Recent advancements in machine learning have transformed the discovery of physical laws, moving from manual derivation to data-driven methods that simultaneously learn both the structure and parameters of governing equations. This shift introduces new challenges regarding the validity of the discovered equations, particularly concerning their uniqueness and, hence, identifiability. While the issue of non-uniqueness has been well-studied in the context of parameter estimation, it remains underexplored for algorithms that recover both structure and parameters simultaneously. Early studies have primarily focused on idealized scenarios with perfect, noise-free data. In contrast, this paper investigates how noise influences the uniqueness and identifiability of physical laws governed by partial differential equations (PDEs). We develop a comprehensive mathematical framework to analyze the uniqueness of PDEs in the presence of noise and introduce new algorithms that account for noise, providing thresholds to assess uniqueness and identifying situations where excessive noise hinders reliable conclusions. Numerical experiments demonstrate the effectiveness of these algorithms in detecting uniqueness despite the presence of noise.

Information Pursuit (IP) is an explainable prediction algorithm that greedily selects a sequence of interpretable queries about the data in order of information gain, updating its posterior at each step based on observed query-answer pairs. The standard paradigm uses hand-crafted dictionaries of potential data queries curated by a domain expert or a large language model after a human prompt. However, in practice, hand-crafted dictionaries are limited by the expertise of the curator and the heuristics of prompt engineering. This paper introduces a novel approach: learning a dictionary of interpretable queries directly from the dataset. Our query dictionary learning problem is formulated as an optimization problem by augmenting IP's variational formulation with learnable dictionary parameters. To formulate learnable and interpretable queries, we leverage the latent space of large vision and language models like CLIP. To solve the optimization problem, we propose a new query dictionary learning algorithm inspired by classical sparse dictionary learning. Our experiments demonstrate that learned dictionaries significantly outperform hand-crafted dictionaries generated with large language models.

The problem of symbolic regression (SR) arises in many different applications, such as identifying physical laws or deriving mathematical equations describing the behavior of financial markets from given data. Various methods exist to address the problem of SR, often based on genetic programming. However, these methods are usually complicated and involve various hyperparameters. In this paper, we present our new approach ParFam that utilizes parametric families of suitable symbolic functions to translate the discrete symbolic regression problem into a continuous one, resulting in a more straightforward setup compared to current state-of-the-art methods. In combination with a global optimizer, this approach results in a highly effective method to tackle the problem of SR. We theoretically analyze the expressivity of ParFam and demonstrate its performance with extensive numerical experiments based on the common SR benchmark suit SRBench, showing that we achieve state-of-the-art results. Moreover, we present an extension incorporating a pre-trained transformer network (DL-ParFam) to guide ParFam, accelerating the optimization process by up to two magnitudes. Our code and results can be found at https://github.com/Philipp238/parfam.

Recent years have seen significant progress in developing spiking neural networks (SNNs) as a potential solution to the energy challenges posed by conventional artificial neural networks (ANNs). However, our theoretical understanding of SNNs remains relatively limited compared to the ever-growing body of literature on ANNs. In this paper, we study a discrete-time model of SNNs based on leaky integrate-and-fire (LIF) neurons, referred to as discrete-time LIF-SNNs, a widely used framework that still lacks solid theoretical foundations. We demonstrate that discrete-time LIF-SNNs realize piecewise constant functions defined on polyhedral regions, and more importantly, we quantify the network size required to approximate continuous functions. Moreover, we investigate the impact of latency (number of time steps) and depth (number of layers) on the complexity of the input space partitioning induced by discrete-time LIF-SNNs. Our analysis highlights the importance of latency and contrasts these networks with ANNs that use piecewise linear activation functions. Finally, we present numerical experiments to support our theoretical findings.

