arXiv subjects

Gitta Kutyniok

Publications and source records attributed to Gitta Kutyniok.

At least 109 records · Page 6Linked to original sources

arXiv2020

Numerical Solution of the Parametric Diffusion Equation by Deep Neural Networks

We perform a comprehensive numerical study of the effect of approximation-theoretical results for neural networks on practical learning problems in the context of numerical analysis. As the underlying model, we study the machine-learning-based solution of parametric partial differential equations. Here, approximation theory predicts that the performance of the model should depend only very mildly on the dimension of the parameter space and is determined by the intrinsic dimension of the solution manifold of the parametric partial differential equation. We use various methods to establish comparability between test-cases by minimizing the effect of the choice of test-cases on the optimization and sampling aspects of the learning problem. We find strong support for the hypothesis that approximation-theoretical effects heavily influence the practical behavior of learning problems in numerical analysis.

math.NA↗

arXiv2020

Interval Neural Networks: Uncertainty Scores

We propose a fast, non-Bayesian method for producing uncertainty scores in the output of pre-trained deep neural networks (DNNs) using a data-driven interval propagating network. This interval neural network (INN) has interval valued parameters and propagates its input using interval arithmetic. The INN produces sensible lower and upper bounds encompassing the ground truth. We provide theoretical justification for the validity of these bounds. Furthermore, its asymmetric uncertainty scores offer additional, directional information beyond what Gaussian-based, symmetric variance estimation can provide. We find that noise in the data is adequately captured by the intervals produced with our method. In numerical experiments on an image reconstruction task, we demonstrate the practical utility of INNs as a proxy for the prediction error in comparison to two state-of-the-art uncertainty quantification methods. In summary, INNs produce fast, theoretically justified uncertainty scores for DNNs that are easy to interpret, come with added information and pose as improved error proxies - features that may prove useful in advancing the usability of DNNs especially in sensitive applications such as health care.

cs.LG↗

arXiv2019

Shearlets as Feature Extractor for Semantic Edge Detection: The Model-Based and Data-Driven Realm

Semantic edge detection has recently gained a lot of attention as an image processing task, mainly due to its wide range of real-world applications. This is based on the fact that edges in images contain most of the semantic information. Semantic edge detection involves two tasks, namely pure edge detecion and edge classification. Those are in fact fundamentally distinct in terms of the level of abstraction that each task requires, which is known as the distracted supervision paradox that limits the possible performance of a supervised model in semantic edge detection. In this work, we will present a novel hybrid method to avoid the distracted supervision paradox and achieve high-performance in semantic edge detection. Our approach is based on a combination of the model-based concept of shearlets, which provides probably optimally sparse approximations of a model-class of images, and the data-driven method of a suitably designed convolutional neural netwok. Finally, we present several applications such as tomographic reconstruction and show that our approach signifiantly outperforms former methods, thereby indicating the value of such hybrid methods for the area in biomedical imaging.

eess.IV↗

arXiv2019

The Mismatch Principle: The Generalized Lasso Under Large Model Uncertainties

We study the estimation capacity of the generalized Lasso, i.e., least squares minimization combined with a (convex) structural constraint. While Lasso-type estimators were originally designed for noisy linear regression problems, it has recently turned out that they are in fact robust against various types of model uncertainties and misspecifications, most notably, non-linearly distorted observation models. This work provides more theoretical evidence for this somewhat astonishing phenomenon. At the heart of our analysis stands the mismatch principle, which is a simple recipe to establish theoretical error bounds for the generalized Lasso. The associated estimation guarantees are of independent interest and are formulated in a fairly general setup, permitting arbitrary sub-Gaussian data, possibly with strongly correlated feature designs; in particular, we do not assume a specific observation model which connects the input and output variables. Although the mismatch principle is conceived based on ideas from statistical learning theory, its actual application area are (high-dimensional) estimation tasks for semi-parametric models. In this context, the benefits of the mismatch principle are demonstrated for a variety of popular problem classes, such as single-index models, generalized linear models, and variable selection. Apart from that, our findings are also relevant to recent advances in quantized and distributed compressed sensing.

math.ST↗

arXiv2019

Extraction of digital wavefront sets using applied harmonic analysis and deep neural networks

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.

eess.IV↗

arXiv2019

The Computational Complexity of Understanding Network Decisions

For a Boolean function $Φ\colon\{0,1\}^d\to\{0,1\}$ and an assignment to its variables $\mathbf{x}=(x_1, x_2, \dots, x_d)$ we consider the problem of finding the 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 $k\leq d$ exist is complete for the complexity class $\mathsf{NP}^{\mathsf{PP}}$ and thus generally unfeasible to solve. We introduce a variant where 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 reduces the complexity to the class $\mathsf{NP}^{\mathsf{BPP}}$. Finally, we show that finding the minimal set of relevant variables can not be reasonably approximated, i.e. with an approximation factor $d^{1-α}$ for $α> 0$, by a polynomial time algorithm unless $\mathsf{P} = \mathsf{NP}$ (this holds even with the probability gap).

