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Recently, it was proved that group equivariance emerges in ensembles of neural networks as the result of full augmentation in the limit of infinitely wide neural networks (neural tangent kernel limit). In this paper, we extend this result significantly. We provide a proof that this emergence does not depend on the neural tangent kernel limit at all. We also consider stochastic settings, and furthermore general architectures. For the latter, we provide a simple sufficient condition on the relation between the architecture and the action of the group for our results to hold. We validate our findings through simple numeric experiments.

Secret key generation (SKG) between authenticated devices is a pivotal task for secure communications. Diffie-Hellman (DH) is de-facto standard but not post-quantum secure. In this paper, we shall invent and analyze a new security primitive that is specifically designed for WPAN. For WPAN, wireless channel-based SKG has been proposed but was not widely deployed due to its critical dependence on the channel's entropy which is uncontrollable. We formulate a different approach: We still exploit channel properties but mainly hinge on the reciprocity of the wireless channel and not on the channel's entropy. The radio advantage comes from the use of full duplex communication. We show that in this situation both legitimate parties can agree on a common secret key even without ever probing the channel at all. At the core is a new bisparse blind deconvolution scheme for which we prove correctness and information-theoretic, i.e. perfect, security. We show that, ultimately, a secret key can be extracted and give a lower bound for the number of secret key bits which is then verified by experiments.

We investigate the optimization of neural networks on symmetric data, and compare the strategy of constraining the architecture to be equivariant to that of using data augmentation. Our analysis reveals that that the relative geometry of the admissible and the equivariant layers, respectively, plays a key role. Under natural assumptions on the data, network, loss, and group of symmetries, we show that compatibility of the spaces of admissible layers and equivariant layers, in the sense that the corresponding orthogonal projections commute, implies that the sets of equivariant stationary points are identical for the two strategies. If the linear layers of the network also are given a unitary parametrization, the set of equivariant layers is even invariant under the gradient flow for augmented models. Our analysis however also reveals that even in the latter situation, stationary points may be unstable for augmented training although they are stable for the manifestly equivariant models., Comment: v3: Completely revised manuscript: New framework for neural nets, new main result (involving compability condition), new experiments, new author. v2: Revised manuscript. Mostly small edits, apart from new experiments (see Appendix E)

We propose an adaptive refinement algorithm to solve total variation regularized measure optimization problems. The method iteratively constructs dyadic partitions of the unit cube based on i) the resolution of discretized dual problems and ii) on the detection of cells containing points that violate the dual constraints. The detection is based on upper-bounds on the dual certificate, in the spirit of branch-and-bound methods. The interest of this approach is that it avoids the use of heuristic approaches to find the maximizers of dual certificates. We prove the convergence of this approach under mild hypotheses and a linear convergence rate under additional non-degeneracy assumptions. These results are confirmed by simple numerical experiments.

The bi-sparse blind deconvolution problem is studied -- that is, from the knowledge of $h*(Qb)$, where $Q$ is some linear operator, recovering $h$ and $b$, which are both assumed to be sparse. The approach rests upon lifting the problem to a linear one, and then applying the hierarchical sparsity framework. In particular, the efficient HiHTP algorithm is proposed for performing the recovery. Then, under a random model on the matrix $Q$, it is theoretically shown that an $s$-sparse $h \in \mathbb{K}^\mu$ and $\sigma$-sparse $b \in \mathbb{K}^n$ with high probability can be recovered when $\mu \succcurlyeq s\log(s)^2\log(\mu)\log(\mu n) + s\sigma \log(n)$., Comment: V2: Completely revised version, entirely different proof, resulting in the recovery guarantee improved by a factor s