Recent advancements in generative models have revolutionized video synthesis and editing. However, the scarcity of diverse, high-quality datasets continues to hinder video-conditioned robotic learning, limiting cross-platform generalization. In this work, we address the challenge of swapping a robotic arm in one video with another— a key step for cross-embodiment learning. Unlike previous methods that depend on paired video demonstrations in the same environmental settings, our proposed framework, RoboSwap, operates on unpaired data from diverse environments, alleviating the data collection needs. RoboSwap introduces a novel video editing pipeline integrating both GANs and diffusion models, combining their isolated advantages. Specifically, we segment robotic arms from their backgrounds and train an unpaired GAN model to translate one robotic arm to another. The translated arm is blended with the original video background and refined with a diffusion model to enhance coherence, motion realism and object interaction. The GAN and diffusion stages are trained independently. Our experiments demonstrate that RoboSwap outperforms state-of-the-art video and image editing models on three benchmarks in terms of both structural coherence and motion consistency, thereby offering a robust solution for generating reliable, cross-embodiment data in robotic learning.

We address the problem of generating long-horizon videos for robotic manipulation tasks. Text-to-video diffusion models have made significant progress in photorealism, language understanding, and motion generation but struggle with long-horizon robotic tasks. Recent works use video diffusion models for high-quality simulation data and predictive rollouts in robot planning. However, these works predict short sequences of the robot achieving one task and employ an autoregressive paradigm to extend to the long horizon, leading to error accumulations in the generated video and in the execution. To overcome these limitations, we propose a novel pipeline that bypasses the need for autoregressive generation. We achieve this through a threefold contribution: 1) we first decompose the high-level goals into smaller atomic tasks and generate keyframes aligned with these instructions. A second diffusion model then interpolates between each of the two generated frames, achieving the long-horizon video. 2) We propose a semantics preserving attention module to maintain consistency between the keyframes. 3) We design a lightweight policy model to regress the robot joint states from generated videos. Our approach achieves state-of-the-art results on two benchmarks in video quality and consistency while outperforming previous policy models on long-horizon tasks.

Proper initialization is critical for Recurrent Neural Networks (RNNs), particularly in long-range reasoning tasks, where repeated application of the same weight matrix can cause vanishing or exploding signals. A common baseline for linear recurrences is Glorot initialization, designed to ensure stable signal propagation---but derived under the infinite-width, fixed-length regime—an unrealistic setting for RNNs processing long sequences. In this work, we show that Glorot initialization is in fact unstable: small positive deviations in the spectral radius are amplified through time and cause the hidden state to explode. Our theoretical analysis demonstrates that sequences of length $t = O(\sqrt{n})$, where $n$ is the hidden width, are sufficient to induce instability. To address this, we propose a simple, dimension-aware rescaling of Glorot that shifts the spectral radius slightly below one, preventing rapid signal explosion or decay. These results suggest that standard initialization schemes may break down in the long-sequence regime, motivating a separate line of theory for stable recurrent initialization.

The lack of trustworthiness is a major downside of deep learning. To mitigate the associated risks clear obligations of deep learning models have been proposed via regulatory guidelines. Therefore, a crucial question is to what extent trustworthy deep learning can be realized. Establishing trust-worthiness requires that the factors influencing an algorithmic computation can be retraced, i.e., the algorithmic implementation is transparent. Motivated by the observation that the current evolution of deep learning models necessitates a change in computing technology, we derive a mathematical framework that enables us to analyze whether a transparent implementation in a given computing model is feasible. We exemplarily apply our trustworthiness framework to analyze deep learning approaches for inverse problems in digital and analog computing models represented by Turing and Blum-Shub-Smale Machines, respectively. Based on previous results, we find that Blum-Shub-Smale Machines have the potential to establish trustworthy solvers for inverse problems under fairly general conditions, whereas, Turing machines cannot guarantee trustworthiness to the same degree. For a longer version of this paper with more details and proofs, we refer to [1].

We introduce $r$-loopy Weisfeiler-Leman ($r$-$\ell$WL), a novel hierarchy of graph isomorphism tests and a corresponding GNN framework, $r$-$\ell$MPNN, that can count cycles up to length $r{+}2$. Most notably, we show that $r$-$\ell$WL can count homomorphisms of cactus graphs. This extends 1-WL, which can only count homomorphisms of trees and, in fact, is incomparable to $k$-WL for any fixed $k$. We empirically validate the expressive and counting power of $r$-$\ell$MPNN on several synthetic datasets and demonstrate the scalability and strong performance on various real-world datasets, particularly on sparse graphs.

