TY - JOUR
AB - We present a novel algorithm that allows us to gain detailed insight into the effects of sparsity in linear and nonlinear optimization, which is of great importance in many scientific areas such as image and signal processing, medical imaging, compressed sensing, and machine learning (e.g., for the training of neural networks). Sparsity is an important feature to ensure robustness against noisy data, but also to find models that are interpretable and easy to analyze due to the small number of relevant terms. It is common practice to enforce sparsity by adding the ℓ1-norm as a weighted penalty term. In order to gain a better understanding and to allow for an informed model selection, we directly solve the corresponding multiobjective optimization problem (MOP) that arises when we minimize the main objective and the ℓ1-norm simultaneously. As this MOP is in general non-convex for nonlinear objectives, the weighting method will fail to provide all optimal compromises. To avoid this issue, we present a continuation method which is specifically tailored to MOPs with two objective functions one of which is the ℓ1-norm. Our method can be seen as a generalization of well-known homotopy methods for linear regression problems to the nonlinear case. Several numerical examples - including neural network training - demonstrate our theoretical findings and the additional insight that can be gained by this multiobjective approach.
AU - Bieker, Katharina
AU - Gebken, Bennet
AU - Peitz, Sebastian
ID - 20731
IS - 11
JF - IEEE Transactions on Pattern Analysis and Machine Intelligence
TI - On the Treatment of Optimization Problems with L1 Penalty Terms via Multiobjective Continuation
VL - 44
ER -
TY - JOUR
AB - AbstractThe kinetic Brownian motion on the sphere bundle of a Riemannian manifold $$\mathbb {M}$$
M
is a stochastic process that models a random perturbation of the geodesic flow. If $$\mathbb {M}$$
M
is an orientable compact constantly curved surface, we show that in the limit of infinitely large perturbation the $$L^2$$
L
2
-spectrum of the infinitesimal generator of a time-rescaled version of the process converges to the Laplace spectrum of the base manifold.
AU - Kolb, Martin
AU - Weich, Tobias
AU - Wolf, Lasse Lennart
ID - 31193
IS - 4
JF - Annales Henri Poincaré
KW - Mathematical Physics
KW - Nuclear and High Energy Physics
KW - Statistical and Nonlinear Physics
SN - 1424-0637
TI - Spectral Asymptotics for Kinetic Brownian Motion on Surfaces of Constant Curvature
VL - 23
ER -
TY - JOUR
AU - Guedes Bonthonneau, Yannick
AU - Weich, Tobias
ID - 35306
IS - 3
JF - Journal of the European Mathematical Society
KW - Applied Mathematics
KW - General Mathematics
SN - 1435-9855
TI - Ruelle–Pollicott resonances for manifolds with hyperbolic cusps
VL - 24
ER -
TY - JOUR
AU - Hesse, Kerstin
AU - Le Gia, Quoc Thong
ID - 34633
JF - Journal of Computational and Applied Mathematics
KW - Applied Mathematics
KW - Computational Mathematics
SN - 0377-0427
TI - L_2 error estimates for polynomial discrete penalized least-squares approximation on the sphere from noisy data
VL - 408
ER -
TY - JOUR
AU - Hanusch, Maximilian
ID - 34817
IS - 1
JF - Communications in Analysis and Geometry
KW - regularity of Lie groups
SN - 1019-8385
TI - Regularity of Lie groups
VL - 30
ER -
TY - GEN
AU - Hanusch, Maximilian
ID - 34856
TI - Analysis 1 und 2 Skript/Buch
ER -
TY - JOUR
AU - Kolb, Martin
AU - Klump, Alexander
ID - 35644
IS - 4
JF - Theory of Probability and its Applications
TI - Uniqueness of the Inverse First Passage Time Problem and the Shape of the Shiryaev boundary
VL - 67
ER -
TY - JOUR
AB - Motivated by the work [6] of Mariusz Bieniek, Krzysztof Burdzy and Soumik Pal we study a Fleming-Viot-type particle system consisting of independently moving particles each driven by generalized Bessel processes on the positive real line. Upon hitting the boundary {0} this particle is killed and an uniformly chosen different one branches into two particles. Using the symmetry of the model and the self similarity property of Bessel processes, we obtain a criterion to decide whether the particles converge to the origin at a finite time. This addresses open problem 1.4 in [6]. Specifically, inspired by [6, Open Problem 1.5], we investigate the case of three moving particles and refine the general result of [6, Theorem 1.1(ii)] extending the regime of drift parameters, where convergence does not occur – even to values, where it does occur when considering the case of only two particles.
