a symbol which is periodic in time and fulfills some further integrability and smoothness conditions. The key to this result is a new Szeg\"o formula for certain pseudo-differential operators. The formula justifies the water-filling principle along time and frequency in terms of the time--continuous time-varying transfer function (the symbol).
As it becomes increasingly apparent that 4G will not be able to meet the emerging demands of futu... more As it becomes increasingly apparent that 4G will not be able to meet the emerging demands of future mobile communication systems, the question what could make up a 5G system, what are the crucial challenges and what are the key drivers is part of intensive, ongoing discussions. Partly due to the advent of compressive sensing, methods that can optimally exploit sparsity in signals have received tremendous attention in recent years. In this paper we will describe a variety of scenarios in which signal sparsity arises naturally in 5G wireless systems. Signal sparsity and the associated rich collection of tools and algorithms will thus be a viable source for innovation in 5G wireless system design. We will discribe applications of this sparse signal processing paradigm in MIMO random access, cloud radio access networks, compressive channel-source network coding, and embedded security. We will also emphasize important open problem that may arise in 5G system design, for which sparsity wi...
In this article new bounds on weighted p-norms of ambiguity functions and Wigner functions are de... more In this article new bounds on weighted p-norms of ambiguity functions and Wigner functions are derived. Such norms occur frequently in several areas of physics and engineering. In pulse optimization for Weyl--Heisenberg signaling in wide-sense stationary uncorrelated scattering channels for example it is a key step to find the optimal waveforms for a given scattering statistics which is a problem also well known in radar and sonar waveform optimizations. The same situation arises in quantum information processing and optical communication when optimizing pure quantum states for communicating in bosonic quantum channels, i.e. find optimal channel input states maximizing the pure state channel fidelity. Due to the non-convex nature of this problem the optimum and the maximizers itself are in general difficult find, numerically and analytically. Therefore upper bounds on the achievable performance are important which will be provided by this contribution. Based on a result due to E. Li...
In this article we show the relation between the theory of pulse shaping for WSSUS channels and t... more In this article we show the relation between the theory of pulse shaping for WSSUS channels and the notion of approximate eigenstructure for time-varying channels. We consider pulse shaping for a general signaling scheme, called Weyl-Heisenberg signaling, which includes OFDM with cyclic prefix and OFDM/OQAM. The pulse design problem in the view of optimal WSSUS--averaged SINR is an interplay between localization and "orthogonality". The localization problem itself can be expressed in terms of eigenvalues of localization operators and is intimately connected to the concept of approximate eigenstructure of LTV channel operators. In fact, on the L_2-level both are equivalent as we will show. The concept of "orthogonality" in turn can be related to notion of tight frames. The right balance between these two sides is still an open problem. However, several statements on achievable values of certain localization measures and fundamental limits on SINR can already be ma...
We consider the time-continuous doubly--dispersive channel with additive Gaussian noise and estab... more We consider the time-continuous doubly--dispersive channel with additive Gaussian noise and establish a capacity formula for the case where the channel operator is represented by a symbol which is periodic in time and fulfills some further integrability, smoothness and oscillation conditions. More precisely, we apply the well-known Holsinger-Gallager model for translating a time-continuous channel for a sequence of time--intervals of increasing length $\alpha\rightarrow\infty$ to a series of equivalent sets of discrete, parallel channels, known at the transmitter. We quantify conditions when this procedure converges. Finally, under periodicity assumptions this result can indeed be justified as the channel capacity in the sense Shannon. The key to this is result is a new Szeg\"o formula for certain pseudo--differential operators with real-valued symbol. The Szeg\"o limit holds if the symbol belongs to the homogeneous Besov space $\dot{B}^1_{\infty,1}$ with respect to its ti...
Sporadic traffic, e.g. in the Internet of Things (IoT), will dramatically increase in the 5G mark... more Sporadic traffic, e.g. in the Internet of Things (IoT), will dramatically increase in the 5G market. Obviously, such traffic should not be crudely integrated into the bulky synchronization procedures of the current 4G LTE cellular systems. A promising approach is to allow for transmission of control signaling and payload "in one shot" in physical layer random access channel. Hereby, a major challenge is to cope with highly asynchronous access as well as high carrier frequency offsets of low-cost IoT devices. We address this challenge by using a waveform design approach based on bi-orthogonal frequency division multiplexing. More specifically, we introduce a novel mathematical approach to analyze such highly asynchronous random access proving superiority of our proposed waveform design. Numerical experiments, using the configuration of the new access method, confirm the analytical results as well as the performance of the waveform.
We consider the problem of sparse signal recovery from noisy measurements. Many of frequently use... more We consider the problem of sparse signal recovery from noisy measurements. Many of frequently used recovery methods rely on some sort of tuning depending on either noise or signal parameters. If no estimates for either of them are available, the noisy recovery problem is significantly harder. The square root LASSO and the least absolute deviation LASSO are known to be noise-blind, in the sense that the tuning parameter can be chosen independent on the noise and the signal. We generalize those recovery methods to the and give a recovery guarantee once the tuning parameter is above a threshold. Moreover, we analyze the effect of a bad chosen tuning parameter mistuning on a theoretic level and prove the optimality of our recovery guarantee. Further, for Gaussian matrices we give a refined analysis of the threshold of the tuning parameter and proof a new relation of the tuning parameter on the dimensions. Indeed, for a certain amount of measurements the tuning parameter becomes independ...
It is well-established that many iterative sparse reconstruction algorithms can be unrolled to yi... more It is well-established that many iterative sparse reconstruction algorithms can be unrolled to yield a learnable neural network for improved empirical performance. A prime example is learned ISTA (LISTA) where weights, step sizes and thresholds are learned from training data. Recently, Analytic LISTA (ALISTA) has been introduced, combining the strong empirical performance of a fully learned approach like LISTA, while retaining theoretical guarantees of classical compressed sensing algorithms and significantly reducing the number of parameters to learn. However, these parameters are trained to work in expectation, often leading to suboptimal reconstruction of individual targets. In this work we therefore introduce Neurally Augmented ALISTA, in which an LSTM network is used to compute step sizes and thresholds individually for each target vector during reconstruction. This adaptive approach is theoretically motivated by revisiting the recovery guarantees of ALISTA. We show that our ap...
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific ... more HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. A Kashin Approach to the Capacity of the
In this paper, we study the capacity and degree-of-freedom (DoF) scaling for the continuous-time ... more In this paper, we study the capacity and degree-of-freedom (DoF) scaling for the continuous-time amplitude limited AWGN channels in radio frequency (RF) and intensity modulated optical communication (OC) channels. More precisely, we study how the capacity varies in terms of the OFDM block transmission time T, bandwidth W, amplitude A, and the noise spectral density N_0. We first find suitable discrete encoding spaces for both cases, and prove that they are convex sets that have a semi-definite programming (SDP) representation. Using tools from convex geometry, we find lower and upper bounds on the volume of these encoding sets, which we exploit to drive pretty sharp lower and upper bounds on the capacity. We also study a practical Tone-Reservation (TR) encoding algorithm and prove that its performance can be characterized by the statistical width of an appropriate convex set. Recently, it has been observed that in high-dimensional estimation problems under constraints such as those ...
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