Optimization problems are a staple of today’s scientific and technical landscape. However, at present, solvers of such problems are almost exclusively run on digital hardware. Using Turing machines as a mathematical model for any type of digital hardware, in this paper, we analyze fundamental limitations of this conceptual approach of solving optimization problems. Since in most applications, the optimizer itself is of significantly more interest than the optimal value of the corresponding function, we will focus on computability of the optimizer. In fact, we will show that in various situations the optimizer is unattainable on Turing machines and consequently on digital computers. Moreover, even worse, there does not exist a Turing machine, which approximates the optimizer itself up to a certain constant error. We prove such results for a variety of well-known problems from very different areas, including artificial intelligence, financial mathematics, and information theory, often deriving the even stronger result that such problems are not Banach-Mazur computable, also not even in an approximate sense.

Image classifiers are known to be difficult to interpret and therefore require explanation methods to understand their decisions. We present ShearletX, a novel mask explanation method for image classifiers based on the shearlet transform - a multiscale directional image representation. Current mask explanation methods are regularized by smoothness constraints that protect against undesirable fine-grained explanation artifacts. However, the smoothness of a mask limits its ability to separate fine-detail patterns, that are relevant for the classifier, from nearby nuisance patterns, that do not affect the classifier. ShearletX solves this problem by avoiding smoothness regularization all together, replacing it by shearlet sparsity constraints. The resulting explanations consist of a few edges, textures, and smooth parts of the original image, that are the most relevant for the decision of the classifier. To support our method, we propose a mathematical definition for explanation artifacts and an information theoretic score to evaluate the quality of mask explanations. We demonstrate the superiority of ShearletX over previous mask based explanation methods using these new metrics, and present exemplary situations where separating fine-detail patterns allows explaining phenomena that were not explainable before.

Physical law learning is the ambiguous attempt at automating the derivation of governing equations with the use of machine learning techniques. This paper shall serve as a first step to build a comprehensive theoretical framework for learning physical laws, aiming to provide reliability to according algorithms. One key problem consists in the fact that the governing equations might not be uniquely determined by the given data. We will study this problem in the common situation that a physical law is described by an ordinary or partial differential equation. For various different classes of differential equations, we provide both necessary and sufficient conditions for a function from a given function class to uniquely determine the differential equation which is governing the phenomenon. We then use our results to determine in extensive numerical experiments whether a function solves a differential equation uniquely.

To foster research and facilitate fair comparisons among recently proposed pathloss radio map prediction methods, we have launched the ICASSP 2023 First Pathloss Radio Map Prediction Challenge. In this short overview paper, we briefly describe the pathloss prediction problem, the provided datasets, the challenge task and the challenge evaluation methodology. Finally, we present the results of the challenge.

Graph neural networks (GNNs) have shown state-of-the-art performances in various applications. However, GNNs often struggle to capture long-range dependencies in graphs due to oversmoothing. In this paper, we generalize the concept of oversmoothing from undirected to directed graphs. To this aim, we extend the notion of Dirichlet energy by considering a directed symmetrically normalized Laplacian. As vanilla graph convolutional networks are prone to oversmooth, we adopt a neural graph ODE framework. Specifically, we propose fractional graph Laplacian neural ODEs, which describe non-local dynamics. We prove that our approach allows propagating information between distant nodes while maintaining a low probability of long-distance jumps. Moreover, we show that our method is more flexible with respect to the convergence of the graph’s Dirichlet energy, thereby mitigating oversmoothing. We conduct extensive experiments on synthetic and real-world graphs, both directed and undirected, demonstrating our method’s versatility across diverse graph homophily levels. Our code is available at https://github.com/RPaolino/fLode

This work bridges two important concepts: the Neural Tangent Kernel (NTK), which captures the evolution of deep neural networks (DNNs) during training, and the Neural Collapse (NC) phenomenon, which refers to the emergence of symmetry and structure in the last-layer features of well-trained classification DNNs. We adopt the natural assumption that the empirical NTK develops a block structure aligned with the class labels, i.e., samples within the same class have stronger correlations than samples from different classes. Under this assumption, we derive the dynamics of DNNs trained with mean squared (MSE) loss and break them into interpretable phases. Moreover, we identify an invariant that captures the essence of the dynamics, and use it to prove the emergence of NC in DNNs with block-structured NTK. We provide large-scale numerical experiments on three common DNN architectures and three benchmark datasets to support our theory.

