A new approach to equations with memory

3We discuss a novel approach to the mathematical analysis of equations with memory, based on the notion of a {\it state}. This is the initial configuration of the system at time $t=0$ which can be unambiguously determined by the knowledge of the dynamics for positive times. As a model, for a nonincreasing convex function $G:\R^+\to\R^+$ such that $$G(0)=\lim_{s\to 0}G(s)>\lim_{s\to\infty}G(s)>0$$ we consider an abstract version of the evolution equation $$\partial_{tt} {\bm u}({\bm x},t)-\Delta\Big[G(0) {\bm u}({\bm x},t) +\int_0^\infty G'(s) {\bm u}({\bm x},t-s) {\rm d} s\Big]=0$$ arising from linear viscoelasticity.Mathematics Subject Classification: 45 - Integral equations, 74 - Mechanics of deformable solids. European Research Council (ERC) code: PE1_12 Mathematical Physics, PE1_20 - Application of mathematics in sciences.openopenFabrizio M.; Giorgi C.; Pata V.Fabrizio, M.; Giorgi, Claudio; Pata, Vittorin

xUnit: Learning a Spatial Activation Function for Efficient Image Restoration

In recent years, deep neural networks (DNNs) achieved unprecedented performance in many low-level vision tasks. However, state-of-the-art results are typically achieved by very deep networks, which can reach tens of layers with tens of millions of parameters. To make DNNs implementable on platforms with limited resources, it is necessary to weaken the tradeoff between performance and efficiency. In this paper, we propose a new activation unit, which is particularly suitable for image restoration problems. In contrast to the widespread per-pixel activation units, like ReLUs and sigmoids, our unit implements a learnable nonlinear function with spatial connections. This enables the net to capture much more complex features, thus requiring a significantly smaller number of layers in order to reach the same performance. We illustrate the effectiveness of our units through experiments with state-of-the-art nets for denoising, de-raining, and super resolution, which are already considered to be very small. With our approach, we are able to further reduce these models by nearly 50% without incurring any degradation in performance.Comment: Conference on Computer Vision and Pattern Recognition (CVPR), 201

GOexpress: an R/Bioconductor package for the identification and visualisation of robust gene ontology signatures through supervised learning of gene expression data

Background: Identification of gene expression profiles that differentiate experimental groups is critical for discovery and analysis of key molecular pathways and also for selection of robust diagnostic or prognostic biomarkers. While integration of differential expression statistics has been used to refine gene set enrichment analyses, such approaches are typically limited to single gene lists resulting from simple two-group comparisons or time-series analyses. In contrast, functional class scoring and machine learning approaches provide powerful alternative methods to leverage molecular measurements for pathway analyses, and to compare continuous and multi-level categorical factors. Results: We introduce GOexpress, a software package for scoring and summarising the capacity of gene ontology features to simultaneously classify samples from multiple experimental groups. GOexpress integrates normalised gene expression data (e.g., from microarray and RNA-seq experiments) and phenotypic information of individual samples with gene ontology annotations to derive a ranking of genes and gene ontology terms using a supervised learning approach. The default random forest algorithm allows interactions between all experimental factors, and competitive scoring of expressed genes to evaluate their relative importance in classifying predefined groups of samples. Conclusions: GOexpress enables rapid identification and visualisation of ontology-related gene panels that robustly classify groups of samples and supports both categorical (e.g., infection status, treatment) and continuous (e.g., time-series, drug concentrations) experimental factors. The use of standard Bioconductor extension packages and publicly available gene ontology annotations facilitates straightforward integration of GOexpress within existing computational biology pipelines.Department of Agriculture, Food and the MarineEuropean Commission - Seventh Framework Programme (FP7)Science Foundation IrelandUniversity College Dubli