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Abstract Background Gene expression is regulated at different molecular levels, including chromatin accessibility, transcription, RNA maturation, and transport. These regulatory mechanisms have strong connections with cellular metabolism. In order to study the cellular system and its functioning, omics data at each molecular level can be generated and efficiently integrated. Here, we propose BRANEnet, a novel multi-omics integration framework for multilayer heterogeneous networks. BRANEnet is an expressive, scalable, and versatile method to learn node embeddings, leveraging random walk information within a matrix factorization framework. Our goal is to efficiently integrate multi-omics data to study different regulatory aspects of multilayered processes that occur in organisms. We evaluate our framework using multi-omics data of Saccharomyces cerevisiae, a well-studied yeast model organism. Results We test BRANEnet on transcriptomics (RNA-seq) and targeted metabolomics (NMR) data for wild-type yeast strain during a heat-shock time course of 0, 20, and 120 min. Our framework learns features for differentially expressed bio-molecules showing heat stress response. We demonstrate the applicability of the learned features for targeted omics inference tasks: transcription factor (TF)-target prediction, integrated omics network (ION) inference, and module identification. The performance of BRANEnet is compared to existing network integration methods. Our model outperforms baseline methods by achieving high prediction scores for a variety of downstream tasks.

The stability of neural networks with respect to adversarial perturbations has been extensively studied. One of the main strategies consist of quantifying the Lipschitz regularity of neural networks. In this paper, we introduce a multivariate Lipschitz constant-based stability analysis of fully connected neural networks allowing us to capture the influence of each input or group of inputs on the neural network stability. Our approach relies on a suitable re-normalization of the input space, with the objective to perform a more precise analysis than the one provided by a global Lipschitz constant. We investigate the mathematical properties of the proposed multivariate Lipschitz analysis and show its usefulness in better understanding the sensitivity of the neural network with regard to groups of inputs. We display the results of this analysis by a new representation designed for machine learning practitioners and safety engineers termed as a Lipschitz star. The Lipschitz star is a graphical and practical tool to analyze the sensitivity of a neural network model during its development, with regard to different combinations of inputs. By leveraging this tool, we show that it is possible to build robust-by-design models using spectral normalization techniques for controlling the stability of a neural network, given a safety Lipschitz target. Thanks to our multivariate Lipschitz analysis, we can also measure the efficiency of adversarial training in inference tasks. We perform experiments on various open access tabular datasets, and also on a real Thales Air Mobility industrial application subject to certification requirements.