cs.CC↗

arXiv2019

A Rate-Distortion Framework for Explaining Neural Network Decisions

We formalise the widespread idea of interpreting neural network decisions as an explicit optimisation problem in a rate-distortion framework. A set of input features is deemed relevant for a classification decision if the expected classifier score remains nearly constant when randomising the remaining features. We discuss the computational complexity of finding small sets of relevant features and show that the problem is complete for $\mathsf{NP}^\mathsf{PP}$, an important class of computational problems frequently arising in AI tasks. Furthermore, we show that it even remains $\mathsf{NP}$-hard to only approximate the optimal solution to within any non-trivial approximation factor. Finally, we consider a continuous problem relaxation and develop a heuristic solution strategy based on assumed density filtering for deep ReLU neural networks. We present numerical experiments for two image classification data sets where we outperform established methods in particular for sparse explanations of neural network decisions.

cs.LG↗

arXiv2019

Error bounds for approximations with deep ReLU neural networks in $W^{s,p}$ norms

We analyze approximation rates of deep ReLU neural networks for Sobolev-regular functions with respect to weaker Sobolev norms. First, we construct, based on a calculus of ReLU networks, artificial neural networks with ReLU activation functions that achieve certain approximation rates. Second, we establish lower bounds for the approximation by ReLU neural networks for classes of Sobolev-regular functions. Our results extend recent advances in the approximation theory of ReLU networks to the regime that is most relevant for applications in the numerical analysis of partial differential equations.

math.FA↗

arXiv2019

On the Transferability of Spectral Graph Filters

This paper focuses on spectral filters on graphs, namely filters defined as elementwise multiplication in the frequency domain of a graph. In many graph signal processing settings, it is important to transfer a filter from one graph to another. One example is in graph convolutional neural networks (ConvNets), where the dataset consists of signals defined on many different graphs, and the learned filters should generalize to signals on new graphs, not present in the training set. A necessary condition for transferability (the ability to transfer filters) is stability. Namely, given a graph filter, if we add a small perturbation to the graph, then the filter on the perturbed graph is a small perturbation of the original filter. It is a common misconception that spectral filters are not stable, and this paper aims at debunking this mistake. We introduce a space of filters, called the Cayley smoothness space, that contains the filters of state-of-the-art spectral filtering methods, and whose filters can approximate any generic spectral filter. For filters in this space, the perturbation in the filter is bounded by a constant times the perturbation in the graph, and filters in the Cayley smoothness space are thus termed linearly stable. By combining stability with the known property of equivariance, we prove that graph spectral filters are transferable.

cs.LG↗

arXiv2019

The Oracle of DLphi

We present a novel technique based on deep learning and set theory which yields exceptional classification and prediction results. Having access to a sufficiently large amount of labelled training data, our methodology is capable of predicting the labels of the test data almost always even if the training data is entirely unrelated to the test data. In other words, we prove in a specific setting that as long as one has access to enough data points, the quality of the data is irrelevant.

cs.LG↗

arXiv2018

Learning sparse representations on the sphere

Many representation systems on the sphere have been proposed in the past, such as spherical harmonics, wavelets, or curvelets. Each of these data representations is designed to extract a specific set of features, and choosing the best fixed representation system for a given scientific application is challenging. In this paper, we show that we can learn directly a representation system from given data on the sphere. We propose two new adaptive approaches: the first is a (potentially multi-scale) patch-based dictionary learning approach, and the second consists in selecting a representation among a parametrized family of representations, the α-shearlets. We investigate their relative performance to represent and denoise complex structures on different astrophysical data sets on the sphere.

