Noncommutative Bell polynomials, quasideterminants and incidence Hopf algebras

Bell polynomials appear in several combinatorial constructions throughout mathematics. Perhaps most naturally in the combinatorics of set partitions, but also when studying compositions of diffeomorphisms on vector spaces and manifolds, and in the study of cumulants and moments in probability theory. We construct commutative and noncommutative Bell polynomials and explain how they give rise to Fa\`a di Bruno Hopf algebras. We use the language of incidence Hopf algebras, and along the way provide a new description of antipodes in noncommutative incidence Hopf algebras, involving quasideterminants. We also discuss M\"obius inversion in certain Hopf algebras built from Bell polynomials.Comment: 37 pages, final version, to appear in IJA

Pulmonary Nodule Classification in Lung Cancer from 3D Thoracic CT Scans Using fastai and MONAI

We construct a convolutional neural network to classify pulmonary nodules as malignant or benign in the context of lung cancer. To construct and train our model, we use our novel extension of the fastai deep learning framework to 3D medical imaging tasks, combined with the MONAI deep learning library. We train and evaluate the model using a large, openly available data set of annotated thoracic CT scans. Our model achieves a nodule classification accuracy of 92.4% and a ROC AUC of 97% when compared to a “ground truth” based on multiple human raters subjective assessment of malignancy. We further evaluate our approach by predicting patient-level diagnoses of cancer, achieving a test set accuracy of 75%. This is higher than the 70% obtained by aggregating the human raters assessments. Class activation maps are applied to investigate the features used by our classifier, enabling a rudimentary level of explainability for what is otherwise close to “black box” predictions. As the classification of structures in chest CT scans is useful across a variety of diagnostic and prognostic tasks in radiology, our approach has broad applicability. As we aimed to construct a fully reproducible system that can be compared to new proposed methods and easily be adapted and extended, the full source code of our work is available at https://github.com/MMIV-ML/Lung-CT-fastai-2020

On the Lie enveloping algebra of a post-Lie algebra

We consider pairs of Lie algebras $g$ and $\bar{g}$, defined over a common vector space, where the Lie brackets of $g$ and $\bar{g}$ are related via a post-Lie algebra structure. The latter can be extended to the Lie enveloping algebra $U(g)$. This permits us to define another associative product on $U(g)$, which gives rise to a Hopf algebra isomorphism between $U(\bar{g})$ and a new Hopf algebra assembled from $U(g)$ with the new product. For the free post-Lie algebra these constructions provide a refined understanding of a fundamental Hopf algebra appearing in the theory of numerical integration methods for differential equations on manifolds. In the pre-Lie setting, the algebraic point of view developed here also provides a concise way to develop Butcher's order theory for Runge--Kutta methods.Comment: 25 page

2D and 3D U-Nets for skull stripping in a large and heterogeneous set of head MRI using fastai

Skull stripping in brain imaging is the removal of the parts of images corresponding to non-brain tissue. Fast and accurate skull stripping is a crucial step for numerous medical brain imaging applications, e.g. registration, segmentation and feature extraction, as it eases subsequent image processing steps. In this work, we propose and compare two novel skull stripping methods based on 2D and 3D convolutional neural networks trained on a large, heterogeneous collection of 2777 clinical 3D T1-weighted MRI images from 1681 healthy subjects. We investigated the performance of the models by testing them on 927 images from 324 subjects set aside from our collection of data, in addition to images from an independent, large brain imaging study: the IXI dataset (n = 556). Our models achieved mean Dice scores higher than 0:978 and Jaccard indices higher than 0:957 on all tests sets, making predictions on new unseen brain MR images in approximately 1.4s for the 3D model and 12.4s for the 2D model. A preliminary exploration of the models’ robustness to variation in the input data showed favourable results when compared to a traditional, well-established skull stripping method. With further research aimed at increasing the models’ robustness, such accurate

2D and 3D U-Nets for skull stripping in a large and heterogeneous set of head MRI using fastai

Skull stripping in brain imaging is the removal of the parts of images corresponding to non-brain tissue. Fast and accurate skull stripping is a crucial step for numerous medical brain imaging applications, e.g. registration, segmentation and feature extraction, as it eases subsequent image processing steps. In this work, we propose and compare two novel skull stripping methods based on 2D and 3D convolutional neural networks trained on a large, heterogeneous collection of 2777 clinical 3D T1-weighted MRI images from 1681 healthy subjects. We investigated the performance of the models by testing them on 927 images from 324 subjects set aside from our collection of data, in addition to images from an independent, large brain imaging study: the IXI dataset (n = 556). Our models achieved mean Dice scores higher than 0:978 and Jaccard indices higher than 0:957 on all tests sets, making predictions on new unseen brain MR images in approximately 1.4s for the 3D model and 12.4s for the 2D model. A preliminary exploration of the models’ robustness to variation in the input data showed favourable results when compared to a traditional, well-established skull stripping method. With further research aimed at increasing the models’ robustness, such accurate and fast skull stripping methods can potentially form a useful component of brain MRI analysis pipelines.publishedVersio

