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Gaussian Processes

The project is an interactive, living book dedicated to reconstructing Archimedes’ Measurement of a Circle (Circuli Dimensio) using the original geometric proportions. By employing SageMath for symbolic validation, alongside Python, Julia and Perl for high-precision numerical calculations using libraries for decimal and arbitrary-precision arithmetic, the project bypasses modern algebraic transformations and demonstrates the accuracy of ancient geometric concepts.

Beyond Archimedes' original derivations, the project bridges the gap to modern approaches for calculating π. It demonstrates how Archimedes' four determination equations can be translated into modern notation like a super-radical, a product formula and a summation formula. To achieve this, the four equations were systematically reduced to a single unified equation. This step opens the door to further developments and serves as the link between geometry and algebra.

Using a specialized mathematical gain function, the system creates a dynamic bridge between ancient geometry and modern infinite series such as those of Ramanujan and Chudnovsky or the π formula of Saha and Sinha. Particular emphasis is placed on this new π formula, as it is more than it appears at first glance. By precisely handling the λ parameter the optimized approach achieves absolute numerical stability in getting hundreds or more of error-free decimal places. It is demonstrated, among other things, that the maximum gain in decimal places is achieved when the chosen value of λ equals the maximum number of iterations.

A key highlight is that the derived concept based on Archimedes allows the fundamental re-examination of every existing approach. E.g. the equations found on Wikipedia have been consolidated into a single equation as well as the BPS algorithm can also be reduced to a single equation. Furthermore, the work of Snellius and Dörrie can also be incorporated into the geometry of Archimedes.

And much more!

CTA is a testing environment for online heterogenous task assignment approaches which involve representation of worker-task pairs as noisy functions.

"n" dimensional input with "m" objectives example of a Pareto front with the BOSS code. In these examples we reproduce the pareto front as produced by mathlab in the link below "https://www.mathworks.com/help/gads/pareto-front-for-two-objectives.html" For 3D inputs and 3 objectives, we try to reproduce these results The difference in these methods is that here we calculate the pareto front using GP functions with an RBF kernel.

Data simulations for Gaussian Process Classification of data from sensors attached to a hypothetical machine.

Source Code for Gabler et al. "Bayesian Optimization with Unknown Constraints in Graphical Skill-Models for Compliant Manipulation Tasks Using an Industrial Robot", Frontiers in Robotics and AI, 2022

Learning in the Wild with Incremental Skeptical Gaussian Processes

This is a pilot project for a future iteration of The Alice Challenge: remote control of an ultracold atoms experiment by experts and citizen scientist. Scientific paper available here: https://www.pnas.org/content/115/48/E11231

Miscellaneous Machine Learning Example Code for sci-kit learn, refactored to PEP8 style

Scalable Gaussian processes in TensorFlow