Deep neural networks have seen tremendous success over the last years. Since the training is performed on digital hardware, in this paper, we analyze what actually can be computed on current hardware platforms modeled as Turing machines, which would lead to inherent restrictions of deep learning. For this, we focus on the class of inverse problems, which, in particular, encompasses any task to reconstruct data from measurements. We prove that finite-dimensional inverse problems are not Banach-Mazur computable for small relaxation parameters. Even more, our results introduce a lower bound on the accuracy that can be obtained algorithmically.

Global Navigation Satellite Systems typically perform poorly in urban environments, where the likelihood of line-of-sight conditions between devices and satellites is low. Therefore, alternative location methods are required to achieve good accuracy. We present LocUNet: A convolutional, end-to-end trained neural network (NN) for the localization task, which is able to estimate the position of a user from the received signal strength (RSS) of a small number of Base Stations (BS). Using estimations of pathloss radio maps of the BSs and the RSS measurements of the users to be localized, LocUNet can localize users with state-of-the-art accuracy and enjoys high robustness to inaccuracies in the estimations of radio maps. The proposed method does not require generating RSS fingerprints of each specific area where the localization task is performed and is suitable for real-time applications. Moreover, two novel datasets that allow for numerical evaluations of RSS and ToA methods in realistic urban environments are presented and made publicly available for the research community. By using these datasets, we also provide a fair comparison of state-of-the-art RSS and ToA-based methods in the dense urban scenario and show numerically that LocUNet outperforms all the compared methods.

This paper deals with the problem of localization in a cellular network in a dense urban scenario. Global Navigation Satellite Systems (GNSS) typically perform poorly in urban environments, where the likelihood of line-of-sight conditions is low, and thus alternative localization methods are required for good accuracy. We present LocUNet: A deep learning method for localization, based merely on Received Signal Strength (RSS) from Base Stations (BSs), which does not require any increase in computation complexity at the user devices with respect to the device standard operations, unlike methods that rely on Time of Arrival (ToA) or Angle of Arrival information. In the proposed method, the user to be localized reports the RSS from BSs to a Central Processing Unit (CPU), which may be located in the cloud. Alternatively, the localization can be performed locally at the user. Using estimated pathloss radio maps of the BSs, LocUNet can localize users with state-of-the-art accuracy and enjoys high robustness to inaccuracies in the radio maps. The proposed method does not require pre-sampling of the environment; and is suitable for real-time applications, thanks to the RadioUNet, a neural network-based radio map estimator. We also introduce two datasets that allow numerical comparisons of RSS and ToA methods in realistic urban environments.

Neural Tangent Kernel (NTK) is widely used to analyze overparametrized neural networks due to the famous result by Jacot et al. (2018): in the infinite-width limit, the NTK is deterministic and constant during training. However, this result cannot explain the behavior of deep networks, since it generally does not hold if depth and width tend to infinity simultaneously. In this paper, we study the NTK of fully-connected ReLU networks with depth comparable to width. We prove that the NTK properties depend significantly on the depth-to-width ratio and the distribution of parameters at initialization. In fact, our results indicate the importance of the three phases in the hyperparameter space identified in Poole et al. (2016): ordered, chaotic and the edge of chaos (EOC). We derive exact expressions for the NTK dispersion in the infinite-depth-and-width limit in all three phases and conclude that the NTK variability grows exponentially with depth at the EOC and in the chaotic phase but not in the ordered phase. We also show that the NTK of deep networks may stay constant during training only in the ordered phase and discuss how the structure of the NTK matrix changes during training.

