Spectral Generalized Multi-Dimensional Scaling

Multidimensional scaling (MDS) is a family of methods that embed a given set of points into a simple, usually flat, domain. The points are assumed to be sampled from some metric space, and the mapping attempts to preserve the distances between each pair of points in the set. Distances in the target space can be computed analytically in this setting. Generalized MDS is an extension that allows mapping one metric space into another, that is, multidimensional scaling into target spaces in which distances are evaluated numerically rather than analytically. Here, we propose an efficient approach for computing such mappings between surfaces based on their natural spectral decomposition, where the surfaces are treated as sampled metric-spaces. The resulting spectral-GMDS procedure enables efficient embedding by implicitly incorporating smoothness of the mapping into the problem, thereby substantially reducing the complexity involved in its solution while practically overcoming its non-convex nature. The method is compared to existing techniques that compute dense correspondence between shapes. Numerical experiments of the proposed method demonstrate its efficiency and accuracy compared to state-of-the-art approaches

On the optimality of shape and data representation in the spectral domain

A proof of the optimality of the eigenfunctions of the Laplace-Beltrami operator (LBO) in representing smooth functions on surfaces is provided and adapted to the field of applied shape and data analysis. It is based on the Courant-Fischer min-max principle adapted to our case. % The theorem we present supports the new trend in geometry processing of treating geometric structures by using their projection onto the leading eigenfunctions of the decomposition of the LBO. Utilisation of this result can be used for constructing numerically efficient algorithms to process shapes in their spectrum. We review a couple of applications as possible practical usage cases of the proposed optimality criteria. % We refer to a scale invariant metric, which is also invariant to bending of the manifold. This novel pseudo-metric allows constructing an LBO by which a scale invariant eigenspace on the surface is defined. We demonstrate the efficiency of an intermediate metric, defined as an interpolation between the scale invariant and the regular one, in representing geometric structures while capturing both coarse and fine details. Next, we review a numerical acceleration technique for classical scaling, a member of a family of flattening methods known as multidimensional scaling (MDS). There, the optimality is exploited to efficiently approximate all geodesic distances between pairs of points on a given surface, and thereby match and compare between almost isometric surfaces. Finally, we revisit the classical principal component analysis (PCA) definition by coupling its variational form with a Dirichlet energy on the data manifold. By pairing the PCA with the LBO we can handle cases that go beyond the scope defined by the observation set that is handled by regular PCA

Graph matching: relax or not?

We consider the problem of exact and inexact matching of weighted undirected graphs, in which a bijective correspondence is sought to minimize a quadratic weight disagreement. This computationally challenging problem is often relaxed as a convex quadratic program, in which the space of permutations is replaced by the space of doubly-stochastic matrices. However, the applicability of such a relaxation is poorly understood. We define a broad class of friendly graphs characterized by an easily verifiable spectral property. We prove that for friendly graphs, the convex relaxation is guaranteed to find the exact isomorphism or certify its inexistence. This result is further extended to approximately isomorphic graphs, for which we develop an explicit bound on the amount of weight disagreement under which the relaxation is guaranteed to find the globally optimal approximate isomorphism. We also show that in many cases, the graph matching problem can be further harmlessly relaxed to a convex quadratic program with only n separable linear equality constraints, which is substantially more efficient than the standard relaxation involving 2n equality and n^2 inequality constraints. Finally, we show that our results are still valid for unfriendly graphs if additional information in the form of seeds or attributes is allowed, with the latter satisfying an easy to verify spectral characteristic

A travessia facilitada

O simples ato de atravessar um obstáculo urbano, nas grandes cidades brasileiras, exige um esforço desnecessário e traz riscos para os cidadãos. As grandes avenidas, rios, linhas férreas e outras grandes fissuras urbanas são parcamente atravessadas por pontes desenhadas exclusivamente para o trânsito motorizado, sem levar em conta o pedestre ou o ciclista. As estruturas dominantes são desprovidas de valores estéticos e drenam a identidade das cidades, subtraindo a qualidade da paisagem. Esse texto apresenta uma proposta de requalificação desses equipamentos.The sole act of crossing an urban obstacle in large Brazilian cities requires unnecessary effort and risk for their citizens. Great avenues, rivers, railway lines and other urban cracks are barely served by bridges designed to serve motorized traffic without considering pedestrians and cyclists. Generic structures are devoid of aesthetic values and subtract the quality of the landscape from its viewers and users. This text leads to a proposal of the redesign and reuse of such equipment

Changes in Students’ Perceptions of Self-Assessment in Courses with Different Approaches to Assessment

The importance of student self-assessment and its contribution to learning in teacher education is well documented in the research literature. However, we still need to better understand when and why self-assessment actually works. This study examines preservice teachers’ perception of self-assessment prior to and following experiencing self-assessment. The study included 135 students studying at two education colleges in Israel. The students attended courses with differing evaluation approaches. The findings show that the experience with self-assessment in the courses with formative evaluation or integrative evaluation encourages the students’ positive perception of self-assessment, in contrast to summative evaluation courses. The study expands our understanding of the importance of student involvement in the evaluation processes, as well as the role of the feedback in the process. These two factors had the greatest impact on the students’ perceptions, as well as on the accuracy of their self-grading

Some notes on fishes in British East Africa and Uganda

Volume: I

Learning Approach and Learning: Exploring a New Technological Learning System

This study furthers the understanding of the connections between learning approaches and learning. The research population embraced 44 males from the Jewish ultraorthodox community, who abide by distinct methods of study. One group follows the very didactic, linear and structured approach of a methodical and gradual order, while the second group follows the multi-directional approach that emphasizes global, abstract thought. The participants, who for ideological reasons hardly use computer technology, were exposed to a new technological learning system. The study employed the qualitative research method, with the research tools including textual analysis, observations and guided in-depth interviews. The findings show that those following the multi-directional method handled the device better from the didactic perspective. The question of how learning and teaching paradigms influence individual study is discussed

Possible Origins of the Complex Topographic Organization of Motor Cortex: Reduction of a Multidimensional Space onto a Two-Dimensional Array

We propose that some of the features of the topographic organization in motor cortex emerge from a competition among several conflicting mapping requisites. These competing requisites include a somatotopic map of the body, a map of hand location in space, and a partitioning of cortex into regions that emphasize different complex, ethologically relevant movements. No one type of map fully explains the topography; instead, all three influences (and perhaps others untested here) interact to form the topography. A standard algorithm (Kohonen network) was used to generate an artificial motor cortex array that optimized local continuity for these conflicting mapping requisites. The resultant hybrid map contained many features seen in actual motor cortex, including the following: a rough, overlapping somatotopy; a posterior strip in which simpler movements were represented and more somatotopic segregation was observed, and an anterior strip in which more complex, multisegmental movements were represented and the somatotopy was less segregated; a clustering of different complex, multisegmental movements into specific subregions of cortex that resembled the arrangement of subregions found in the monkey; three hand representations arranged on the cortex in a manner similar to the primary motor, dorsal premotor, and ventral premotor hand areas in the monkey; and maps of hand location that approximately matched the maps observed in the monkey