Riemannian geometry of Grassmann manifolds with a view on algorithmic computation

We give simple formulas for the canonical metric, gradient, Lie derivative, Riemannian connection, parallel translation, geodesics and distance on the Grassmann manifold of p-planes in ℝn. In these formulas, p-planes are represented as the column space of n × p matrices. The Newton method on abstract Riemannian manifolds proposed by Smith is made explicit on the Grassmann manifold. Two applications - computing an invariant subspace of a matrix and the mean of subspaces - are worked out.

A Grassman-Rayleigh quotient iteration for computing invariant subspaces

The classical Rayleigh Quotient Iteration (RQI) computes a 1-dimensional invariant subspace of a symmetric matrix A with cubic convergence. We propose a generalization of the RQI which computes a p-dimensional invariant subspace of A. The geometry of the algorithm on the Grassmann manifold Gr(p,n) is developed to show cubic convergence and to draw connections with recently proposed Newton algorithms on Riemannian manifolds.

Manopt, a matlab toolbox for optimization on manifolds

Optimization on manifolds is a rapidly developing branch of nonlinear optimization. Its focus is on problems where the smooth geometry of the search space can be leveraged to design effcient numerical algorithms. In particular, optimization on manifolds is well-suited to deal with rank and orthogonality constraints. Such structured constraints appear pervasively in machine learning applications, including low-rank matrix completion, sensor network localization, camera network registration, independent component analysis, metric learning, dimensionality reduction and so on. The Manopt toolbox, available at www.manopt.org, is a user-friendly, documented piece of software dedicated to simplify experimenting with state of the art Riemannian optimization algorithms. By dealing internally with most of the differential geometry, the package aims particularly at lowering the entrance barrier. © 2014 Nicolas Boumal.

Fixed-rank matrix factorizations and Riemannian low-rank optimization

Motivated by the problem of learning a linear regression model whose parameter is a large fixed-rank non-symmetric matrix, we consider the optimization of a smooth cost function defined on the set of fixed-rank matrices. We adopt the geometric framework of optimization on Riemannian quotient manifolds. We study the underlying geometries of several well-known fixed-rank matrix factorizations and then exploit the Riemannian quotient geometry of the search space in the design of a class of gradient descent and trust-region algorithms. The proposed algorithms generalize our previous results on fixed-rank symmetric positive semidefinite matrices, apply to a broad range of applications, scale to high-dimensional problems, and confer a geometric basis to recent contributions on the learning of fixed-rank non-symmetric matrices. We make connections with existing algorithms in the context of low-rank matrix completion and discuss the usefulness of the proposed framework. Numerical experiments suggest that the proposed algorithms compete with state-of-the-art algorithms and that manifold optimization offers an effective and versatile framework for the design of machine learning algorithms that learn a fixed-rank matrix. © 2013 Springer-Verlag Berlin Heidelberg.

Probabilistic graphical models for semi-supervised traffic classification

Traffic classification using machine learning continues to be an active research area. The majority of work in this area uses off-the-shelf machine learning tools and treats them as black-box classifiers. This approach turns all the modelling complexity into a feature selection problem. In this paper, we build a problem-specific solution to the traffic classification problem by designing a custom probabilistic graphical model. Graphical models are a modular framework to design classifiers which incorporate domain-specific knowledge. More specifically, our solution introduces semi-supervised learning which means we learn from both labelled and unlabelled traffic flows. We show that our solution performs competitively compared to previous approaches while using less data and simpler features. Copyright © 2010 ACM.

The effect of inclined soil layers on surface vibration from underground railways using a semi-analytical approach

Ground vibration due to underground railways is a significant source of disturbance for people living or working near the subways. The numerical models used to predict vibration levels have inherent uncertainty which must be understood to give confidence in the predictions. A semi-analytical approach is developed herein to investigate the effect of soil layering on the surface vibration of a halfspace where both soil properties and layer inclination angles are varied. The study suggests that both material properties and inclination angle of the layers have significant effect ( ± 10dB) on the surface vibration response. © 2009 IOP Publishing Ltd.

