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Dissertation presented as the partial requirement for obtaining a Master's degree in Data Science and Advanced Analytics The present dissertation focuses on proposing and describing a collaborative filtering approach for Music Recommender Systems. Music Recommender Systems, which are part of a broader class of Recommender Systems, refer to the task of automatically filtering data to predict the songs that are more likely to match a particular profile. So far, academic researchers have proposed a variety of machine learning approaches for determining which tracks to recommend to users. The most sophisticated among them consist, often, on complex learning techniques which can also require considerable computational resources. However, recent research studies proved that more simplistic approaches based on nearest neighbors could lead to good results, often at much lower computational costs, representing a viable alternative solution to the Music Recommender System problem. Throughout this thesis, we conduct offline experiments on a freely-available collection of listening histories from real users, each one containing several different music tracks. We extract a subset of 10 000 songs to assess the performance of the proposed system, comparing it with a Popularity-based model approach. Furthermore, we provide a conceptual overview of the recommendation problem, describing the state-of-the-art methods, and presenting its current challenges. Finally, the last section is dedicated to summarizing the essential conclusions and presenting possible future improvements.

International audience; Discrete gradient vector fields are combinatorial structures that can be used for accelerating the homology computation of CW complexes, such as simplicial or cubical complexes, by reducing their number of cells. Consequently, they provide a bound for the Betti numbers (the most basic homological information). A discrete gradient vector field can eventually reduce the complex to its minimal form, having as many cells of each dimension as its corresponding Betti number, but this is not guaranteed. Moreover, finding an optimal discrete gradient vector field is an NP-hard problem. We describe here a generalization, which we call Homological Discrete Vector Field (HDVF), which can overcome these limitations by allowing cycles under a certain algebraic condition. In this work we define the HDVF and its associated reduction, we study how to efficiently compute a HDVF, we establish the relation between the HDVF and other concepts in computational homology and we estimate the average complexity of its computation. We also introduce five basic operations for modifying a HDVF, which can also be applied to discrete gradient vector fields.

In this work, a new method to determine and correct the linear drift for any crystalline orientation in a single-column-resolved high-resolution scanning transmission electron microscopy (HR-STEM) image, which is based on angle measurements in the Fourier space, is presented. This proposal supposes a generalization and the improvement of a previous work that needs the presence of two symmetrical planes in the crystalline orientation to be applicable. Now, a mathematical derivation of the drift effect on two families of asymmetric planes in the reciprocal space is inferred. However, though it was not possible to find an analytical solution for all conditions, a simple formula was derived to calculate the drift effect that is exact for three specific rotation angles. Taking this into account, an iterative algorithm based on successive rotation/drift correction steps is devised to remove drift distortions in HR-STEM images. The procedure has been evaluated using a simulated micrograph of a monoclinic material in an orientation where all the reciprocal lattice vectors are different. The algorithm only needs four iterations to resolve a 15° drift angle in the image. Ministerio de Economía y Competitividad MAT2016-77491-C2-2-R Ministerio de Ciencia, Innovación y Universidades TEC2017-86102-C2-2-R

Image processing is a fundamental operation in many real time applications, where lots of parallelism can be extracted. Segmenting the image into different connected components is the most known operations, but there are many others like extracting the region adjacency graph (RAG) of these regions, or searching for features points, being invariant to rotations, scales, brilliant changes, etc. Most of these algorithms part from the basis of Tracing-type approaches or scan/raster methods. This fact necessarily implies a data dependence between the processing of one pixel and the previous one, which prevents using a pure parallel approach. In terms of time complexity, this means that linear order O(N) (N being the number of pixels) cannot be cut down. In this paper, we describe a novel approach based on the building of a pure Topological framework, which allows to implement fully parallel algorithms. Concerning topological analysis, a first stage is computed in parallel for every pixel, thus conveying the local neighboring conditions. Then, they are extended in a second parallel stage to the necessary global relations (e.g. to join all the pixels of a connected component). This combinatorial optimization process can be seen as the compression of the whole image to just one pixel. Using this final representation, every region can be related with the rest, which yields to pure topological construction of other image operations. Besides, complex data structures can be avoided: all the processing can be done using matrixes (with the same indexation as the original image) and element-wise operations. The time complexity order of our topological approach for a m×n pixel image is near O(log(m+n)), under the assumption that a processing element exists for each pixel. Results for a multicore processor show very good scalability until the memory bandwidth bottleneck is reached, both for bigger images and for much optimized implementations. The inherent parallelism of our approach points to the direction that even better results will be obtained in other less classical computing architectures.

In this paper we design a new family of relations between (co)homology classes, working with coefficients in a field and starting from an AT-model (Algebraic Topological Model) AT(C) of a finite cell complex C These relations are induced by elementary relations of type “to be in the (co)boundary of” between cells. This high-order connectivity information is embedded into a graph-based representation model, called Second Order AT-Region-Incidence Graph (or AT-RIG) of C. This graph, having as nodes the different homology classes of C, is in turn, computed from two generalized abstract cell complexes, called primal and dual AT-segmentations of C. The respective cells of these two complexes are connected regions (set of cells) of the original cell complex C, which are specified by the integral operator of AT(C). In this work in progress, we successfully use this model (a) in experiments for discriminating topologically different 3D digital objects, having the same Euler characteristic and (b) in designing a parallel algorithm for computing potentially significant (co)homological information of 3D digital objects. Ministerio de Economía y Competitividad MTM2016-81030-P Ministerio de Economía y Competitividad TEC2012-37868-C04-02