Metrics for more than two points at once

The conventional definition of a topological metric over a space specifies properties that must be obeyed by any measure of "how separated" two points in that space are. Here it is shown how to extend that definition, and in particular the triangle inequality, to concern arbitrary numbers of points. Such a measure of how separated the points within a collection are can be bootstrapped, to measure "how separated" from each other are two (or more) collections. The measure presented here also allows fractional membership of an element in a collection. This means it directly concerns measures of ``how spread out" a probability distribution over a space is. When such a measure is bootstrapped to compare two collections, it allows us to measure how separated two probability distributions are, or more generally, how separated a distribution of distributions is.Comment: 8 page

Faithful transformation of quasi-isotropic to Weyl-Papapetrou coordinates: A prerequisite to compare metrics

We demonstrate how one should transform correctly quasi-isotropic coordinates to Weyl-Papapetrou coordinates in order to compare the metric around a rotating star that has been constructed numerically in the former coordinates with an axially symmetric stationary metric that is given through an analytical form in the latter coordinates. Since a stationary metric associated with an isolated object that is built numerically partly refers to a non-vacuum solution (interior of the star) the transformation of its coordinates to Weyl-Papapetrou coordinates, which are usually used to describe vacuum axisymmetric and stationary solutions of Einstein equations, is not straightforward in the non-vacuum region. If this point is \textit{not} taken into consideration, one may end up to erroneous conclusions about how well a specific analytical metric matches the metric around the star, due to fallacious coordinate transformations.Comment: 18 pages, 2 figure

Reducing variability in along-tract analysis with diffusion profile realignment

Diffusion weighted MRI (dMRI) provides a non invasive virtual reconstruction of the brain's white matter structures through tractography. Analyzing dMRI measures along the trajectory of white matter bundles can provide a more specific investigation than considering a region of interest or tract-averaged measurements. However, performing group analyses with this along-tract strategy requires correspondence between points of tract pathways across subjects. This is usually achieved by creating a new common space where the representative streamlines from every subject are resampled to the same number of points. If the underlying anatomy of some subjects was altered due to, e.g. disease or developmental changes, such information might be lost by resampling to a fixed number of points. In this work, we propose to address the issue of possible misalignment, which might be present even after resampling, by realigning the representative streamline of each subject in this 1D space with a new method, coined diffusion profile realignment (DPR). Experiments on synthetic datasets show that DPR reduces the coefficient of variation for the mean diffusivity, fractional anisotropy and apparent fiber density when compared to the unaligned case. Using 100 in vivo datasets from the HCP, we simulated changes in mean diffusivity, fractional anisotropy and apparent fiber density. Pairwise Student's t-tests between these altered subjects and the original subjects indicate that regional changes are identified after realignment with the DPR algorithm, while preserving differences previously detected in the unaligned case. This new correction strategy contributes to revealing effects of interest which might be hidden by misalignment and has the potential to improve the specificity in longitudinal population studies beyond the traditional region of interest based analysis and along-tract analysis workflows.Comment: v4: peer-reviewed round 2 v3 : deleted some old text from before peer-review which was mistakenly included v2 : peer-reviewed version v1: preprint as submitted to journal NeuroImag

Assessing architectural evolution: A case study

This is the post-print version of the Article. The official published can be accessed from the link below - Copyright @ 2011 SpringerThis paper proposes to use a historical perspective on generic laws, principles, and guidelines, like Lehman’s software evolution laws and Martin’s design principles, in order to achieve a multi-faceted process and structural assessment of a system’s architectural evolution. We present a simple structural model with associated historical metrics and visualizations that could form part of an architect’s dashboard. We perform such an assessment for the Eclipse SDK, as a case study of a large, complex, and long-lived system for which sustained effective architectural evolution is paramount. The twofold aim of checking generic principles on a well-know system is, on the one hand, to see whether there are certain lessons that could be learned for best practice of architectural evolution, and on the other hand to get more insights about the applicability of such principles. We find that while the Eclipse SDK does follow several of the laws and principles, there are some deviations, and we discuss areas of architectural improvement and limitations of the assessment approach

Visualising Basins of Attraction for the Cross-Entropy and the Squared Error Neural Network Loss Functions

Quantification of the stationary points and the associated basins of attraction of neural network loss surfaces is an important step towards a better understanding of neural network loss surfaces at large. This work proposes a novel method to visualise basins of attraction together with the associated stationary points via gradient-based random sampling. The proposed technique is used to perform an empirical study of the loss surfaces generated by two different error metrics: quadratic loss and entropic loss. The empirical observations confirm the theoretical hypothesis regarding the nature of neural network attraction basins. Entropic loss is shown to exhibit stronger gradients and fewer stationary points than quadratic loss, indicating that entropic loss has a more searchable landscape. Quadratic loss is shown to be more resilient to overfitting than entropic loss. Both losses are shown to exhibit local minima, but the number of local minima is shown to decrease with an increase in dimensionality. Thus, the proposed visualisation technique successfully captures the local minima properties exhibited by the neural network loss surfaces, and can be used for the purpose of fitness landscape analysis of neural networks.Comment: Preprint submitted to the Neural Networks journa