This work develops a flexible and mathematically sound framework for the design and analysis of graph scattering networks with variable branching ratios and generic functional calculus filters.Spectrally-agnostic stability guarantees for node- and graph-level perturbations are derived; the vertex-set non-preserving case is treated by utilizing recently developed mathematical-physics based tools. Energy propagation through the network layers is investigated and related to truncation stability. New methods of graph-level feature aggregation are introduced and stability of the resulting composite scattering architectures is established. Finally, scattering transforms are extended to edge- and higher order tensorial input. Theoretical results are complemented by numerical investigations: Suitably chosen scattering networks conforming to the developed theory perform better than traditional graph-wavelet based scattering approaches in social network graph classification tasks andsignificantly outperform other graph-based learning approaches to regression of quantum-chemical energies on QM$7$.

Message passing neural networks (MPNN) have seen a steep rise in popularity since their introduction as generalizations of convolutional neural networks to graph-structured data, and are now considered state-of-the-art tools for solving a large variety of graph-focused problems. We study the generalization error of MPNNs in graph classification and regression. We assume that graphs of different classes are sampled from different random graph models. We show that, when training a MPNN on a dataset sampled from such a distribution, the generalization gap increases in the complexity of the MPNN, and decreases, not only with respect to the number of training samples, but also with the average number of nodes in the graphs. This shows how a MPNN with high complexity can generalize from a small dataset of graphs, as long as the graphs are large. The generalization bound is derived from a uniform convergence result, that shows that any MPNN, applied on a graph, approximates the MPNN applied on the geometric model that the graph discretizes.

This work provides the first theoretical study on the ability of graph Message Passing Neural Networks (gMPNNs) ---such as Graph Neural Networks (GNNs)--- to perform inductive out-of-distribution (OOD) link prediction tasks, where deployment (test) graph sizes are larger than training graphs. We first prove non-asymptotic bounds showing that link predictors based on permutation-equivariant (structural) node embeddings obtained by gMPNNs can converge to a random guess as test graphs get larger. We then propose a theoretically-sound gMPNN that outputs structural pairwise (2-node) embeddings and prove non-asymptotic bounds showing that, as test graphs grow, these embeddings converge to embeddings of a continuous function that retains its ability to predict links OOD. Empirical results on random graphs show agreement with our theoretical results.

For a d-ary Boolean function Φ: {0, 1}d → {0, 1} and an assignment to its variables x = (x1, x2, . . . , xd) we consider the problem of finding those subsets of the variables that are sufficient to determine the function value with a given probability δ. This is motivated by the task of interpreting predictions of binary classifiers described as Boolean circuits, which can be seen as special cases of neural networks. We show that the problem of deciding whether such subsets of relevant variables of limited size k ≤ d exist is complete for the complexity class NPPP and thus, generally, unfeasible to solve. We then introduce a variant, in which it suffices to check whether a subset determines the function value with probability at least δ or at most δ − γ for 0 < γ < δ. This promise of a probability gap reduces the complexity to the class NPBPP. Finally, we show that finding the minimal set of relevant variables cannot be reasonably approximated, i.e. with an approximation factor d1−α for α > 0, by a polynomial time algorithm unless P = NP. This holds even with the promise of a probability gap.

This paper focuses on spectral graph convolutional neural networks (ConvNets), where filters are defined as elementwise multiplication in the frequency domain of a graph. In machine learning settings where the data set consists of signals defined on many different graphs, the trained ConvNet should generalize to signals on graphs unseen in the training set. It is thus important to transfer ConvNets between graphs. Transferability, which is a certain type of generalization capability, can be loosely defined as follows: if two graphs describe the same phenomenon, then a single filter or ConvNet should have similar repercussions on both graphs. This paper aims at debunking the common misconception that spectral filters are not transferable. We show that if two graphs discretize the same “continuous” space, then a spectral filter or ConvNet has approximately the same repercussion on both graphs. Our analysis is more permissive than the standard analysis. Transferability is typically described as the robustness of the filter to small graph perturbations and re-indexing of the vertices. Our analysis accounts also for large graph perturbations. We prove transferability between graphs that can have completely different dimensions and topologies, only requiring that both graphs discretize the same underlying space in some generic sense.