Equivariance of linear neural network layers is well studied. In this work, we relax the equivariance condition to only be true in a projective sense. We propose a way to construct a projectively equivariant neural network through building a standard equivariant network where the linear group representations acting on each intermediate feature space are "multiplicatively modified lifts" of projective group representations. By theoretically studying the relation of projectively and linearly equivariant linear layers, we show that our approach is the most general possible when building a network out of linear layers. The theory is showcased in two simple experiments., Comment: v3: Another significant rewrite. Accepted for publication in TMLR. v2: Significant rewrite. The title has been changed: "neural network" -> "network". More general description of projectively equivariant linear layers, with new proposed architectures, and a completely new accompanying experiment section, as a result

Compressive Sensing has well boosted massive random access protocols over the last decade. In this paper we apply an orthogonal FFT basis as it is used in OFDM, but subdivide its image into so-called sub-channels and let each sub-channel take only a fraction of the load. In a random fashion the subdivision is consecutively applied over a suitable number of time-slots. Within the time-slots the users will not change their sub-channel assignment and send in parallel the data. Activity detection is carried out jointly across time-slots in each of the sub-channels. For such system design we derive three rather fundamental results: i) First, we prove that the subdivision can be driven to the extent that the activity in each sub-channel is sparse by design. An effect that we call sparsity capture effect. ii) Second, we prove that effectively the system can sustain any overload situation relative to the FFT dimension, i.e. detection failure of active and non-active users can be kept below any desired threshold regardless of the number of users. The only price to pay is delay, i.e. the number of time-slots over which cross-detection is performed. We achieve this by jointly exploring the effect of measure concentration in time and frequency and careful system parameter scaling. iii) Third, we prove that parallel to activity detection active users can carry one symbol per pilot resource and time-slot so it supports so-called one-shot messaging. The key to proving these results are new concentration results for sequences of randomly sub-sampled FFTs detecting the sparse vectors "en bloc". Eventually, we show by simulations that the system is scalable resulting in a coarsely 20-fold capacity increase compared to standard OFDM.

We characterize the class of image plane transformations which realize rigid camera motions and call these transformations `rigidity preserving'. In particular, 2D translations of pinhole images are not rigidity preserving. Hence, when using CNNs for 3D inference tasks, it can be beneficial to modify the inductive bias from equivariance towards translations to equivariance towards rigidity preserving transformations. We investigate how equivariance with respect to rigidity preserving transformations can be approximated in CNNs, and test our ideas on both 6D object pose estimation and visual localization. Experimentally, we improve on several competitive baselines., Comment: v2: Substantially revised version. Among other things, experiments with the PixLoc model added

In this paper, we are concerned with rotation equivariance on 2D point cloud data. We describe a particular set of functions able to approximate any continuous rotation equivariant and permutation invariant function. Based on this result, we propose a novel neural network architecture for processing 2D point clouds and we prove its universality for approximating functions exhibiting these symmetries. We also show how to extend the architecture to accept a set of 2D-2D correspondences as indata, while maintaining similar equivariance properties. Experiments are presented on the estimation of essential matrices in stereo vision., Comment: CVPR 2022 camera ready

The blind deconvolution problem amounts to reconstructing both a signal and a filter from the convolution of these two. It constitutes a prominent topic in mathematical and engineering literature. In this work, we analyze a sparse version of the problem: The filter $h\in \mathbb{R}^\mu$ is assumed to be $s$-sparse, and the signal $b \in \mathbb{R}^n$ is taken to be $\sigma$-sparse, both supports being unknown. We observe a convolution between the filter and a linear transformation of the signal. Motivated by practically important multi-user communication applications, we derive a recovery guarantee for the simultaneous demixing and deconvolution setting. We achieve efficient recovery by relaxing the problem to a hierarchical sparse recovery for which we can build on a flexible framework. At the same time, for this we pay the price of some sub-optimal guarantees compared to the number of free parameters of the problem. The signal model we consider is sufficiently general to capture many applications in a number of engineering fields. Despite their practical importance, we provide first rigorous performance guarantees for efficient and simple algorithms for the bi-sparse and generalized demixing setting. We complement our analytical results by presenting results of numerical simulations. We find evidence that the sub-optimal scaling $s^2\sigma \log(\mu)\log(n)$ of our derived sufficient condition is likely overly pessimistic and that the observed performance is better described by a scaling proportional to $ s\sigma$ up to log-factors., Comment: 6 pages, 5 figures

