Geometrical complexity of data approximators

There are many methods developed to approximate a cloud of vectors embedded in high-dimensional space by simpler objects: starting from principal points and linear manifolds to self-organizing maps, neural gas, elastic maps, various types of principal curves and principal trees, and so on. For each type of approximators the measure of the approximator complexity was developed too. These measures are necessary to find the balance between accuracy and complexity and to define the optimal approximations of a given type. We propose a measure of complexity (geometrical complexity) which is applicable to approximators of several types and which allows comparing data approximations of different types.Comment: 10 pages, 3 figures, minor correction and extensio

Driven by Compression Progress: A Simple Principle Explains Essential Aspects of Subjective Beauty, Novelty, Surprise, Interestingness, Attention, Curiosity, Creativity, Art, Science, Music, Jokes

I argue that data becomes temporarily interesting by itself to some self-improving, but computationally limited, subjective observer once he learns to predict or compress the data in a better way, thus making it subjectively simpler and more beautiful. Curiosity is the desire to create or discover more non-random, non-arbitrary, regular data that is novel and surprising not in the traditional sense of Boltzmann and Shannon but in the sense that it allows for compression progress because its regularity was not yet known. This drive maximizes interestingness, the first derivative of subjective beauty or compressibility, that is, the steepness of the learning curve. It motivates exploring infants, pure mathematicians, composers, artists, dancers, comedians, yourself, and (since 1990) artificial systems.Comment: 35 pages, 3 figures, based on KES 2008 keynote and ALT 2007 / DS 2007 joint invited lectur

Mobile learning: benefits of augmented reality in geometry teaching

As a consequence of the technological advances and the widespread use of mobile devices to access information and communication in the last decades, mobile learning has become a spontaneous learning model, providing a more flexible and collaborative technology-based learning. Thus, mobile technologies can create new opportunities for enhancing the pupils’ learning experiences. This paper presents the development of a game to assist teaching and learning, aiming to help students acquire knowledge in the field of geometry. The game was intended to develop the following competences in primary school learners (8-10 years): a better visualization of geometric objects on a plane and in space; understanding of the properties of geometric solids; and familiarization with the vocabulary of geometry. Findings show that by using the game, students have improved around 35% the hits of correct responses to the classification and differentiation between edge, vertex and face in 3D solids.This research was supported by the Arts and Humanities Research Council Design Star CDT (AH/L503770/1), the Portuguese Foundation for Science and Technology (FCT) projects LARSyS (UID/EEA/50009/2013) and CIAC-Research Centre for Arts and Communication.info:eu-repo/semantics/publishedVersio

Geometrical Insights for Implicit Generative Modeling

Learning algorithms for implicit generative models can optimize a variety of criteria that measure how the data distribution differs from the implicit model distribution, including the Wasserstein distance, the Energy distance, and the Maximum Mean Discrepancy criterion. A careful look at the geometries induced by these distances on the space of probability measures reveals interesting differences. In particular, we can establish surprising approximate global convergence guarantees for the $1$-Wasserstein distance,even when the parametric generator has a nonconvex parametrization.Comment: this version fixes a typo in a definitio

Web-based learning in the field of empirical research methods

This study focuses on the development of a complex web-based learning environment aimed at promoting the acquisition of applicable knowledge in the context of studying empirical research methods at university. This learning environment was then modified further on an empirical basis. The main focus of the present article is to describe the conceptualisation of the learning environment and research activities which were guided by an integrative research paradigm. The learning environment consisted of highly structured, complex texts in which the process of empirical research was illustrated in a detailed manner. By combining these texts with other instructional measures, the learning environment is given a flexible hypertext-structure. The effectiveness of the learning environment as a whole was investigated in three studies (two evaluation studies in the field and one experimental study in the laboratory). It was demonstrated that the additional instructional measures (e.g. a specific feedback-guidance and time-management measures) were not effective. The importance of cognitive, motivational and emotional learning prerequisites for the successful utilisation of the learning environment was highlighted. The implementation of special training and additional preparatory modules is recommended in order to optimise the fit between students' prerequisites and learning environmIm Zentrum der vorliegenden Arbeit steht zum einen die Konzeptualisierung einer Lernumgebung zur Förderung des Erwerbs anwendbaren Wissens im Kontext der universitären Ausbildung in empirischen Forschungsmethoden. Zum anderen werden ausgehend von einem integrativen Forschungsparadigma Forschungsaktivitäten beschrieben, die die empirische Basis zur Weiterentwicklung der Lernumgebung bereitstellen. Die Lernumgebung besteht aus hoch strukturierten, komplexen Texten, in welchen der Prozess empirischer Forschung auf detaillierte Weise veranschaulicht wird. Diese Texte wurden mit anderen instruktionalen Maßnahmen kombiniert, wodurch die Lernumgebung eine flexible, hypertextartige Struktur bekam. Die Effektivität der gesamten Lernumgebung wurde im Rahmen dreier empirischer Studien untersucht, von denen zwei als Evaluationsstudien im Feld durchgeführt wurden; die dritte war eine experimentelle Laborstudie. Es wurde gezeigt, dass die zusätzlichen instruktionalen Maßnahmen (z. B. eine spezifische Feedback-Anleitung und eine Zeitmanagement-Maßnahme) nicht wirksam waren. Die Bedeutung kognitiver, motivationaler und emotionaler Lernvoraussetzungen für die erfolgreiche Nutzung der Lernumgebung konnte nachgewiesen werden. Um die Passung zwischen den Eingangsvoraussetzungen der Studierenden und der Lernumgebung zu verbessern, wurde die Implementation eines speziellen Trainings und eines zusätzlichen vorbereitenden Moduls vorgeschlag