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121. Boom and bust cycles due to pseudospectra of matrices with unimodular spectra

Author

Bhat, Junaid Majeed, Bensa, Jaš, and Žnidarič, Marko

Subjects
Mathematical Physics, Condensed Matter - Statistical Mechanics, Condensed Matter - Strongly Correlated Electrons, Quantum Physics
Abstract

We discuss dynamics obtained by increasing powers of non-normal matrices that are roots of the identity, and therefore have all eigenvalues on the unit circle. Naively, one would expect that the expectation value of such powers cannot grow as one increases the power. We demonstrate that, rather counterintuitively, a completely opposite behavior is possible. In the limit of infinitely large matrices one can have an exponential growth. For finite matrices this exponential growth is a part of repeating cycles of exponential growths followed by exponential decays. The effect can occur if the spectrum is different than the pseudospectrum, with the exponential growth rate being given by the pseudospectrum. We show that this effect appears in a class of transfer matrices appearing in studies of two-dimensional non-interacting systems, for a matrix describing the Ehrenfest urn, as well as in previously observed purity dynamics in a staircase random circuit., Comment: 10 pages, 5 figures

Published
2024

122. Inevitability of Polarization in Geometric Opinion Exchange

Author

Alidou, Abdou Majeed, Baligács, Júlia, Hahn-Klimroth, Max, Hązła, Jan, Hintze, Lukas, and Scheftelowitsch, Olga

Subjects
Computer Science - Social and Information Networks, Economics - Theoretical Economics
Abstract

Polarization and unexpected correlations between opinions on diverse topics (including in politics, culture and consumer choices) are an object of sustained attention. However, numerous theoretical models do not seem to convincingly explain these phenomena. This paper is motivated by a recent line of work, studying models where polarization can be explained in terms of biased assimilation and geometric interplay between opinions on various topics. The agent opinions are represented as unit vectors on a multidimensional sphere and updated according to geometric rules. In contrast to previous work, we focus on the classical opinion exchange setting, where the agents update their opinions in discrete time steps, with a pair of agents interacting randomly at every step. The opinions are updated according to an update rule belonging to a general class. Our findings are twofold. First, polarization appears to be ubiquitous in the class of models we study, requiring only relatively modest assumptions reflecting biased assimilation. Second, there is a qualitative difference between two-dimensional dynamics on the one hand, and three or more dimensions on the other. Accordingly, we prove almost sure polarization for a large class of update rules in two dimensions. Then, we prove polarization in three and more dimensions in more limited cases and try to shed light on central difficulties that are absent in two dimensions.

Published
2024

123. CodeAid: Evaluating a Classroom Deployment of an LLM-based Programming Assistant that Balances Student and Educator Needs

Author

Kazemitabaar, Majeed, Ye, Runlong, Wang, Xiaoning, Henley, Austin Z., Denny, Paul, Craig, Michelle, and Grossman, Tovi

Subjects
Computer Science - Human-Computer Interaction, Computer Science - Artificial Intelligence
Abstract

Timely, personalized feedback is essential for students learning programming. LLM-powered tools like ChatGPT offer instant support, but reveal direct answers with code, which may hinder deep conceptual engagement. We developed CodeAid, an LLM-powered programming assistant delivering helpful, technically correct responses, without revealing code solutions. CodeAid answers conceptual questions, generates pseudo-code with line-by-line explanations, and annotates student's incorrect code with fix suggestions. We deployed CodeAid in a programming class of 700 students for a 12-week semester. A thematic analysis of 8,000 usages of CodeAid was performed, further enriched by weekly surveys, and 22 student interviews. We then interviewed eight programming educators to gain further insights. Our findings reveal four design considerations for future educational AI assistants: D1) exploiting AI's unique benefits; D2) simplifying query formulation while promoting cognitive engagement; D3) avoiding direct responses while encouraging motivated learning; and D4) maintaining transparency and control for students to asses and steer AI responses., Comment: CHI 2024 Paper - The paper includes 17 pages, 8 figures, 2 tables, along with a 2-page appendix

Published
2024
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