It is well known that CS can boost massive random access protocols. Usually, the protocols operate in some overloaded regime where the sparsity can be exploited. In this paper, we consider a different approach by taking an orthogonal FFT base, subdivide its image into appropriate sub-channels and let each subchannel take only a fraction of the load. To show that this approach can actually achieve the full capacity we provide i) new concentration inequalities, and ii) devise a sparsity capture effect, i.e where the sub-division can be driven such that the activity in each each sub-channel is sparse by design. We show by simulations that the system is scalable resulting in a coarsely 30-fold capacity increase., Comment: 5 pages, IEEE Statistical Signal Processing Workshop (SSP) 2021, Track: Massive Machine-Type Communications (invited paper)

A new family of operators, coined hierarchical measurement operators, is introduced and discussed within the well-known hierarchical sparse recovery framework. Such operator is a composition of block and mixing operations and notably contains the Kronecker product as a special case. Results on their hierarchical restricted isometry property (HiRIP) are derived, generalizing prior work on recovery of hierarchically sparse signals from Kronecker-structured linear measurements. Specifically, these results show that, very surprisingly, sparsity properties of the block and mixing part can be traded against each other. The measurement structure is well-motivated by a massive random access channel design in communication engineering. Numerical evaluation of user detection rates demonstrate the huge benefit of the theoretical framework., Comment: 5 pages, 2 figures. arXiv admin note: text overlap with arXiv:2005.10379

Compressed sensing is a paradigm within signal processing that provides the means for recovering structured signals from linear measurements in a highly efficient manner. Originally devised for the recovery of sparse signals, it has become clear that a similar methodology would also carry over to a wealth of other classes of structured signals. In this work, we provide an overview over the theory of compressed sensing for a particularly rich family of such signals, namely those of hierarchically structured signals. Examples of such signals are constituted by blocked vectors, with only few non-vanishing sparse blocks. We present recovery algorithms based on efficient hierarchical hard-thresholding. The algorithms are guaranteed to converge, in a stable fashion both with respect to measurement noise as well as to model mismatches, to the correct solution provided the measurement map acts isometrically restricted to the signal class. We then provide a series of results establishing the required condition for large classes of measurement ensembles. Building upon this machinery, we sketch practical applications of this framework in machine-type communications and quantum tomography., Comment: This book chapter is a report on findings within the DFG-funded priority program `Compressed Sensing in Information Processing' (CoSIP)

A new class of measurement operators, coined hierarchical measurement operators, and prove results guaranteeing the efficient, stable and robust recovery of hierarchically structured signals from such measurements. We derive bounds on their hierarchical restricted isometry properties based on the restricted isometry constants of their constituent matrices, generalizing and extending prior work on Kronecker-product measurements. As an exemplary application, we apply the theory to two communication scenarios. The fast and scalable HiHTP algorithm is shown to be suitable for solving these types of problems and its performance is evaluated numerically in terms of sparse signal recovery and block detection capability., Comment: 22 pages, 5 figures. v3: Significant rework of previous version. Section on incoherent blocks added, as well as more numerical experiments. v4: Revision of the manuscript. Several previous mistakes have been corrected

We analyze an exchange algorithm for the numerical solution total-variation regularized inverse problems over the space M($\Omega$) of Radon measures on a subset $\Omega$ of R d. Our main result states that under some regularity conditions, the method eventually converges linearly. Additionally, we prove that continuously optimizing the amplitudes of positions of the target measure will succeed at a linear rate with a good initialization. Finally, we propose to combine the two approaches into an alternating method and discuss the comparative advantages of this approach.

