Efficient Neural Network Verification via Layer-based Semidefinite Relaxations and Linear Cuts

We introduce an efficient and tight layer-based semidefinite relaxation for verifying local robustness of neural networks. The improved tightness is the result of the combination between semidefinite relaxations and linear cuts. We obtain a computationally efficient method by decomposing the semidefinite formulation into layerwise constraints. By leveraging on chordal graph decompositions, we show that the formulation here presented is provably tighter than current approaches. Experiments on a set of benchmark networks show that the approach here proposed enables the verification of more instances compared to other relaxation methods. The results also demonstrate that the SDP relaxation here proposed is one order of magnitude faster than previous SDP methods.

High Transcriptional Error Rates Vary as a Function of Gene Expression Level

Errors in gene transcription can be costly, and organisms have evolved to prevent their occurrence or mitigate their costs. The simplest interpretation of the drift barrier hypothesis suggests that species with larger population sizes would have lower transcriptional error rates. However, Escherichia coli seems to have a higher transcriptional error rate than species with lower effective population sizes, for example Saccharomyces cerevisiae. This could be explained if selection in E. coli were strong enough to maintain adaptations that mitigate the consequences of transcriptional errors through robustness, on a gene by gene basis, obviating the need for low transcriptional error rates and associated costs of global proofreading. Here, we note that if selection is powerful enough to evolve local robustness, selection should also be powerful enough to locally reduce error rates. We therefore predict that transcriptional error rates will be lower in highly abundant proteins on which selection is strongest. However, we only expect this result when error rates are high enough to significantly impact fitness. As expected, we find such a relationship between expression and transcriptional error rate for non-C→U errors in E. coli (especially G→A), but not in S. cerevisiae. We do not find this pattern for C→U changes in E. coli, presumably because most deamination events occurred during sample preparation, but do for C→U changes in S. cerevisiae, supporting the interpretation that C→U error rates estimated with an improved protocol, and which occur at rates comparable with E. coli non-C→U errors, are biological.

High transcriptional error rates vary as a function of gene expression level

Errors in gene transcription can be costly, and organisms have evolved to prevent their occurrence or mitigate their costs. The simplest interpretation of the drift barrier hypothesis suggests that species with larger population sizes would have lower transcriptional error rates. However, Escherichia coli seems to have a higher transcriptional error rate than species with lower effective population sizes, e.g. Saccharomyces cerevisiae. This could be explained if selection in E. coli were strong enough to maintain adaptations that mitigate the consequences of transcriptional errors through robustness, on a gene by gene basis, obviating the need for low transcriptional error rates and associated costs of global proofreading. Here we note that if selection is powerful enough to evolve local robustness, selection should also be powerful enough to locally reduce error rates. We therefore predict that transcriptional error rates will be lower in highly abundant proteins on which selection is strongest. However, we only expect this result when error rates are high enough to significantly impact fitness. As expected, we find such a relationship between expression and transcriptional error rate for non C➔U errors in E. coli (especially G➔A), but not in S. cerevisiae. We do not find this pattern for C➔U changes in E. coli, presumably because most deamination events occurred during sample preparation, but do for C➔U changes in S. cerevisiae, supporting the interpretation that C➔U error rates estimated with an improved protocol, and which occur at rates comparable to E. coli non C➔U errors, are biological.

ROBUSTNESS ANALYSIS IN A TODIM-BASED MULTICRITERIA EVALUATION MODEL OF RENTAL PROPERTIES

A new robustness analysis framework is proposed where robustness of a solution in a decision aiding process is measured as the distance from that solution to an expected outcome, chosen by the decision-aiding analyst. The framework is explained by the application of the TODIM method of multicriteria decision aiding to the problem of predicting rental ranges for properties in a Chilean city. Therefore, the robustness concern concentrates on changes in criteria weights as well as in trade-off rates, as they are defined in the method. Two main contributions are introduced: a local robustness measure, defined in terms of a distance among rankings; and a global robustness measure, as an adaptation of the minimax-regret rule to select a global robust solution, i.e. a ranking produced by TODIM.