arXiv subjects

Sebastian Mayer

Publications and source records attributed to Sebastian Mayer.

9 recordsLinked to original sources

arXiv2023

Studying Evolutionary Solution Adaption Using a Flexibility Benchmark Based on a Metal Cutting Process

We consider optimizing for different production requirements from the viewpoint of a bio-inspired framework for system flexibility that allows us to study the ability of an algorithm to transfer solutions from previous optimization tasks, which also relates to dynamic evolutionary optimization. Optimizing manufacturing process parameters is typically a multi-objective problem with often contradictory objectives such as production quality and production time. If production requirements change, process parameters have to be optimized again. Since optimization usually requires costly simulations based on, for example, the Finite Element method, it is of great interest to have means to reduce the number of evaluations needed for optimization. Based on the extended Oxley model for orthogonal metal cutting, we introduce a multi-objective optimization benchmark where different materials define related optimization tasks. We use the benchmark to study the flexibility of NSGA-II, which we extend by two variants: 1) varying goals, which optimizes solutions for two tasks simultaneously to obtain in-between source solutions expected to be more adaptable, and 2) active-inactive genotype, which accommodates different possibilities that can be activated or deactivated. Results show that adaption, i.e. transferring a solution from a previous optimization task, with standard NSGA-II greatly reduces the number of evaluations required for optimization to a target goal in comparison to starting from scratch. The proposed variants further improve the adaption costs, although further work is needed towards making the methods advantageous for real applications.

cs.NE↗

arXiv2022

The elements of flexibility for task-performing systems

What makes living systems flexible so that they can react quickly and adapt easily to changing environments? This question has not only engaged biologists for decades but is also of great interest to computer scientists and engineers who seek inspiration from nature to increase the flexibility of task-performing systems such as machine learning systems, robots, or manufacturing systems. In this paper, we give a broad overview of design features of living systems that are known to promote flexibility. We call these design features the "elements of flexibility". Moreover, to facilitate interdisciplinary, bio-inspired research that brings the elements of flexibility to man-made task-performing systems, we introduce a general formalism for system flexibility optimization. The formalism is intended to (i) provide a common language to communicate ideas about system flexibility among researchers with different backgrounds, (ii) help to understand and compare existing research on system flexibility, e.g., in transfer learning or manufacturing flexibility, and (iii) provide a basis for a general theory of system flexibility optimization.

eess.SY↗

arXiv2019

Entropy numbers of finite dimensional mixed-norm balls and function space embeddings with small mixed smoothness

We study the embedding $\text{id}: \ell_p^b(\ell_q^d) \to \ell_r^b(\ell_u^d)$ and prove matching bounds for the entropy numbers $e_k(\text{id})$ provided that $0<p<r\leq \infty$ and $0<q\leq u\leq \infty$. Based on this finding, we establish optimal dimension-free asymptotic rates for the entropy numbers of embeddings of Besov and Triebel-Lizorkin spaces of small dominating mixed smoothness which settles an open question in the literature. Both results rely on a novel covering construction recently found by Edmunds and Netrusov.

math.FA↗

arXiv2019

Informed Machine Learning -- A Taxonomy and Survey of Integrating Knowledge into Learning Systems

Despite its great success, machine learning can have its limits when dealing with insufficient training data. A potential solution is the additional integration of prior knowledge into the training process which leads to the notion of informed machine learning. In this paper, we present a structured overview of various approaches in this field. We provide a definition and propose a concept for informed machine learning which illustrates its building blocks and distinguishes it from conventional machine learning. We introduce a taxonomy that serves as a classification framework for informed machine learning approaches. It considers the source of knowledge, its representation, and its integration into the machine learning pipeline. Based on this taxonomy, we survey related research and describe how different knowledge representations such as algebraic equations, logic rules, or simulation results can be used in learning systems. This evaluation of numerous papers on the basis of our taxonomy uncovers key methods in the field of informed machine learning.