The problem of excessive pilot overhead required for uplink massive MIMO channel estimation is well known, let alone when it is considered along with wideband (OFDM) transmissions. Towards channel estimators that are both efficient and require low-training overhead, compressive sensing (CS) approaches have been increasingly popular, exploiting the sparse nature of the physical channel. However, no analytical insights regarding the overhead required for reliable channel estimation in wideband massive MIMO are available. By observing that the wideband massive MIMO channel can be represented by a vector that is not simply sparse but has well defined structural properties, referred to as hierarchical sparsity, we propose low complexity channel estimators for the multiuser scenario that take this property into account. By employing the framework of the hierarchical restricted isometry property, rigorous performance guarantees for these algorithms are provided suggesting concrete design goals for the user pilot sequences. For a specific design, we analytically characterize the scaling of the required pilot overhead with increasing number of antennas and bandwidth, revealing that, as long as the number of antennas is sufficiently large, it is independent of the per user channel sparsity level as well as the number of active users. Hence, surprisingly, in contrast to the classical setting, pilot overhead can be shifted into spatial dimensions not affecting crucial bandwidth constraints thereby increasing the overall system capacity. These analytical insights are verified by simulation results demonstrating also the superiority of the proposed algorithm over conventional CS algorithms that ignore the hierarchical sparsity property., Comment: 14 two-column pages; submitted to IEEE Trans. Wireless Commun

The problem of wideband massive MIMO channel estimation is considered. Targeting for low complexity algorithms as well as small training overhead, a compressive sensing (CS) approach is pursued. Unfortunately, due to the Kronecker-type sensing (measurement) matrix corresponding to this setup, application of standard CS algorithms and analysis methodology does not apply. By recognizing that the channel possesses a special structure, termed hierarchical sparsity, we propose an efficient algorithm that explicitly takes into account this property. In addition, by extending the standard CS analysis methodology to hierarchical sparse vectors, we provide a rigorous analysis of the algorithm performance in terms of estimation error as well as number of pilot subcarriers required to achieve it. Small training overhead, in turn, means higher number of supported users in a cell and potentially improved pilot decontamination. We believe, that this is the first paper that draws a rigorous connection between the hierarchical framework and Kronecker measurements. Numerical results verify the advantage of employing the proposed approach in this setting instead of standard CS algorithms., Comment: 8 pages, 5 figures

In this paper we develop a general theory of compressed sensing for analog signals, in close similarity to prior results for vectors in finite dimensional spaces that are sparse in a given orthonormal basis. The signals are modeled by functions in a reproducing kernel Hilbert space. Sparsity is defined as the minimal number of terms in expansions based on the kernel functions. Minimizing this number is under certain conditions equivalent to minimizing an atomic norm, the pre-dual of the supremum norm for functions in the Hilbert space. The norm minimizer is shown to exist based on a compactness argument. Recovery based on minimizing the atomic norm is robust and stable, so it provides controllable accuracy for recovery when the signal is only approximately sparse and the measurement is corrupted by noise. As applications of the theory, we include results on the recovery of sparse bandlimited functions and functions that have a sparse inverse short-time Fourier transform.

This work treats the recovery of sparse, binary signals through box-constrained basis pursuit using biased measurement matrices. Using a probabilistic model, we provide conditions under which the recovery of both sparse and saturated binary signals is very likely. In fact, we also show that under the same condition, the solution of the boxed-constrained basis pursuit program can be found using boxed-constrained least-squares.

The estimation of direction of arrivals with help of $TV$-minimization is studied. Contrary to prior work in this direction, which has only considered certain antenna placement designs, we consider general antenna geometries. Applying the soft-recovery framework, we are able to derive a theoretic guarantee for a certain direction of arrival to be approximately recovered. We discuss the impact of the recovery guarantee for a few concrete antenna designs. Additionally, numerical simulations supporting the findings of the theoretical part are performed.

We propose a scheme utilizing ideas from infinite dimensional compressed sensing for thermal source localization. Using the soft recovery framework of one of the authors, we provide rigorous theoretical guarantees for the recovery performance. In particular, we extend the framework in order to also include noisy measurements. Further, we conduct numerical experiments, showing that our proposed method has strong performance, in a wide range of settings. These include scenarios with few sensors, off-grid source positioning and high noise levels, both in one and two dimensions.

