A recent proof of NP-hardness of Euclidean sum-of-squares clustering, due to Drineas et al.

Finding modules, or clusters, in networks currently attracts much attention in several domains. The most studied criterion for doing so, due to Newman and Girvan [Phys. Rev. E 69, 026113 (2004)], is modularity maximization. Many heuristics have been proposed for maximizing modularity and yield rapidly near optimal solution or sometimes optimal ones but without a guarantee of optimality. There are few exact algorithms, prominent among which is a paper by Xu [Eur. Phys. J. B 60, 231 (2007)]. Modularity maximization can also be expressed as a clique partitioning problem and the row generation algorithm of Grötschel and Wakabayashi [Math. Program. 45, 59 (1989)] applied. We propose to extend both of these algorithms using the powerful column generation methods for linear and non linear integer programming. Performance of the four resulting algorithms is compared on problems from the literature. Instances with up to 512 entities are solved exactly. Moreover, the computing time of previously solved problems are reduced substantially.

This article reports the evolution of the literature on Dynamic Data Envelopment Analysis (DDEA) models from 1996 to 2016. Systematic searches in the databases Scopus and Web of Science were performed to outline the state of the art. The results enabled the establishment of DDEA studies as the scope of this article, analyzing the transition elements to represent temporal interdependence. The categorization of these studies enabled the mapping of the evolution of the DDEA literature and identification of the relationships between models. The three most widely adopted studies to conduct DDEA research were classified as structuring models. Mapping elucidated the literature behavior through three phases and showed an increase in publications with applications in recent years. The analysis of applications indicated that most studies address evaluations in the agriculture and farming, banking and energy sectors and consider the facilities as transition elements between analysis periods.

Given a set of entities associated with points in Euclidean space, minimum sum-of-squares clustering (MSSC) consist in partitioning this set into clusters such that the sum of squared distances from each point to the centroid of its cluster is minimized. A column generation algorithm for MSSC was given in du Merle et al. [15]. The bottleneck of that algorithm is resolution of the auxiliary problem of finding a column with negative reduced cost. We propose a new way to solve this auxiliary problem based on geometric arguments. This greatly improves the efficiency of the whole algorithm and leads to exact solution of instances with over 2300 entities, i.e., more than 10 times as much as previously done.

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In order to process efficiently ever-higher dimensional data such as images, sentences, or audio recordings, one needs to find a proper way to reduce the dimensionality of such data. In this regard, SVD-based methods including PCA and Isomap have been extensively used. Recently, a neural network alternative called autoencoder has been proposed and is often preferred for its higher flexibility. This work aims to show that PCA is still a relevant technique for dimensionality reduction in the context of classification. To this purpose, we evaluated the performance of PCA compared to Isomap, a deep autoencoder, and a variational autoencoder. Experiments were conducted on three commonly used image datasets: MNIST, Fashion-MNIST, and CIFAR-10. The four different dimensionality reduction techniques were separately employed on each dataset to project data into a low-dimensional space. Then a k-NN classifier was trained on each projection with a cross-validated random search over the number of neighbours. Interestingly, our experiments revealed that k-NN achieved comparable accuracy on PCA and both autoencoders' projections provided a big enough dimension. However, PCA computation time was two orders of magnitude faster than its neural network counterparts.