In this paper we propose a highly efficient and very accurate deep learning method for estimating the propagation pathloss from a point x (transmitter location) to any point y on a planar domain. For applications such as user-cell site association and device-to-device link scheduling, an accurate knowledge of the pathloss function for all pairs of transmitter-receiver locations is very important. Commonly used statistical models approximate the pathloss as a decaying function of the distance between transmitter and receiver. However, in realistic propagation environments characterized by the presence of buildings, street canyons, and objects at different heights, such radial-symmetric functions yield very misleading results. In this paper we show that properly designed and trained deep neural networks are able to learn how to estimate the pathloss function, given an urban environment, in a very accurate and computationally efficient manner. Our proposed method, termed RadioUNet, learns from a physical simulation dataset, and generates pathloss estimations that are very close to the simulations, but are much faster to compute for real-time applications. Moreover, we propose methods for transferring what was learned from simulations to real-life. Numerical results show that our method significantly outperforms previously proposed methods.

In this paper we propose a highly efficient and very accurate method for estimating the propagation pathloss from a point x to all points y on the 2D plane. Our method, termed RadioUNet, is a deep neural network. For applications such as user-cell site association and device-to-device (D2D) link scheduling, an accurate knowledge of the pathloss function for all pairs of locations is very important. Commonly used statistical models approximate the pathloss as a decaying function of the distance between the points. However, in realistic propagation environments characterized by the presence of buildings, street canyons, and objects at different heights, such radial-symmetric functions yield very misleading results. In this paper we show that properly designed and trained deep neural networks are able to learn how to estimate the pathloss function, given an urban environment, very accurately and extremely quickly. Our proposed method generates pathloss estimations that are very close to estimations given by physical simulation, but much faster. Moreover, experimental results show that our method significantly outperforms previously proposed methods based on radial basis function interpolation and tensor completion.

Microlocal analysis provides deep insight into singularity structures and is often crucial for solving inverse problems, predominately, in imaging sciences. Of particular importance is the analysis of wavefront sets and the correct extraction of those. In this paper, we introduce the first algorithmic approach to extract the wavefront set of images, which combines data-based and model-based methods. Based on a celebrated property of the shearlet transform to unravel information on the wavefront set, we extract the wavefront set of an image by first applying a discrete shearlet transform and then feeding local patches of this transform to a deep convolutional neural network trained on labeled data. The resulting algorithm outperforms all competing algorithms in edge-orientation and ramp-orientation detection.

Applications such as magnetic resonance imaging acquire imaging data by point samples of their Fourier transform. This raises the question of balancing the efficiency of the sampling strategies with the approximation accuracy of an associated reconstruction procedure. In this paper, we introduce a novel sampling-reconstruction scheme based on a random anisotropic sampling pattern and a compressed sensing--type reconstruction strategy with a variant of dualizable shearlet frames as sparsifying representation system. For this scheme, we prove asymptotic almost optimality in an approximation theoretic sense for cartoon-like functions as a model class for the imaging data. Finally, we present numerical experiments showing the superiority of our scheme over other approaches.

BACKGROUND: High-throughput proteomics techniques, such as mass spectrometry (MS)-based approaches, produce very high-dimensional data-sets. In a clinical setting one is often interested in how mass spectra differ between patients of different classes, for example spectra from healthy patients vs. spectra from patients having a particular disease. Machine learning algorithms are needed to (a) identify these discriminating features and (b) classify unknown spectra based on this feature set. Since the acquired data is usually noisy, the algorithms should be robust against noise and outliers, while the identified feature set should be as small as possible. RESULTS: We present a new algorithm, Sparse Proteomics Analysis (SPA), based on the theory of compressed sensing that allows us to identify a minimal discriminating set of features from mass spectrometry data-sets. We show (1) how our method performs on artificial and real-world data-sets, (2) that its performance is competitive with standard (and widely used) algorithms for analyzing proteomics data, and (3) that it is robust against random and systematic noise. We further demonstrate the applicability of our algorithm to two previously published clinical data-sets.