We study the solutions of infinite dimensional linear inverse problems over Banach spaces. The regularizer is defined as the total variation of a linear mapping of the function to recover, while the data fitting term is a near arbitrary convex function. The first contribution is about the solu-tion's structure: we show that under suitable assumptions, there always exist an m-sparse solution, where m is the number of linear measurements of the signal. Our second contribution is about the computation of the solution. While most existing works first discretize the problem, we show that exacts solutions of the infinite dimensional problem can be obtained by solving two consecutive finite dimensional convex programs. These results extend recent advances in the understanding of total-variation reg-ularized problems.

This paper describes a dual certificate condition on a linear measurement operator $A$ (defined on a Hilbert space $\mathcal{H}$ and having finite-dimensional range) which guarantees that an atomic norm minimization, in a certain sense, will be able to approximately recover a structured signal $v_0 \in \mathcal{H}$ from measurements $Av_0$. Put very streamlined, the condition implies that peaks in a sparse decomposition of $v_0$ are close the the support of the atomic decomposition of the solution $v^*$. The condition applies in a relatively general context - in particular, the space $\mathcal{H}$ can be infinite-dimensional. The abstract framework is applied to several concrete examples, one example being super-resolution. In this process, several novel results which are interesting on its own are obtained.

We propose and analyze a solution to the problem of recovering a block sparse signal with sparse blocks from linear measurements. Such problems naturally emerge inter alia in the context of mobile communication, in order to meet the scalability and low complexity requirements of massive antenna systems and massive machine-type communication. We introduce a new variant of the Hard Thresholding Pursuit (HTP) algorithm referred to as HiHTP. We provide both a proof of convergence and a recovery guarantee for noisy Gaussian measurements that exhibit an improved asymptotic scaling in terms of the sampling complexity in comparison with the usual HTP algorithm. Furthermore, hierarchically sparse signals and Kronecker product structured measurements naturally arise together in a variety of applications. We establish the efficient reconstruction of hierarchically sparse signals from Kronecker product measurements using the HiHTP algorithm. Additionally, we provide analytical results that connect our recovery conditions to generalized coherence measures. Again, our recovery results exhibit substantial improvement in the asymptotic sampling complexity scaling over the standard setting. Finally, we validate in numerical experiments that for hierarchically sparse signals, HiHTP performs significantly better compared to HTP., Comment: 11+4 pages, 5 figures. V3: Incomplete funding information corrected and minor typos corrected. V4: Change of title and additional author Axel Flinth. Included new results on Kronecker product measurements and relations of HiRIP to hierarchical coherence measures. Improved presentation of general hierarchically sparse signals and correction of minor typos

This article provides a new type of analysis of a compressed-sensing based technique for recovering column-sparse matrices, namely minimization of the $\ell_{1,2}$-norm. Rather than providing conditions on the measurement matrix which guarantees the solution of the program to be exactly equal to the ground truth signal (which already has been thoroughly investigated), it presents a condition which guarantees that the solution is approximately equal to the ground truth. Soft recovery statements of this kind are to the best knowledge of the author a novelty in Compressed Sensing. Apart from the theoretical analysis, we present two heuristic proposes how this property of the $\ell_{1,2}$-program can be utilized to design algorithms for recovery of matrices which are sparse and have low rank at the same time.

