Róbert Rajkó

Nicolas Gillis, Róbert Rajkó

Given a nonnegative matrix factorization, $R$, and a factorization rank, $r$, Exact nonnegative matrix factorization (Exact NMF) decomposes $R$ as the product of two nonnegative matrices, $C$ and $S$ with $r$ columns, such as $R = CS^\top$. A central research topic in the literature is the conditions under which such a decomposition is unique/identifiable, up to trivial ambiguities. In this paper, we focus on partial identifiability, that is, the uniqueness of a subset of columns of $C$ and $S$. We start our investigations with the data-based uniqueness (DBU) theorem from the chemometrics literature. The DBU theorem analyzes all feasible solutions of Exact NMF, and relies on sparsity conditions on $C$ and $S$. We provide a mathematically rigorous theorem of a recently published restricted version of the DBU theorem, relying only on simple sparsity and algebraic conditions: it applies to a particular solution of Exact NMF (as opposed to all feasible solutions) and allows us to guarantee the partial uniqueness of a single column of $C$ or $S$. Second, based on a geometric interpretation of the restricted DBU theorem, we obtain a new partial identifiability result. This geometric interpretation also leads us to another partial identifiability result in the case $r=3$. Third, we show how partial identifiability results can be used sequentially to guarantee the identifiability of more columns of $C$ and $S$. We illustrate these results on several examples, including one from the chemometrics literature.

Rudolf Ferenc, István Siket, Péter Hegedűs et al.

Forecasting defect proneness of source code has long been a major research concern. Having an estimation of those parts of a software system that most likely contain bugs may help focus testing efforts, reduce costs, and improve product quality. Many prediction models and approaches have been introduced during the past decades that try to forecast bugged code elements based on static source code metrics, change and history metrics, or both. However, there is still no universal best solution to this problem, as most suitable features and models vary from dataset to dataset and depend on the context in which we use them. Therefore, novel approaches and further studies on this topic are highly necessary. In this paper, we employ a chemometric approach - Partial Least Squares with Discriminant Analysis (PLS-DA) - for predicting bug prone Classes in Java programs using static source code metrics. To our best knowledge, PLS-DA has never been used before as a statistical approach in the software maintenance domain for predicting software errors. In addition, we have used rigorous statistical treatments including bootstrap resampling and randomization (permutation) test, and evaluation for representing the software engineering results. We show that our PLS-DA based prediction model achieves superior performances compared to the state-of-the-art approaches (i.e. F-measure of 0.44-0.47 at 90% confidence level) when no data re-sampling applied and comparable to others when applying up-sampling on the largest open bug dataset, while training the model is significantly faster, thus finding optimal parameters is much easier. In terms of completeness, which measures the amount of bugs contained in the Java Classes predicted to be defective, PLS-DA outperforms every other algorithm: it found 69.3% and 79.4% of the total bugs with no re-sampling and up-sampling, respectively.