arXiv subjects

Leon Frischauf

Publications and source records attributed to Leon Frischauf.

6 recordsLinked to original sources

arXiv2024

Spectral Function Space Learning and Numerical Linear Algebra Networks for Solving Linear Inverse Problems

We consider solving a probably ill-conditioned linear operator equation, where the operator is not modeled by physical laws but is specified via training pairs (consisting of images and data) of the input-output relation of the operator. We derive a stable method for computing the operator, which consists of first a Gram-Schmidt orthonormalization of images and a principal component analysis of the data. This two-step algorithm provides a spectral decomposition of the linear operator. Moreover, we show that both Gram-Schmidt and principal component analysis can be written as a deep neural network, which relates this procedure to de-and encoder networks. Therefore, we call the two-step algorithm a linear algebra network. Finally, we provide numerical simulations showing the strategy is feasible for reconstructing spectral functions and for solving operator equations without explicitly exploiting the physical model.

math.NA↗

arXiv2024

Classification with neural networks with quadratic decision functions

Neural networks with quadratic decision functions have been introduced as alternatives to standard neural networks with affine linear ones. They are advantageous when the objects or classes to be identified are compact and of basic geometries like circles, ellipses etc. In this paper we investigate the use of such ansatz functions for classification. In particular we test and compare the algorithm on the MNIST dataset for classification of handwritten digits and for classification of subspecies. We also show, that the implementation can be based on the neural network structure in the software Tensorflow and Keras, respectively.

cs.LG↗

arXiv2023

Quadratic neural networks for solving inverse problems

In this paper we investigate the solution of inverse problems with neural network ansatz functions with generalized decision functions. The relevant observation for this work is that such functions can approximate typical test cases, such as the Shepp-Logan phantom, better, than standard neural networks. Moreover, we show that the convergence analysis of numerical methods for solving inverse problems with shallow generalized neural network functions leads to more intuitive convergence conditions, than for deep affine linear neural networks.

math.NA↗

arXiv2022

Universal approximation properties of shallow quadratic neural networks

In this paper we study shallow neural network functions which are linear combinations of compositions of activation and quadratic functions, replacing standard affine linear functions, often called neurons. We show the universality of this approximation and prove convergence rates results based on the theory of wavelets and statistical learning. We show for simple test cases that this ansatz requires a smaller numbers of neurons than standard affine linear neural networks. Moreover, we investigate the efficiency of this approach for clustering tasks with the MNIST data set. Similar observations are made when comparing deep (multi-layer) networks.

math.NA↗

arXiv2021

Diffusion Tensor Regularization with Metric Double Integrals

In this paper we propose a variational regularization method for denoising and inpainting of diffusion tensor magnetic resonance images. We consider these images as manifold-valued Sobolev functions, i.e. in an infinite dimensional setting, which are defined appropriately. The regularization functionals are defined as double integrals, which are equivalent to Sobolev semi-norms in the Euclidean setting. We extend the analysis of Ciak, Melching and Scherzer "Regularization with Metric Double Integrals of Functions with Values in a Set of Vectors", in: Journal of Mathematical Imaging and Vision (2019) concerning stability and convergence of the variational regularization methods by a uniqueness result, apply them to diffusion tensor processing, and validate our model in numerical examples with synthetic and real data.

math.OC↗

arXiv2021

Data driven reconstruction using frames and Riesz bases

We study the problem of regularization of inverse problems adopting a purely data driven approach, by using the similarity to the method of regularization by projection. We provide an application of a projection algorithm, utilized and applied in frames theory, as a data driven reconstruction procedure in inverse problems, generalizing the algorithm proposed by the authors in Inverse Problems 36 (2020), n. 12, 125009, based on an orthonormalization procedure for the training pairs. We show some numerical experiments, comparing the different methods.

math.NA↗