stat.ML↗

arXiv2019

The recovery of ridge functions on the hypercube suffers from the curse of dimensionality

A multivariate ridge function is a function of the form $f(x) = g(a^{\scriptscriptstyle T} x)$, where $g$ is univariate and $a \in \mathbb{R}^d$. We show that the recovery of an unknown ridge function defined on the hypercube $[-1,1]^d$ with Lipschitz-regular profile $g$ suffers from the curse of dimensionality when the recovery error is measured in the $L_\infty$-norm, even if we allow randomized algorithms. If a limited number of components of $a$ is substantially larger than the others, then the curse of dimensionality is not present and the problem is weakly tractable provided the profile $g$ is sufficiently regular.

math.NA↗

arXiv2015

Counting via entropy: new preasymptotics for the approximation numbers of Sobolev embeddings

In this paper, we reveal a new connection between approximation numbers of periodic Sobolev type spaces, where the smoothness weights on the Fourier coefficients are induced by a (quasi-)norm $\|\cdot\|$ on $\mathbb{R}^d$, and entropy numbers of the embedding $\textrm{id}: \ell_{\|\cdot\|}^d \to \ell_\infty^d$. This connection yields preasymptotic error bounds for approximation numbers of isotropic Sobolev spaces, spaces of analytic functions, and spaces of Gevrey type in $L_2$ and $H^1$, which find application in the context of Galerkin methods. Moreover, we observe that approximation numbers of certain Gevrey type spaces behave preasymptotically almost identical to approximation numbers of spaces of dominating mixed smoothness. This observation can be exploited, for instance, for Galerkin schemes for the electronic Schr\"odinger equation, where mixed regularity is present.

math.NA↗

arXiv2015

Entropy numbers of spheres in Banach and quasi-Banach spaces

We prove sharp upper bounds on the entropy numbers $e_k(S^{d-1}_p,\ell_q^d)$ of the $p$-sphere in $\ell_q^d$ in the case $k \geq d$ and $0< p \leq q \leq \infty$. In particular, we close a gap left open in recent work of the second author, T. Ullrich and J. Vybiral. We also investigate generalizations to spheres of general finite-dimensional quasi-Banach spaces.

math.NA↗

arXiv2013

Entropy and sampling numbers of classes of ridge functions

We study properties of ridge functions $f(x)=g(a\cdot x)$ in high dimensions $d$ from the viewpoint of approximation theory. The considered function classes consist of ridge functions such that the profile $g$ is a member of a univariate Lipschitz class with smoothness $\alpha > 0$ (including infinite smoothness), and the ridge direction $a$ has $p$-norm $\|a\|_p \leq 1$. First, we investigate entropy numbers in order to quantify the compactness of these ridge function classes in $L_{\infty}$. We show that they are essentially as compact as the class of univariate Lipschitz functions. Second, we examine sampling numbers and face two extreme cases. In case $p=2$, sampling ridge functions on the Euclidean unit ball faces the curse of dimensionality. It is thus as difficult as sampling general multivariate Lipschitz functions, a result in sharp contrast to the result on entropy numbers. When we additionally assume that all feasible profiles have a first derivative uniformly bounded away from zero in the origin, then the complexity of sampling ridge functions reduces drastically to the complexity of sampling univariate Lipschitz functions. In between, the sampling problem's degree of difficulty varies, depending on the values of $\alpha$ and $p$. Surprisingly, we see almost the entire hierarchy of tractability levels as introduced in the recent monographs by Novak and Wo\'zniakowski.

math.NA↗

arXiv2004

Immediate and Virtual Basins of Newton's Method for Entire Functions

We investigate the well known Newton method to find roots of entire holomorphic functions. Our main result is that the immediate basin of attraction for every root is simply connected and unbounded. We also introduce ``virtual immediate basins'' in which the dynamics converges to infinity; we prove that these are simply connected as well.

math.DS↗