arXiv subjects

Alex Goeßmann

Publications and source records attributed to Alex Goeßmann.

4 recordsLinked to original sources

arXiv2021

An asymptotic analysis of separating pointlike and $C^β$-curvelike singularities

In this paper, we present a theoretical analysis of separating images consisting of pointlike and $C^{ β}$-curvelike structures, where $β\in (1,2] $. Our approach is based on $l_1$-minimization, in which the sparsity of the desired solution is exploited by two sparse representation systems. It is well known that for such components wavelets provide an optimally sparse representation for point singularities, whereas $α$-shearlet type with $α$=$\frac{2}β$ might be best adapted to the $C^β$-curvilinear singularities. In our analysis, we first propose a reconstruction framework with a theoretical guarantee on convergence, which is extended to use general frames instead of Parseval frames. We then construct a dual pair of bandlimited $α$-shearlets which possesses a good time and frequency localization. Finally, we apply the result to derive an asymptotic accuracy of the reconstructions. In addition, we show that it is possible to separate these two components as long as $α<2$, i.e., bandlimited $α$-shearlets which range from wavelet to shearlet type do not coincide with wavelets in the sense of isotropic fashion.

math.FA↗

arXiv2021

Representations of molecules and materials for interpolation of quantum-mechanical simulations via machine learning

Computational study of molecules and materials from first principles is a cornerstone of physics, chemistry, and materials science, but limited by the cost of accurate and precise simulations. In settings involving many simulations, machine learning can reduce these costs, often by orders of magnitude, by interpolating between reference simulations. This requires representations that describe any molecule or material and support interpolation. We comprehensively review and discuss current representations and relations between them, using a unified mathematical framework based on many-body functions, group averaging, and tensor products. For selected state-of-the-art representations, we compare energy predictions for organic molecules, binary alloys, and Al-Ga-In sesquioxides in numerical experiments controlled for data distribution, regression method, and hyper-parameter optimization.

physics.comp-ph↗

arXiv2020

The Restricted Isometry of ReLU Networks: Generalization through Norm Concentration

While regression tasks aim at interpolating a relation on the entire input space, they often have to be solved with a limited amount of training data. Still, if the hypothesis functions can be sketched well with the data, one can hope for identifying a generalizing model. In this work, we introduce with the Neural Restricted Isometry Property (NeuRIP) a uniform concentration event, in which all shallow $\mathrm{ReLU}$ networks are sketched with the same quality. To derive the sample complexity for achieving NeuRIP, we bound the covering numbers of the networks in the Sub-Gaussian metric and apply chaining techniques. In case of the NeuRIP event, we then provide bounds on the expected risk, which hold for networks in any sublevel set of the empirical risk. We conclude that all networks with sufficiently small empirical risk generalize uniformly.

stat.ML↗

arXiv2019

Group Field Theory and Holographic Tensor Networks: Dynamical Corrections to the Ryu-Takayanagi formula

We introduce group field theory networks as a generalization of spin networks and of (symmetric) random tensor networks and provide a statistical computation of the Rényi entropy for a bipartite network state using the partition function of a simple interacting group field theory. The expectation value of the entanglement entropy is calculated by an expansion into stranded Feynman graphs and is shown to be captured by a Ryu- Takayanagi formula. For a simple interacting group field theory, we can prove the linear corrections, given by a polynomial perturbation of the Gaussian measure, to be negligible for a broad class of networks.

hep-th↗