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We propose a novel class of deep stochastic predictors for classifying metric data on graphs within the PAC‐Bayes risk certification paradigm. Classifiers are realized as linearly parametrized deep assignment flows with random initial conditions. Building on the recent PAC‐Bayes literature and data‐dependent priors, this approach enables (i) to use risk bounds as training objectives for learning posterior distributions on the hypothesis space and (ii) to compute tight out‐of‐sample risk certificates of randomized classifiers more efficiently than related work. Comparison with empirical test set errors illustrates the performance and practicality of this self‐certifying classification method. [ABSTRACT FROM AUTHOR]

Like many other types of cancer, colorectal cancer (CRC) develops through multiple pathways of carcinogenesis. This is also true for colorectal carcinogenesis in Lynch syndrome (LS), the most common inherited CRC syndrome. However, a comprehensive understanding of the distribution of these pathways of carcinogenesis, which allows for tailored clinical treatment and even prevention, is still lacking. We suggest a linear dynamical system modeling the evolution of different pathways of colorectal carcinogenesis based on the involved driver mutations. The model consists of different components accounting for independent and dependent mutational processes. We define the driver gene mutation graphs and combine them using the Cartesian graph product. This leads to matrix components built by the Kronecker sum and product of the adjacency matrices of the gene mutation graphs enabling a thorough mathematical analysis and medical interpretation. Using the Kronecker structure, we developed a mathematical model which we applied exemplarily to the three pathways of colorectal carcinogenesis in LS. Beside a pathogenic germline variant in one of the DNA mismatch repair (MMR) genes, driver mutations in APC, CTNNB1, KRAS and TP53 are considered. We exemplarily incorporate mutational dependencies, such as increased point mutation rates after MMR deficiency, and based on recent experimental data, biallelic somatic CTNNB1 mutations as common drivers of LS-associated CRCs. With the model and parameter choice, we obtained simulation results that are in concordance with clinical observations. These include the evolution of MMR-deficient crypts as early precursors in LS carcinogenesis and the influence of variants in MMR genes thereon. The proportions of MMR-deficient and MMR-proficient APC-inactivated crypts as first measure for the distribution among the pathways in LS-associated colorectal carcinogenesis are compatible with clinical observations. The approach provides a modular framework for modeling multiple pathways of carcinogenesis yielding promising results in concordance with clinical observations in LS CRCs. Author summary: Cancer is a disease caused by alterations of the genome. The alterations can affect each component of the genome, whereas only some lead to a change in the functioning of the cell. As there are several of those so-called driver mutations, there are different possibilities in which order they can occur. It is currently assumed that the order of driver mutations is linked to the course of cancer and thus to clinical treatment and even prevention. However, cells with a driver mutation, which carry a risk to grow out to a tumor, are clinically invisible for a long time. This means the early carcinogenesis is a hidden process. Mathematical models allow testing related medical hypotheses to obtain a better understanding of the underlying biological processes. We proposed a mathematical model for different molecular pathways of carcinogenesis based on a linear dynamical system. Thereby, we used the Kronecker structure, a specific structure which allows for a thorough mathematical analysis and medical interpretation. The model consists of multiple components to account for independent and dependent mutational processes. For the presented work, we focused on cancer development in the colon. However, modifications of the model could be applied to other organs. [ABSTRACT FROM AUTHOR]

We introduce the linear assignment flow as an approximation of the full nonlinear assignment flow, which is a method for contextual data labeling on arbitrary graphs. The linear assignment flow is a dynamical system evolving on the tangent space of a statistical manifold. It is numerically determined using exponential integrators and Krylov subspace approximation, for which we provide error estimates. The approximation property of the linear assignment flow is illustrated by a numerical experiment. This work is supplemented by two papers on variational modeling and unsupervised labeling [1]. [ABSTRACT FROM AUTHOR]