Riemannian Sparse Coding for Positive Definite Matrices

International audienceInspired by the great success of sparse coding for vector valued data, our goal is to represent symmetric positive definite (SPD) data matrices as sparse linear combinations of atoms from a dictionary, where each atom itself is an SPD matrix. Since SPD matrices follow a non-Euclidean (in fact a Riemannian) geometry, existing sparse coding techniques for Euclidean data cannot be directly extended. Prior works have approached this problem by defining a sparse coding loss function using either extrinsic similarity measures (such as the log-Euclidean distance) or kernelized variants of statistical measures (such as the Stein divergence, Jeffrey's divergence, etc.). In contrast, we propose to use the intrinsic Riemannian distance on the manifold of SPD matrices. Our main contribution is a novel mathematical model for sparse coding of SPD matrices; we also present a computationally simple algorithm for optimizing our model. Experiments on several computer vision datasets showcase superior classification and retrieval performance compared with state-of-the-art approaches

Filter-based DIRECT method for constrained global optimization

This paper presents a DIRECT-type method that uses a filter methodology to assure convergence to a feasible and optimal solution of nonsmooth and nonconvex constrained global optimization problems. The filter methodology aims to give priority to the selection of hyperrectangles with feasible center points, followed by those with infeasible and non-dominated center points and finally by those that have infeasible and dominated center points. The convergence properties of the algorithm are analyzed. Preliminary numerical experiments show that the proposed filter-based DIRECT algorithm gives competitive results when compared with other DIRECT-type methods.The authors would like to thank two anonymous referees and the Associate Editor for their valuable comments and suggestions to improve the paper. This work has been supported by COMPETE: POCI-01-0145-FEDER-007043 and FCT - Fundac¸ao para a Ciência e Tecnologia within the projects UID/CEC/00319/2013 and ˆ UID/MAT/00013/2013.info:eu-repo/semantics/publishedVersio

Investigation of eighth-grade students' understanding of the slope of the linear function

This study aimed to investigate eighth-grade students' difficulties and misconceptions and their performance of translation between the different representation modes related to the slope of linear functions. The participants were 115 Turkish eighth-grade students in a city in the eastern part of the Black Sea region of Turkey. Data was collected with an instrument consisting of seven written questions and a semi-structured interview protocol conducted with six students. Students' responses to questions were categorized and scored. Quantitative data was analyzed using the SPSS 17.0 statistical packet program with cross tables and one-way ANOVA. Qualitative data obtained from interviews was analyzed using descriptive analytical techniques. It was found that students' performance in articulating the slope of the linear function using its algebraic representation form was higher than their performance in using transformation between graphical and algebraic representation forms. It was also determined that some of them had difficulties and misunderstood linear function equations, graphs, and slopes and could not comprehend the connection between slope and the x- and y-intercepts

Dificultades para codificar, relacionar y categorizar problemas verbales algebraicos: dos estudios con estudiantes de secundaria y profesores en formación

En resolución de problemas verbales por transferencia, la activación de problemas ya conocidos que sirvan de guía, depende de las analogías percibidas entre éstos y el problema a resolver. Se desarrollan dos estudios relacionados para analizar en qué características se basan los estudiantes para codificar problemas y detectar sus analogías, en tareas de categorización (sorting). Se utilizaron técnicas cuantitativas y cualitativas combinadas. Primero se analizó cómo los estudiantes de secundaria son influidos por diferentes variables características de problemas de ciencias. Una gran proporción de sujetos no fue capaz de percibir las analogías y diferencias adecuadas entre problemas. El segundo estudio trató de avanzar una explicación de estos resultados. El nivel académico y la familiaridad con las temáticas fueron factores significativos, pero los futuros profesores participantes mostraron demasiadas dificultades, alertando sobre la conveniencia de revisar algunos supuestos instruccionales habituales

An investigation of the purposes of the measurement and assessment practice of primary school teachers within the context of the new mathematics curriculum [Sinif ö?retmenlerinin ölçme-de?erlendirme uygulama amaçlarinin yeni matematik ö?retimi programi kapsaminda incelenmesi]

The aim of this study is to examine the purpose of conducting the measurement and assessment practice of 4th and 5th grade teachers within the context of 2005 Primary School Mathematics Curriculum. To this aim, online questionnaire was administered to 512 fourth and fifth grade teachers who were randomly selected from 7 geographical regions and 15 cities in Turkey. Additionally, a case study was conducted with eight of them selected from Trabzon through interviews, observations and using document analysis methods. The collected data were analyzed using SPSS 15.0 and NVivo 8.0 programmes. The findings obtained from the questionnaire showed that elementary teachers conduct in a stable manner with the formative and summative assessment practice. However, findings obtained from case study revealed that there are some inconsistencies between the statements and classroom applications of elementary teachers, and that diagnostic and formative assessment practice, which were prescribed in 2005 mathematics curriculum, are not effectually conducted by them

Comparing the probability-related misconceptions of pupils at different education levels [Usporedba pogrješnih predod?aba o konceptima vjerojatnosti kod u?enika razli?itog stupnja obrazovanja]

The aim of the paper is to compare and evaluate the probability-related misconceptions of pupils at different education levels. A cross-sectional/age study was thus conducted with 540 pupils in 5th-8th grades. An instrument, comprising six questions on the concepts of compound events, probability of an event and probability comparisons, was used. Data were analyzed using descriptive statistical techniques (SPSS 15.0) and the chi-square test. This study showed that the percentage of correct answers increased when the level of education increased, whereas the misconceptions about the concept of compound events I decreased; also, the percentage of correct answers decreased and the misconceptions about the concept of compound events II increased. In the questions related to the concept of probability of an event and probability comparisons, as the level of education increased both the percentage of correct answers and the misconceptions increased. Therefore, it can be suggested that pupils' misconceptions regarding probability concepts vary, which depends on the characteristics of such concepts and pupils' age and education level

The effect of computer-assisted instruction on 7th grade students’ achievement and attitudes toward mathematics: The case of the topic “vertical circular cylinder” [Utjecaj nastave uz pomo? ra?unala na postignu?a i stavove u?enika sedmog razreda prema matematici: Slu?aj obrade teme „Uspravni kru?ni valjak”]

The purpose of this study was to investigate the effect of computer-assisted instruction (CAI) regarding concepts of “surface area and volume of vertical circular cylinder” on seventh grade students’ achievement and attitudes toward mathematics. A pre-test and post-test control group with quasi-experimental study design was used. This research was carried out with 49 seventh grade Turkish students (24 students in the experimental group, and 25 students in the control group). The students in the experimental group were taught with CAI, while the students in the control group were taught in the traditional way (textbookbased direct instruction). The “Mathematics Achievement Test (MAT)” and the “Mathematics Attitude Scale (MAS)” were administered to both groups as a pretest and a post-test. The data were analyzed with multivariate analysis of variance (MANCOVA). The results indicated that CAI significantly improved students’ mathematics achievement compared to the results achieved through traditional instruction. However, it was found that short-term CAI implementation did not make a significant difference on students’ attitudes toward mathematics. © 2015 FACTEACHEREDUCATION. All rights reserved