We consider the logarithmic Fekete problem, which consists of placing a fixed number of points on the unit sphere in \(\mathbb {R}^d\), in such a way that the product of all pairs of mutual Euclidean distances is maximized or, equivalently, so that their logarithmic energy is minimized. Using tools from Computational Algebraic Geometry, we find and classify all critical configurations for this problem when considering at most six points in every dimension d. In particular, our approach gives new proofs of several key results appearing in the literature, with the benefit of using a unified approach.

By February 2021, Uruguay was experiencing the first wave of the COVID-19 pandemic, while many countries were already suffering the second wave. Several countries took various measures to prevent the saturation of the health system, ranging from closure of restaurants and suspension of classes to nighttime traffic restrictions. In this paper, we explore the effect of mobility restriction measures on the infection incidence in countries that are in some way similar to Uruguay: they have between one and twelve million inhabitants, a reasonable testing effort and they had the epidemic under control at some point. For these countries, we study mobility indexes provided by Google, an index on governmental measures compiled by the University of Oxford, and the daily new cases per 100,000 inhabitants. First, we observed that the mobility reported by Google is directly related to government measures: the higher the level of restrictive measures, the lower the mobility index. Then, we analyze the influence of mobility reduction on the growth/decrease speed of the 7-day average of new cases per 100,000 inhabitants (P7) and show that high levels of mobility reduction lead to a decrease in the index. Finally, we related the required duration of mobility restrictions with the P7 maximum and also point out the risk of lifting the measures too early.

The use of different evaluation measures for classification tasks have gained a significant amount of attention in the past decade, specially for those problems with multiple and imbalanced classes [1], [2]. However, the optimization of classifiers with respect to these measures is still heuristic, using ad-hoc rules with classical accuracy-optimized classifiers. We propose a classifier designed specifically to optimize one of the possible measures, namely, the so-called G-mean. Nevertheless, the technique is general, and it can be used to optimize generic evaluation measures. The optimization algorithm to train the classifier is described, and the numerical scheme is tested showing its usability and robustness. The code is publicly available, as well as the datasets used along this paper.

Graph matching-aligning a pair of graphs to minimize their edge disagreements-has received wide-spread attention from both theoretical and applied communities over the past several decades, including combinatorics, computer vision, and connectomics. Its attention can be partially attributed to its computational difficulty. Although many heuristics have previously been proposed in the literature to approximately solve graph matching, very few have any theoretical support for their performance. A common technique is to relax the discrete problem to a continuous problem, therefore enabling practitioners to bring gradient-descent-type algorithms to bear. We prove that an indefinite relaxation (when solved exactly) almost always discovers the optimal permutation, while a common convex relaxation almost always fails to discover the optimal permutation. These theoretical results suggest that initializing the indefinite algorithm with the convex optimum might yield improved practical performance. Indeed, experimental results illuminate and corroborate these theoretical findings, demonstrating that excellent results are achieved in both benchmark and real data problems by amalgamating the two approaches.

Early detection and treatment of psychiatric disorders on children has shown significant impact in their subsequent development and quality of life. The assessment of psychopathology in childhood is commonly carried out by performing long comprehensive interviews such as the widely used Preschool Age Psychiatric Assessment (PAPA). Unfortunately, the time required to complete a full interview is too long to apply it at the scale of the actual population at risk, and most of the population goes undiagnosed or is diagnosed significantly later than desired. In this work, we aim to learn from unique and very rich previously collected PAPA examples the inter-correlations between different questions in order to provide a reliable risk analysis in the form of a much shorter interview. This helps to put such important risk analysis at the hands of regular practitioners, including teachers and family doctors. We use for this purpose the alternating decision trees algorithm, which combines decision trees with boosting to produce small and interpretable decision rules. Rather than a binary prediction, the algorithm provides a measure of confidence in the classification outcome. This is highly desirable from a clinical perspective, where it is preferable to abstain a decision on the low-confidence cases and recommend further screening. In order to prevent over-fitting, we propose to use network inference analysis to predefine a set of candidate question with consistent high correlation with the diagnosis. We report encouraging results with high levels of prediction using two independently collected datasets. The length and accuracy of the developed method suggests that it could be a valuable tool for preliminary evaluation in everyday care.

We present a computer-aided detection pipeline for polyp detection in Computer tomographic colonography. The first stage of the pipeline consists of a simple colon segmentation technique that enhances polyps, which is followed by an adaptive-scale candidate polyp delineation, in order to capture the appropriate polyp size. In the last step, candidates are classified based on new texture and geometric features that consider both the information in the candidate polyp location and its immediate surrounding area. The system is tested with ground truth data, including flat and small polyps which are hard to detect even with optical colonoscopy. We achieve 100% sensitivity for polyps larger than 6 mm in size with just 0.9 false positives per case, and 93% sensitivity with 2.8 false positives per case for polyps larger than 3 mm in size.

Graph matching is a challenging problem with very important applications in a wide range of fields, from image and video analysis to biological and biomedical problems. We propose a robust graph matching algorithm inspired in sparsity-related techniques. We cast the problem, resembling group or collaborative sparsity formulations, as a non-smooth convex optimization problem that can be efficiently solved using augmented Lagrangian techniques. The method can deal with weighted or unweighted graphs, as well as multimodal data, where different graphs represent different types of data. The proposed approach is also naturally integrated with collaborative graph inference techniques, solving general network inference problems where the observed variables, possibly coming from different modalities, are not in correspondence. The algorithm is tested and compared with state-of-the-art graph matching techniques in both synthetic and real graphs. We also present results on multimodal graphs and applications to collaborative inference of brain connectivity from alignment-free functional magnetic resonance imaging (fMRI) data.

Graphical models are a very useful tool to describe and understand natural phenomena, from gene expression to climate change and social interactions. The topological structure of these graphs/networks is a fundamental part of the analysis, and in many cases the main goal of the study. However, little work has been done on incorporating prior topological knowledge onto the estimation of the underlying graphical models from sample data. In this work we propose extensions to the basic joint regression model for network estimation, which explicitly incorporate graph-topological constraints into the corresponding optimization approach. The first proposed extension includes an eigenvector centrality constraint, thereby promoting this important prior topological property. The second developed extension promotes the formation of certain motifs, triangle-shaped ones in particular, which are known to exist for example in genetic regulatory networks. The presentation of the underlying formulations, which serve as examples of the introduction of topological constraints in network estimation, is complemented with examples in diverse datasets demonstrating the importance of incorporating such critical prior knowledge.