astro-ph.IM↗

arXiv2018

Optimal Approximation with Sparsely Connected Deep Neural Networks

We derive fundamental lower bounds on the connectivity and the memory requirements of deep neural networks guaranteeing uniform approximation rates for arbitrary function classes in $L^2(\mathbb R^d)$. In other words, we establish a connection between the complexity of a function class and the complexity of deep neural networks approximating functions from this class to within a prescribed accuracy. Additionally, we prove that our lower bounds are achievable for a broad family of function classes. Specifically, all function classes that are optimally approximated by a general class of representation systems---so-called \emph{affine systems}---can be approximated by deep neural networks with minimal connectivity and memory requirements. Affine systems encompass a wealth of representation systems from applied harmonic analysis such as wavelets, ridgelets, curvelets, shearlets, $α$-shearlets, and more generally $α$-molecules. Our central result elucidates a remarkable universality property of neural networks and shows that they achieve the optimum approximation properties of all affine systems combined. As a specific example, we consider the class of $α^{-1}$-cartoon-like functions, which is approximated optimally by $α$-shearlets. We also explain how our results can be extended to the case of functions on low-dimensional immersed manifolds. Finally, we present numerical experiments demonstrating that the standard stochastic gradient descent algorithm generates deep neural networks providing close-to-optimal approximation rates. Moreover, these results indicate that stochastic gradient descent can actually learn approximations that are sparse in the representation systems optimally sparsifying the function class the network is trained on.

cs.LG↗

arXiv2018

Hierarchical Sparse Channel Estimation for Massive MIMO

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.

cs.IT↗

arXiv2018

Compressed Sensing for Analog Signals

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.

math.FA↗

arXiv2017

Estimation of Angles of Arrival Through Superresolution -- A Soft Recovery Approach for General Antenna Geometries

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.

eess.SP↗

arXiv2017

A Haar Wavelet-Based Perceptual Similarity Index for Image Quality Assessment

In most practical situations, the compression or transmission of images and videos creates distortions that will eventually be perceived by a human observer. Vice versa, image and video restoration techniques, such as inpainting or denoising, aim to enhance the quality of experience of human viewers. Correctly assessing the similarity between an image and an undistorted reference image as subjectively experienced by a human viewer can thus lead to significant improvements in any transmission, compression, or restoration system. This paper introduces the Haar wavelet-based perceptual similarity index (HaarPSI), a novel and computationally inexpensive similarity measure for full reference image quality assessment. The HaarPSI utilizes the coefficients obtained from a Haar wavelet decomposition to assess local similarities between two images, as well as the relative importance of image areas. The consistency of the HaarPSI with the human quality of experience was validated on four large benchmark databases containing thousands of differently distorted images. On these databases, the HaarPSI achieves higher correlations with human opinion scores than state-of-the-art full reference similarity measures like the structural similarity index (SSIM), the feature similarity index (FSIM), and the visual saliency-based index (VSI). Along with the simple computational structure and the short execution time, these experimental results suggest a high applicability of the HaarPSI in real world tasks.

cs.CV↗

arXiv2017

Anisotropic Multiscale Systems on Bounded Domains

We provide a construction of multiscale systems on a bounded domain $Ω\subset \mathbb{R}^2$ coined boundary shearlet systems, which satisfy several properties advantageous for applications to imaging science and the numerical analysis of partial differential equations. More precisely, we construct boundary shearlet systems that form frames for the Sobolev spaces $H^s(Ω),s\in \mathbb{N} \cup \{0\},$ with controllable frame bounds and admit optimally sparse approximations for functions, which are smooth apart from a curve-like discontinuity. We show that the constructed systems allow incorporating boundary conditions. Furthermore, for $s \geq 0$ and $f\in H^s(Ω)$ we prove that weighted $\ell^2$ norms of the $L^2-$analysis coefficients of $f$ are equivalent to its $H^s(Ω)$ norm. This yields in particular, that the reweighted systems are frames also for $H^{-s}(Ω)$. Moreover, we demonstrate numerically, that the associated $L^2-$synthesis operator is also stable as a map to $H^s(Ω)$ which, in combination with the previous result, strongly indicates that these systems constitute so-called Gelfand frames for $(H^s(Ω), L^2(Ω), H^{-s}(Ω))$.

math.FA↗

arXiv2017

Shearlet-based compressed sensing for fast 3D cardiac MR imaging using iterative reweighting

High-resolution three-dimensional (3D) cardiovascular magnetic resonance (CMR) is a valuable medical imaging technique, but its widespread application in clinical practice is hampered by long acquisition times. Here we present a novel compressed sensing (CS) reconstruction approach using shearlets as a sparsifying transform allowing for fast 3D CMR (3DShearCS). Shearlets are mathematically optimal for a simplified model of natural images and have been proven to be more efficient than classical systems such as wavelets. Data is acquired with a 3D Radial Phase Encoding (RPE) trajectory and an iterative reweighting scheme is used during image reconstruction to ensure fast convergence and high image quality. In our in-vivo cardiac MRI experiments we show that the proposed method 3DShearCS has lower relative errors and higher structural similarity compared to the other reconstruction techniques especially for high undersampling factors, i.e. short scan times. In this paper, we further show that 3DShearCS provides improved depiction of cardiac anatomy (measured by assessing the sharpness of coronary arteries) and two clinical experts qualitatively analyzed the image quality.

cs.CV↗