Gradient descent during the learning process of a neural network can be subject to many instabilities. The spectral density of the Jacobian is a key component for analyzing stability. Following the works of Pennington et al., such Jacobians are modeled using free multiplicative convolutions from Free Probability Theory (FPT).We present a reliable and very fast method for computing the associated spectral densities, for given architecture and initialization. This method has a controlled and proven convergence. Our technique is based on an homotopy method: it is an adaptative Newton-Raphson scheme which chains basins of attraction. We find contiguous lilypad-like basins and step from one to the next, heading towards the objective.In order to demonstrate the relevance of our method we show that the relevant FPT metrics computed before training are highly correlated to final test losses – up to 85%. We also give evidence that a very desirable feature for neural networks is the hyperbolicity of their Jacobian at initialization, while remaining at the edge of chaos.

MHC-I associated peptides (MAPs) play a central role in the elimination of virus-infected and neoplastic cells by CD8 T cells. However, accurately predicting the MAP repertoire remains difficult, because only a fraction of the transcriptome generates MAPs. In this study, we investigated whether codon arrangement (usage and placement) regulates MAP biogenesis. We developed an artificial neural network called Codon Arrangement MAP Predictor (CAMAP), predicting MAP presentation solely from mRNA sequences flanking the MAP-coding codons (MCCs), while excluding the MCC per se. CAMAP predictions were significantly more accurate when using original codon sequences than shuffled codon sequences which reflect amino acid usage. Furthermore, predictions were independent of mRNA expression and MAP binding affinity to MHC-I molecules and applied to several cell types and species. Combining MAP ligand scores, transcript expression level and CAMAP scores was particularly useful to increase MAP prediction accuracy. Using an in vitro assay, we showed that varying the synonymous codons in the regions flanking the MCCs (without changing the amino acid sequence) resulted in significant modulation of MAP presentation at the cell surface. Taken together, our results demonstrate the role of codon arrangement in the regulation of MAP presentation and support integration of both translational and post-translational events in predictive algorithms to ameliorate modeling of the immunopeptidome.