Semi-insulating layer for novel high voltage polysilicon thin film transistors

This work describes the deposition and characterisation of semi-insulating oxygen-doped silicon films for the development of high voltage polycrystalline silicon (poly-Si) circuitry on glass. The performance of a novel poly-Si High Voltage Thin Film Transistor (HVTFT) structure, incorporating a layer of semi-insulating material, has been investigated using a two dimensional device simulator. The semi-insulating layer increases the operating voltage of the HVTFT structure by linearising the potential distribution in the device offset region. A glass compatible semi-insulating layer, suitable for HVTFT applications, has been deposited by the Plasma Enhanced Chemical Vapour Deposition (PECVD) technique from silane (SiH4), nitrous oxide (N2O) and helium (He) gas mixtures. The as-deposited films are furnace annealed at 600°C which is the maximum process temperature. By varying the N2O/SiH4 ratio the conductivity of the annealed films can be accurately controlled up to a maximum of around 10-7 Ω-1cm-1. Helium dilution of the reactant gases improves both film uniformity and reproducibility. Raman analysis shows the as-deposited and annealed films to be completely amorphous. A model for the microstructure of these Semi-Insulating Amorphous Oxygen-Doped Silicon (SIAOS) films is proposed to explain the observed physical and electrical properties.

Low-temperature (≤600 °C) semi-insulating oxygen-doped silicon films by the PECVD technique for large-area power applications

This work describes the deposition, annealing and characterisation of semi-insulating oxygen-doped silicon films at temperatures compatible with polysilicon circuitry on glass. The semi-insulating layers are deposited by the plasma enhanced chemical vapour deposition technique from silane (SiH4), nitrous oxide (N2O) and helium (He) gas mixtures at a temperature of 350 °C. The as-deposited films are then furnace annealed at 600 °C which is the maximum process temperature. Raman analysis shows the as-deposited and annealed films to be completely amorphous. The most important deposition variable is the N2O SiH4 gas ratio. By varying the N2O SiH4 ratio the conductivity of the annealed films can be accurately controlled, for the first time, down to a minimum of ≈10-7Ω-1cm-1 where they exhibit a T -1 4 temperature dependence indicative of a hopping conduction mechanism. Helium dilution of the reactant gases is shown to improve both film uniformity and reproducibility. A model for the microstructure of these semi-insulating amorphous oxygen-doped silicon films is proposed to explain the observed physical and electrical properties. © 1995.

Low temperature (≤ 600°C) semi-insulating oxygen-doped silicon films by the PECVD technique for large area power applications

This work describes the annealing and characterisation of semi-insulating oxygen-doped silicon films deposited by the Plasma Enhanced Chemical Vapour Deposition (PECVD) technique from silane (SiH4), nitrous oxide (N2O) and helium (He) gas mixtures. The maximum process temperature is chosen to be compatible with large area polycrystalline silicon (poly-Si) circuitry on glass. The most important deposition variable is shown to be the N2O SiH4 gas ratio. Helium dilution results in improved film uniformity and reproducibility. Raman analysis shows the 'as-deposited' and annealed films to be completely amorphous. A model for the microstructure of these Semi-Insulating Amorphous Oxygen-doped Silicon (SIAOS) films is proposed to explain the observed physical and electrical properties. © 1995.

Semi-analytic slope delay model for CMOS switch-level timing verification

A novel slope delay model for CMOS switch-level timing verification is presented. It differs from conventional methods in being semianalytic in character. The model assumes that all input waveforms are trapezoidal in overall shape, but that they vary in their slope. This simplification is quite reasonable and does not seriously affect precision, but it facilitates rapid solution. The model divides the stages in a switch-level circuit into two types. One corresponds to the logic gates, and the other corresponds to logic gates with pass transistors connected to their outputs. Semianalytic modeling for both cases is discussed.