Post-Lie Algebras and Isospectral Flows

In this paper we explore the Lie enveloping algebra of a post-Lie algebra derived from a classical $R$-matrix. An explicit exponential solution of the corresponding Lie bracket flow is presented. It is based on the solution of a post-Lie Magnus-type differential equation

Predicting conversion to Alzheimer’s disease in individuals with Mild Cognitive Impairment using clinically transferable features

Patients with Mild Cognitive Impairment (MCI) have an increased risk of Alzheimer’s disease (AD). Early identification of underlying neurodegenerative processes is essential to provide treatment before the disease is well established in the brain. Here we used longitudinal data from the ADNI database to investigate prediction of a trajectory towards AD in a group of patients defined as MCI at a baseline examination. One group remained stable over time (sMCI, n = 357) and one converted to AD (cAD, n = 321). By running two independent classification methods within a machine learning framework, with cognitive function, hippocampal volume and genetic APOE status as features, we obtained a cross-validation classification accuracy of about 70%. This level of accuracy was confirmed across different classification methods and validation procedures. Moreover, the sets of misclassified subjects had a large overlap between the two models. Impaired memory function was consistently found to be one of the core symptoms of MCI patients on a trajectory towards AD. The prediction above chance level shown in the present study should inspire further work to develop tools that can aid clinicians in making prognostic decisions.publishedVersio

Functional activity level reported by an informant is an early predictor of Alzheimer’s disease

Background Loss of autonomy in day-to-day functioning is one of the feared outcomes of Alzheimer’s disease (AD), and relatives may have been worried by subtle behavioral changes in ordinary life situations long before these changes are given medical attention. In the present study, we ask if such subtle changes should be given weight as an early predictor of a future AD diagnosis. Methods Longitudinal data from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) were used to define a group of adults with a mild cognitive impairment (MCI) diagnosis remaining stable across several visits (sMCI, n=360; 55-91 years at baseline), and a group of adults who over time converted from having an MCI diagnosis to an AD diagnosis (cAD, n=320; 55-88 years at baseline). Eleven features were used as input in a Random Forest (RF) binary classifier (sMCI vs. cAD) model. This model was tested on an unseen holdout part of the dataset, and further explored by three different permutation-driven importance estimates and a comprehensive post hoc machine learning exploration. Results The results consistently showed that measures of daily life functioning, verbal memory function, and a volume measure of hippocampus were the most important predictors of conversion from an MCI to an AD diagnosis. Results from the RF classification model showed a prediction accuracy of around 70% in the test set. Importantly, the post hoc analyses showed that even subtle changes in everyday functioning noticed by a close informant put MCI patients at increased risk for being on a path toward the major cognitive impairment of an AD diagnosis. Conclusion The results showed that even subtle changes in everyday functioning should be noticed when reported by relatives in a clinical evaluation of patients with MCI. Information of these changes should also be included in future longitudinal studies to investigate different pathways from normal cognitive aging to the cognitive decline characterizing different stages of AD and other neurodegenerative disorders.publishedVersio

Synthesizing skin lesion images using CycleGANs – a case study

Generative adversarial networks (GANs) have seen some success as a way to synthesize training data for supervised machine learning models. In this work, we design two novel approaches for synthetic image generation based on CycleGANs, aimed at generating realistic-looking, class-specific dermoscopic skin lesion images. We evaluate the images’ usefulness as additional training data for a convolutional neural network trained to perform a difficult lesion classification task. We are able to generate visually striking images, but their value for augmenting the classifier’s training data set is low. This is in-line with other researcher’s investigations into similar GAN models, indicating the need for further research into forcing GAN models to produce samples further from the training data distribution, and to find ways of guiding the image generation using feedback from the ultimate classification objective

Backward error analysis and the substitution law for Lie group integrators

Butcher series are combinatorial devices used in the study of numerical methods for differential equations evolving on vector spaces. More precisely, they are formal series developments of differential operators indexed over rooted trees, and can be used to represent a large class of numerical methods. The theory of backward error analysis for differential equations has a particularly nice description when applied to methods represented by Butcher series. For the study of differential equations evolving on more general manifolds, a generalization of Butcher series has been introduced, called Lie--Butcher series. This paper presents the theory of backward error analysis for methods based on Lie--Butcher series.Comment: Minor corrections and additions. Final versio