The Netflix problem (from machine learning) asks the following. Given a ratings matrix in which each entry $(i,j)$ represents the rating of movie $j$ by customer $i$, if customer $i$ has watched movie $j$, and is otherwise missing, we would like to predict the remaining entries in order to make good recommendations to customers on what to watch next. The remaining entries are predicted so as to minimize the {\it rank} of the completed matrix. In this survey we study a more general problem, in which instead of knowing specific matrix elements, we know linear relations on such elements. We describe applications of these results to embeddings of graphs in surfaces (more precisely, embeddings with rotation systems, and embeddings modulo 2).

When designing a public transportation network in a country, one may want to minimise the sum of travel duration of all inhabitants. This corresponds to a purely utilitarian view and does not involve any fairness consideration, as the resulting network will typically benefit the capital city and/or large central cities while leaving some peripheral cities behind. On the other hand, a more egalitarian view will allow some people to travel between peripheral cities without having to go through a central city. We define a model, propose algorithms for computing solution networks, and report on experiments based on real data.

In this paper, we propose a new method for the augmentation of numeric and mixed datasets. The method generates additional data points by utilizing cross-validation resampling and latent variable modeling. It is particularly efficient for datasets with moderate to high degrees of collinearity, as it directly utilizes this property for generation. The method is simple, fast, and has very few parameters, which, as shown in the paper, do not require specific tuning. It has been tested on several real datasets; here, we report detailed results for two cases, prediction of protein in minced meat based on near infrared spectra (fully numeric data with high degree of collinearity) and discrimination of patients referred for coronary angiography (mixed data, with both numeric and categorical variables, and moderate collinearity). In both cases, artificial neural networks were employed for developing the regression and the discrimination models. The results show a clear improvement in the performance of the models; thus for the prediction of meat protein, fitting the model to the augmented data resulted in a reduction in the root mean squared error computed for the independent test set by 1.5 to 3 times.

It has been shown that machine translation models usually generate poor translations for named entities that are infrequent in the training corpus. Earlier named entity translation methods mainly focus on phonetic transliteration, which ignores the sentence context for translation and is limited in domain and language coverage. To address this limitation, we propose DEEP, a DEnoising Entity Pre-training method that leverages large amounts of monolingual data and a knowledge base to improve named entity translation accuracy within sentences. Besides, we investigate a multi-task learning strategy that finetunes a pre-trained neural machine translation model on both entity-augmented monolingual data and parallel data to further improve entity translation. Experimental results on three language pairs demonstrate that \method results in significant improvements over strong denoising auto-encoding baselines, with a gain of up to 1.3 BLEU and up to 9.2 entity accuracy points for English-Russian translation.