Understanding Generalization via Set Theory

Generalization is at the core of machine learning models. However, the definition of generalization is not entirely clear. We employ set theory to introduce the concepts of algorithms, hypotheses, and dataset generalization. We analyze the properties of dataset generalization and prove a theorem on surrogate generalization procedures. This theorem leads to our generalization method. Through a generalization experiment on the MNIST dataset, we obtain 13,541 sample bases. When we use the entire training set to evaluate the model's performance, the models achieve an accuracy of 99.945%. However, if we shift the sample bases or modify the neural network structure, the performance experiences a significant decline. We also identify consistently mispredicted samples and find that they are all challenging examples. The experiments substantiated the accuracy of the generalization definition and the effectiveness of the proposed methods. Both the set-theoretic deduction and the experiments help us better understand generalization.Comment: 14 page

Single-realization recovery of a random Schr\"odinger equation with unknown source and potential

In this paper, we study an inverse scattering problem associated with the stationary Schr\"odinger equation where both the potential and the source terms are unknown. The source term is assumed to be a generalised Gaussian random distribution of the microlocally isotropic type, whereas the potential function is assumed to be deterministic. The well-posedness of the forward scattering problem is first established in a proper sense. It is then proved that the rough strength of the random source can be uniquely recovered, independent of the unknown potential, by a single realisation of the passive scattering measurement. We develop novel techniques to completely remove a restrictive geometric condition in our earlier study [25], at an unobjectionable cost of requiring the unknown potential to be deterministic. The ergodicity is used to establish the single realization recovery, and the asymptotic arguments in our analysis are based on techniques from the theory of pseudo-differential operators and the stationary phase principle.Comment: 28 page

L'expérience de la fragilité et de l'instabilité de la relation interpersonnelle en performance

Mon travail de création à la maîtrise s'est principalement développé autour de la question des relations interpersonnelles dans une pratique de la performance. J'y interroge la fragilité et l'instabilité des relations que nous entretenons avec les autres par le biais de propositions artistiques m'amenant à partager des expériences de rencontre avec les spectateurs. Ce texte présente le processus de création du travail artistique en retraçant le chemin de mes inspirations et l'émergence d'une conception personnelle de la relation à l'autre. J'y explique aussi le choix de la performance comme moyen d'expression artistique. Ce texte est divisé en trois parties qui permettent d'illustrer mes réflexions sur mon travail. Plus précisément, je spécifierai pourquoi et comment je réalise mon projet artistique sur ce thème subtil de la relation