AU - Kolb, Martin
AU - Liesenfeld, Matthias
ID - 35649
IS - 27
JF - Electronic Journal of Probability
TI - On non-extinction in a Fleming-Viot-type particle model with Bessel drift
ER -
TY - JOUR
AB - We consider autoregressive sequences Xn = aXn−1 + ξn and
Mn = max{aMn−1 , ξn} with a constant a ∈ (0, 1) and with positive, in-
dependent and identically distributed innovations {ξk }. It is known that if
P(ξ1 > x) ∼ d
log x with some d ∈ (0, − log a) then the chains {Xn} and {Mn}
are null recurrent. We investigate the tail behaviour of recurrence times in this
case of logarithmically decaying tails. More precisely, we show that the tails
of recurrence times are regularly varying of index −1 − d/ log a. We also prove
limit theorems for {Xn} and {Mn} conditioned to stay over a fixed level x0.
Furthermore, we study tail asymptotics for recurrence times of {Xn} and {Mn}
in the case when these chains are positive recurrent and the tail of log ξ1 is
subexponential.
AU - Denisov, Denis
AU - Hinrichs, Günter
AU - Kolb, Martin
AU - Wachtel, Vitali
ID - 35650
JF - Electronic Journal of Probability
TI - Persistence of autoregressive sequences with logarithmic tails
VL - 27
ER -
TY - JOUR
AU - Liebendörfer, Michael
AU - Göller, Robin
AU - Gildehaus, Lara
AU - Kortemeyer, Jörg
AU - Biehler, Rolf
AU - Hochmuth, Reinhard
AU - Ostsieker, Laura
AU - Rode, Jana
AU - Schaper, Niclas
ID - 35685
IS - 5
JF - International Journal of Mathematical Education in Science and Technology
KW - Applied Mathematics
KW - Education
KW - Mathematics (miscellaneous)
SN - 0020-739X
TI - The role of learning strategies for performance in mathematics courses for engineers
VL - 53
ER -
TY - JOUR
AU - Biehler, Rolf
AU - Langlotz, Hubert
AU - Zappe, Wilfried
ID - 35718
IS - 232
JF - mathematik lehren
TI - Alles normal?! Daten auf Normalverteilung prüfen–mit schulischen Mitteln
VL - 2022
ER -
TY - CHAP
AB - The data age has led to curriculum developments promoting the relevance of stochastics in mathematics education. This has resulted in an increased demand for teacher professional development (PD) courses, and thus in a rising number of facilitators for this purpose. Their expertise, however, remains under-researched, even though facilitators are the crucial factor in PD success. Our study aims at revealing how one aspect of facilitator expertise, orientation, is connected to other aspects: views on content goals, perception of possible challenges, and attitude towards teaching resources. The explorations reveal differing orientations and learning goals with identical teaching resources for two facilitators who had also been working as a team for the re-design of the PD course for years.
AU - Nieszporek, Ralf
AU - Griese, Birgit
AU - Biehler, Rolf
ED - Peters, S. A.
ED - Zapata-Cardona, L.
ED - Bonafini, F.
ED - Fan, A.
ID - 35696
T2 - Bridging the Gap: Empowering and Educating Today’s Learners in Statistics. Proceedings of the Eleventh International Conference on Teaching Statistics
TI - Facilitators’ Orientations Towards Learning Goals, Perceived Challenges, and Teaching Resources for a PD Course on Conditional Probability
ER -
TY - CONF
AB - Fundamental ideas (Burrill & Biehler, 2011) have been specified to structure the curriculum across different age levels and to focus on and revisit the most important ideas. Such ideas must be based on an epistemological analysis of the scientific domain of statistics and its applications. Because these domains are changing, fundamental ideas have to be revisited from time to time. In particular, data science and machine learning have led to new methods and applications in society that must be considered for updating fundamental ideas.
AU - Biehler, Rolf
ED - Peters, S. A.
ED - Zapata-Cardona, L.
ED - Bonafini, F.
ED - Fan, A.
ID - 34919
T2 - Bridging the Gap: Empowering and Educating Today’s Learners in Statistics. Proceedings of the Eleventh International Conference on Teaching Statistics
TI - Revisiting Fundamental Ideas for Statistics Education From the Perspective of Machine Learning and Its Applications
ER -
TY - JOUR
AB - A very warm welcome to this Special Issue of the Statistics Education Research Journal (SERJ) on data science education. Our hope is to give an overview of selected theoretical thoughts and empirical studies on data science education from a statistics education research perspective. Data science education is rapidly developing but research into data science education is still in its infancy. The current issue presents a snapshot of this developing field.