This paper concerns solving the sparse deconvolution and demixing problem using $\ell_{1,2}$-minimization. We show that under a certain structured random model, robust and stable recovery is possible. The results extend results of Ling and Strohmer [Self Calibration and Biconvex Compressive Sensing, Inverse Problems, 2015], and in particular theoretically explain certain experimental findings from that paper. Our results do not only apply to the deconvolution and demixing problem, but to recovery of column-sparse matrices in general., Comment: Changes in v2: A few errors were fixed, resulting in slightly different results. Also, some efforts were made to increase readability. Changes in v3: Version accepted for publication

During the last decade, the paradigm of compressed sensing has gained significant importance in the signal processing community. While the original idea was to utilize sparsity assumptions to design powerful recovery algorithms of vectors $x \in \mathbb{R}^d$, the concept has been extended to cover many other types of problems. A noteable example is low-rank matrix recovery. Many methods used for recovery rely on solving convex programs. A particularly nice trait of compressed sensing is its geometrical intuition. In recent papers, a classical optimality condition has been used together with tools from convex geometry and probability theory to prove beautiful results concerning the recovery of signals from Gaussian measurements. In this paper, we aim to formulate a geometrical condition for stability and robustness, i.e. for the recovery of approximately structured signals from noisy measurements. We will investigate the connection between the new condition with the notion of restricted singular values, classical stability and robustness conditions in compressed sensing, and also to important geometrical concepts from complexity theory. We will also prove the maybe somewhat surprising fact that for many convex programs, exact recovery of a signal $x_0$ immediately implies some stability and robustness when recovering signals close to $x_0$., Comment: Changes in v2: Some typos and minor errors were corrected. The statements of the paper are the same up to the values of some constants. Also, the terms "stable" and "robust" were interchanged. Changes in v3: Peer reviewed version. Section 2.1 was added, and Section 3.3 was completely revised. Also, further typos etc. were corrected. Changes in v4: An issue with references not appearing was fixed

Compressed sensing deals with the recovery of sparse signals from linear measurements. Without any additional information, it is possible to recover an $s$-sparse signal using $m \gtrsim s \log(d/s)$ measurements in a robust and stable way. Some applications provide additional information, such as on the location of the support of the signal. Using this information, it is conceivable the threshold amount of measurements can be lowered. A proposed algorithm for this task is \emph{weighted $\ell_1$-minimization}. Put shortly, one modifies standard $\ell_1$-minimization by assigning different weights to different parts of the index set $[1, \dots d]$. The task of choosing the weights is however non-trivial. This paper provides a complete answer to the question of an optimal choice of the weights. In fact, it is shown that it is possible to directly calculate unique weights that are optimal in the sense that the threshold amount of measurements needed for exact recovery is minimized. The proof uses recent results about the connection between convex geometry and compressed sensing-type algorithms., Comment: Changes in version 2: Peer reviewed version. Several minor errors and typos were corrected. Also, the numerical experiments have been thoroughly revised

The suboptimal performance of wavelets with regard to the approximation of multivariate data gave rise to new representation systems, specifically designed for data with anisotropic features. Some prominent examples of these are given by ridgelets, curvelets, and shearlets, to name a few. The great variety of such so-called directional systems motivated the search for a common framework, which unites many under one roof and enables a simultaneous analysis, for example with respect to approximation properties. Building on the concept of parabolic molecules, the recently introduced framework of $\alpha$-molecules does in fact include the previous mentioned systems. Until now however it is confined to the bivariate setting, whereas nowadays one often deals with higher dimensional data. This motivates the extension of this unifying theory to dimensions larger than 2, put forward in this work. In particular, we generalize the central result that the cross-Gramian of any two systems of $\alpha$-molecules will to some extent be localized. As an exemplary application, we investigate the sparse approximation of video signals, which are instances of 3D data. The multivariate theory allows us to derive almost optimal approximation rates for a large class of representation systems.

Compressed sensing investigates the recovery of sparse signals from linear measurements. But often, in a wide range of applications, one is given only the absolute values (squared) of the linear measurements. Recovering such signals (not necessarily sparse) is known as the phase retrieval problem. We consider this problem in the case when the measurements are time-frequency shifts of a suitably chosen generator, i.e. coming from a Gabor frame. We prove an easily checkable injectivity condition for recovery of any signal from all $N^2$ time-frequency shifts, and for recovery of sparse signals, when only some of those measurements are given.