Simulated Data for Linear Regression with Structured and Sparse Penalties: Introducing pylearn-simulate

A currently very active field of research is how to incorporate structure and prior knowledge in machine learning methods. It has lead to numerous developments in the field of non-smooth convex minimization. With recently developed methods it is possible to perform an analysis in which the computed model can be linked to a given structure of the data and simultaneously do variable selection to find a few important features in the data. However, there is still no way to unambiguously simulate data to test proposed algorithms, since the exact solutions to such problems are unknown. The main aim of this paper is to present a theoretical framework for generating simulated data. These simulated data are appropriate when comparing optimization algorithms in the context of linear regression problems with sparse and structured penalties. Additionally, this approach allows the user to control the signal-to-noise ratio, the correlation structure of the data and the optimization problem to which they are the solution. The traditional approach is to simulate random data without taking into account the actual model that will be fit to the data. But when using such an approach it is not possible to know the exact solution of the underlying optimization problem. With our contribution, it is possible to know the exact theoretical solution of a penalized linear regression problem, and it is thus possible to compare algorithms without the need to use, e.g., cross-validation. We also present our implementation, the Python package pylearn-simulate, available at https://github.com/neurospin/pylearn-simulate and released under the BSD 3clause license. We describe the package and give examples at the end of the paper

Predictive modelling of gas assisted electron and ion beam induced etching and deposition

University of Technology Sydney. Faculty of Science.While the field of experimental micrometre scale EBIED / IBIED (“electron beam chemistry” or “ion beam chemistry”) has been growing in recent years, the 3D simulation of these systems at real scales has been non-existent. This type of simulation is important for it is only in three dimensions that interesting asymmetric and patterning phenomena can be tracked. There are a couple of difficulties in these types of simulations. One is solving the diffusion of adsorbate concentrations in the system. Accurate simulation of diffusion on general 2D surfaces is non-trivial, (even on 1D curves), and can require unnatural re-parametrization of the surface (re-meshing). Another difficulty is that simulations have generally been atomistic and limited in scale. The key to providing large scale 3D simulations comes from applying new, mathematically robust, computer-science methods based on implicit surfaces to this field. In this thesis, the issues above are addressed in a couple of different ways. In one case, diffusion over a complex surface was reduced to piecewise axially symmetric equations. Later, implicit methods for solving adsorbate kinetics continuum equations and evolving the surface are implemented, the closest point method and the level set method respectively. The development of the tools themselves is a non-trivial exercise as there are few software libraries for the level set method and none for the closest point method. These tools were then used independently to simulate etching and diffusion, as well as in concert to demonstrate the ability to simulate 3D deposition in the mass transport limited and reaction rate limited regimes

Generalized Buffon\u27s experiment

Buffonov pokus jedan je, od danas više poznatih, načina približnog izračuna vrijednosti broja $pi$. Ovim radom Buffonov je pokus proširen i prilagođen izračunu približne vrijednosti broja e.Buffon\u27s experiment is one of, today more famous, methods of aproximate calculating a value of number $pi$. In this article Buffon\u27s experiment is enhanced and adapted to aproximate calculating a value of number e

Composition of Stochastic Transition Systems Based on Spans and Couplings

Conventional approaches for parallel composition of stochastic systems relate probability measures of the individual components in terms of product measures. Such approaches rely on the assumption that components interact stochastically independent, which might be too rigid for modeling real world systems. In this paper, we introduce a parallel-composition operator for stochastic transition systems that is based on couplings of probability measures and does not impose any stochastic assumptions. When composing systems within our framework, the intended dependencies between components can be determined by providing so-called spans and span couplings. We present a congruence result for our operator with respect to a standard notion of bisimilarity and develop a general theory for spans, exploiting deep results from descriptive set theory. As an application of our general approach, we propose a model for stochastic hybrid systems called stochastic hybrid motion automata

Flowers in three dimensions and beyond

textPattern formation in buckled membranes was studied along with the morphology of flowers formed at the tip of silicon nanowires and ripples formed in suspended graphene sheets. Nash's perturbation method was tested for a simple case where initial and final metrics embed smoothly and there is a smooth path from one surface to another and was found to work successfully. The method was tested in more realistic conditions where a smooth path was not known and the method failed. Cylindrical flower-like membranes with a metric of negative Gaussian curvature were simulated in three and four dimensions. These four dimensional flowers had 2 orders of magnitude less energy than their three dimensional counterparts. Simulations were used to show that the addition of a fourth spatial dimension did not relieve all bending energy from the cylindrical membranes. Patterns formed at the tip of silicon nanowires were studied and found to be of the Dense Branching Morphology type. The rate of branching is dependent on the curvature of the gold bubble on which they are grown. Graphene was simulated using the modified embedded atom method potential and buckles were found to form if the carbon bonds were stretched. An energy functional was found for the energy of a sheet with a metric different from that of flat space.Physic