AU - Biehler, Rolf
AU - De Veaux, Richard
AU - Engel, Joachim
AU - Kazak, Sibel
AU - Frischemeier, Daniel
ID - 34920
IS - 2
JF - Statistics Education Research Journal
KW - Education
KW - Statistics and Probability
SN - 1570-1824
TI - Editorial: Research on Data Science Education
VL - 21
ER -
TY - JOUR
AU - Biehler, Rolf
AU - Griese, Birgit
ID - 35926
IS - 232
JF - mathematik lehren
TI - Modellieren im Stochastikunterricht: Annahmen hinterfragen, Ergebnisse validieren
VL - 2022
ER -
TY - CONF
AU - Hoffmann, Max
AU - Biehler, Rolf
ED - Halverscheid, Stefan
ED - Kersten, Ina
ED - Schmidt-Thieme, Barbara
ID - 31367
T2 - Bedarfsgerechte fachmathematische Lehramtsausbildung. Analyse, Zielsetzungen und Konzepte unter heterogenen Voraussetzungen
TI - Schnittstellenaufgaben in der Analysis I zur Verknüpfung von Schul- und Hochschulmathematik - Aufgabenbeispiele und Ergebnisse einer Evaluationsstudie.
ER -
TY - CONF
AU - Biehler, Rolf
AU - Hoffmann, Max
ED - Isaev, Viktor
ED - Eichler, Andreas
ED - Loose, Frank
ID - 31365
SN - 2197-8751
T2 - Professionsorientierte Fachwissenschaft. Kohärenzstiftende Lerngelegenheiten für das Lehramtsstudium Mathematik
TI - Fachwissen als Grundlage fachdidaktischer Urteilskompetenz – Beispiele für die Herstellung konzeptueller Bezüge zwischen fachwissenschaftlicher und fachdidaktischer Lehre im gymnasialen Lehramtsstudium
ER -
TY - DATA
AU - Schürmann, Mirko
AU - Büdenbender-Kuklinski, C.
AU - Lankeit, Elisa
AU - Liebendörfer, Michael
AU - Hochmuth, R.
AU - Biehler, Rolf
AU - Schaper, N.
ID - 35698
TI - Dokumentation der Erhebungsinstrumente des Projekts WiGeMath
ER -
TY - JOUR
AB - AbstractWhen the COVID-19 pandemic began, many universities switched to fully online teaching. This unexpected switching to online teaching was challenging for both teachers and students, and restrictions that were put in place because of pandemic made this challenge even greater. However, new ways of teaching might also open new opportunities for students’ learning. The research question driving our study was as follows: how do students regulate their learning and specifically their choice of resources and peer learning in university mathematics classes that are fully taught online as offered during the COVID-19 pandemic? We report on a longitudinal, qualitative study in which students recorded a brief audio diary twice a week over one whole semester (14 weeks). We focused on three students who completed 70 interviews in total and finished the semester with varying degrees of success. The results show how the students structured their studying (e.g., the roles that deadlines or synchronous teaching events played). They illustrate the strengths and limitations of digital materials provided by the lecturer and the use of complementary media. Further, the pandemic uncovered the double-edged role of simple, often anonymous exchanges (e.g., via Discord servers), with few binding forces for either side, and the significance of stable learning partnerships for students’ success. Our research highlights aspects that should be focal points when comparing traditional instruction and online instruction during the pandemic from a self-regulatory perspective. Practical implications refer to how these aspects can be combined sensibly in fully online courses, but also in blended learning contexts.
AU - Liebendörfer, Michael
AU - Kempen, Leander
AU - Schukajlow, Stanislaw
ID - 37470
JF - ZDM – Mathematics Education
KW - General Mathematics
KW - Education
SN - 1863-9690
TI - First-year university students' self-regulated learning during the COVID-19 pandemic: a qualitative longitudinal study
ER -
TY - JOUR
AB - AbstractAs earlier research results suggest that many mathematics teaching students criticize a missing relevance in their studies, we explore explanations and interrelationships of their relevance assessments. We aim at finding out how one could support the students in attributing relevance to their study programs. A two-fold model for relevance assessments in mathematics teacher education is proposed, consisting of relevance content and relevance reasons. We investigate students' relevance perceptions of mathematical topics and of topics’ complexities, as well as their rating of individual and societal/ vocational relevance reasons, all in relation to their perception of the relevance of their overall program of study. Contrary to earlier research findings, our results suggest that mathematics teaching students already do attribute relevance to many content areas and that a preparation for the teaching profession is not the only reason for them to assign relevance. There also seem to be many students who would attribute relevance if they could develop as individuals and pursue their interests. We suggest that giving students opportunities to set individual priorities in their studies could hence support their relevance assessments. As low relevance assessments seem to be connected to students’ motivational problems, students might profit from motivational support, as well.
AU - Büdenbender-Kuklinski, Christiane
AU - Hochmuth, Reinhard
AU - Liebendörfer, Michael
ID - 37472
JF - International Journal of Research in Undergraduate Mathematics Education
KW - Education
KW - Mathematics (miscellaneous)
SN - 2198-9745
TI - Exploring the Perceived Relevance of University Mathematics Studies by First-Semester Teaching Students
ER -