Wavelets and their associated transforms are highly efficient when approximating and analyzing one-dimensional signals. However, multivariate signals such as images or videos typically exhibit curvilinear singularities, which wavelets are provably deficient in sparsely approximating and also in analyzing in the sense of, for instance, detecting their direction. Shearlets are a directional representation system extending the wavelet framework, which overcomes those deficiencies. Similar to wavelets, shearlets allow a faithful implementation and fast associated transforms. In this article, we will introduce a comprehensive carefully documented software package coined ShearLab 3D (www.ShearLab.org) and discuss its algorithmic details. This package provides MATLAB code for a novel faithful algorithmic realization of the 2D and 3D shearlet transform (and their inverses) associated with compactly supported universal shearlet systems incorporating the option of using CUDA. We will present extensive numerical experiments in 2D and 3D concerning denoising, inpainting, and feature extraction, comparing the performance of ShearLab 3D with similar transform-based algorithms such as curvelets, contourlets, or surfacelets. In the spirit of reproducible research, all scripts are accessible on www.ShearLab.org.

Scalable frames are frames with the property that the frame vectors can be rescaled resulting in tight frames. However, if a frame is not scalable, one has to aim for an approximate procedure. For this, in this paper we introduce three novel quantitative measures of the closeness to scalability for frames in finite dimensional real Euclidean spaces. Besides the natural measure of scalability given by the distance of a frame to the set of scalable frames, another measure is obtained by optimizing a quadratic functional, while the third is given by the volume of the ellipsoid of minimal volume containing the symmetrized frame. After proving that these measures are equivalent in a certain sense, we establish bounds on the probability of a randomly selected frame to be scalable. In the process, we also derive new necessary and sufficient conditions for a frame to be scalable.

Recently introduced inpainting algorithms using a combination of applied harmonic analysis and compressed sensing have turned out to be very successful. One key ingredient is a carefully chosen representation system which provides (optimally) sparse approximations of the original image. Due to the common assumption that images are typically governed by anisotropic features, directional representation systems have often been utilized. One prominent example of this class are shearlets, which have the additional benefit of allowing faithful implementations. Numerical results show that shearlets significantly outperform wavelets in inpainting tasks. One of those software packages, ShearLab, even offers the flexibility of using a different parameter for each scale, which is not yet covered by shearlet theory. In this paper, we first introduce universal shearlet systems which are associated with an arbitrary scaling sequence, thereby modeling the previously mentioned flexibility. In addition, this novel construction allows for a smooth transition between wavelets and shearlets and therefore enables us to analyze them in a uniform fashion. For a large class of such scaling sequences, we first prove that the associated universal shearlet systems form band-limited Parseval frames for $L^2(\mathbb{R}^2)$ consisting of Schwartz functions. Second, we analyze the inpainting performance of this class of universal shearlet systems within a distributional model situation using an $\ell^{1}$-analysis minimization algorithm for reconstruction. Our main result states that, provided that the scaling sequence is comparable to the size of the (scale-dependent) gap, asymptotically perfect inpainting is achieved.

This paper proposes an image interpolation algorithm exploiting sparse representation for natural images. It involves three steps: (a) obtaining an initial estimate of the high resolution image using linear methods like FIR filtering, (b) promoting sparsity in a selected dictionary through thresholding and (c) extracting high frequency information from the approximation and adding it to the initial estimate. For the sparse modeling, a shearlet dictionary is chosen to yield a multi-scale directional representation. The proposed algorithm is compared to several state-of-the-art methods to assess its objective as well as subjective performance. Compared to the cubic spline interpolation method, an average PSNR gain of around 0.7 dB is observed over a dataset of 200 images.

Natural images are typically a composition of cartoon and texture structures. One common task is to separate such an image into two single images, one containing the cartoon part and the other containing the texture part. Recently, a powerful class of algorithms using sparse approximation and $\ell_1$ minimization has been introduced to resolve this problem, and numerous inspiring empirical results have already been obtained. In this paper we provide a theoretical study of the separation of a combination of cartoon and texture structures in a continuum model situation using this class of algorithms. The methodology we consider expands the image in a combined dictionary consisting of a curvelet frame and a Gabor frame and minimizes the $\ell_1$ norm. Sparse approximation properties then force the cartoon components into the curvelet coefficients and the texture components into the Gabor coefficients, thereby separating the image. Utilizing the fact that the coefficients are clustered geometrically, we prove that at sufficiently fine scales arbitrarily precise separation is possible. For this analysis, as a model for cartoon we consider a compactly supported function which is $C^2$ apart from a $C^2$ discontinuity curve. As a model for texture we consider locally oscillatory patterns generated by a Gabor system associated with a fixed appropriate size of the Gabor window, which is linked---satisfying an energy matching condition---to the scale of the curvelet system. In accordance with the continuum domain setting, the main ingredients of our analysis are clustered/geometric sparsity and a phase space viewpoint.

Multivariate problems are typically governed by anisotropic features such as edges in images. A common bracket of most of the various directional representation systems which have been proposed to deliver sparse approximations of such features is the utilization of parabolic scaling. One prominent example is the shearlet system. Our objective in this paper is threefold: We first develop a digital shearlet theory which is rationally designed in the sense that it is the digitization of the existing shearlet theory for continuous data. This implies that shearlet theory provides a unified treatment of both the continuum and digital realms. Second, we analyze the utilization of pseudo-polar grids and the pseudo-polar Fourier transform for digital implementations of parabolic scaling algorithms. We derive an isometric pseudo-polar Fourier transform by careful weighting of the pseudo-polar grid, allowing exploitation of its adjoint for the inverse transform. This leads to a digital implementation of the shearlet transform; an accompanying MATLAB toolbox called ShearLab (www.ShearLab.org) is provided. And, third, we introduce various quantitative measures for digital parabolic scaling algorithms in general, allowing one to tune parameters and objectively improve the implementation as well as compare different directional transform implementations. The usefulness of such measures is exemplarily demonstrated for the digital shearlet transform.

Sparse representations have emerged as a powerful tool in signal and information processing, culminated by the success of new acquisition and processing techniques such as compressed sensing (CS). Fusion frames are very rich new signal representation methods that use collections of subspaces instead of vectors to represent signals. This work combines these exciting fields to introduce a new sparsity model for fusion frames. Signals that are sparse under the new model can be compressively sampled and uniquely reconstructed in ways similar to sparse signals using standard CS. The combination provides a promising new set of mathematical tools and signal models useful in a variety of applications. With the new model, a sparse signal has energy in very few of the subspaces of the fusion frame, although it does not need to be sparse within each of the subspaces it occupies. This sparsity model is captured using a mixedl1/l2norm for fusion frames. A signal sparse in a fusion frame can be sampled using very few random projections and exactly reconstructed using a convex optimization that minimizes this mixedl1/l2norm. The provided sampling conditions generalize coherence and RIP conditions used in standard CS theory. It is demonstrated that they are sufficient to guarantee sparse recovery of any signal sparse in our model. More over, a probabilistic analysis is provided using a stochastic model on the sparse signal that shows that under very mild conditions the probability of recovery failure decays exponentially with in creasing dimension of the subspaces.

Frames have established themselves as a means to derive redundant, yet stable decompositions of a signal for analysis or transmission, while also promoting sparse expansions. However, when the signal dimension is large, the computation of the frame measurements of a signal typically requires a large number of additions and multiplications, and this makes a frame decomposition intractable in applications with limited computing budget. To address this problem, in this paper, we focus on frames in finite-dimensional Hilbert spaces and introduce sparsity for such frames as a new paradigm. In our terminology, a sparse frame is a frame whose elements have a sparse representation in an orthonormal basis, thereby enabling low-complexity frame decompositions. To introduce a precise meaning of optimality, we take the sum of the numbers of vectors needed from this orthonormal basis when expanding each frame vector as sparsity measure. We then analyze the recently introduced algorithm Spectral Tetris for construction of unit norm tight frames and prove that the tight frames generated by this algorithm are in fact optimally sparse with respect to the standard unit vector basis. Finally, we show that even the generalization of Spectral Tetris for the construction of unit norm frames associated with a given frame operator produces optimally sparse frames.