Convergence Thresholds of Newton's Method for Monotone Polynomial Equations

Monotone systems of polynomial equations (MSPEs) are systems of fixed-point equations $X_1 = f_1(X_1, ..., X_n),$ $..., X_n = f_n(X_1, ..., X_n)$ where each $f_i$ is a polynomial with positive real coefficients. The question of computing the least non-negative solution of a given MSPE $\vec X = \vec f(\vec X)$ arises naturally in the analysis of stochastic models such as stochastic context-free grammars, probabilistic pushdown automata, and back-button processes. Etessami and Yannakakis have recently adapted Newton's iterative method to MSPEs. In a previous paper we have proved the existence of a threshold $k_{\vec f}$ for strongly connected MSPEs, such that after $k_{\vec f}$ iterations of Newton's method each new iteration computes at least 1 new bit of the solution. However, the proof was purely existential. In this paper we give an upper bound for $k_{\vec f}$ as a function of the minimal component of the least fixed-point $\mu\vec f$ of $\vec f(\vec X)$. Using this result we show that $k_{\vec f}$ is at most single exponential resp. linear for strongly connected MSPEs derived from probabilistic pushdown automata resp. from back-button processes. Further, we prove the existence of a threshold for arbitrary MSPEs after which each new iteration computes at least $1/w2^h$ new bits of the solution, where $w$ and $h$ are the width and height of the DAG of strongly connected components.Comment: version 2 deposited February 29, after the end of the STACS conference. Two minor mistakes correcte

A new iterative procedure for removing impulse noise.

by Hu, Chen.Thesis (M.Phil.)--Chinese University of Hong Kong, 2004.Includes bibliographical references (leaves 36-39).Abstracts in English and Chinese.Chapter 1 --- Introduction --- p.1Chapter 1.1 --- Noise Model --- p.1Chapter 1.1.1 --- Impulse Noise --- p.1Chapter 1.2 --- Removing Impulse Noise --- p.2Chapter 1.2.1 --- Nonlinear Filter --- p.3Chapter 1.2.2 --- Variational Method --- p.4Chapter 1.3 --- Organization of the Dissertation --- p.5Chapter 2 --- Review of ACWMF and DPVM --- p.7Chapter 2.1 --- Review of ACWMF --- p.7Chapter 2.2 --- Review of DPVM --- p.9Chapter 2.2.1 --- Minimization Scheme --- p.9Chapter 3 --- Two-Phase Iterative Method --- p.12Chapter 3.1 --- Introduction --- p.12Chapter 3.2 --- Two-Phase Scheme --- p.13Chapter 3.2.1 --- Detection Phase --- p.13Chapter 3.2.2 --- Restoration Phase --- p.13Chapter 3.2.3 --- Summary of the Algorithm --- p.14Chapter 4 --- Nonlinear Equation Solver --- p.16Chapter 4.1 --- Introduction --- p.16Chapter 4.2 --- Newton's Method --- p.17Chapter 4.2.1 --- Newton's Method --- p.17Chapter 4.2.2 --- Order of Convergence --- p.17Chapter 4.3 --- Secant Method --- p.19Chapter 4.3.1 --- Secant Method --- p.19Chapter 4.3.2 --- Order of Convergence --- p.19Chapter 4.4 --- Secant-like Method --- p.21Chapter 4.4.1 --- Secant-like Method --- p.21Chapter 4.4.2 --- Order of Convergence --- p.24Chapter 5 --- Numerical Experiments --- p.27Chapter 5.1 --- Removing Noise --- p.27Chapter 5.2 --- Complexity of Algorithm --- p.33Chapter 6 --- Concluding Remarks --- p.35Bibliography --- p.3

Hybrid Poissoflolynomial Objective Functions for Tomographic Image Reconstruction from Transmission Scans

This paper describes rapidly converging algorithms for computing attenuation maps from Poisson transmission measurements using penalized-likelihood objective functions. We demonstrate that an under-relaxed cyclic coordinate-ascent algorithm converges faster than the convex algorithm of Lange (see ibid., vol.4, no.10, p.1430-1438, 1995), which in turn converges faster than the expectation-maximization (EM) algorithm for transmission tomography. To further reduce computation, one could replace the log-likelihood objective with a quadratic approximation. However, we show with simulations and analysis that the quadratic objective function leads to biased estimates for low-count measurements. Therefore we introduce hybrid Poisson/polynomial objective functions that use the exact Poisson log-likelihood for detector measurements with low counts, but use computationally efficient quadratic or cubic approximations for the high-count detector measurements. We demonstrate that the hybrid objective functions reduce computation time without increasing estimation bias.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/86023/1/Fessler100.pd

Learning with matrix factorizations

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2004.Includes bibliographical references (p. 157-164).Matrices that can be factored into a product of two simpler matrices can serve as a useful and often natural model in the analysis of tabulated or high-dimensional data. Models based on matrix factorization (Factor Analysis, PCA) have been extensively used in statistical analysis and machine learning for over a century, with many new formulations and models suggested in recent years (Latent Semantic Indexing, Aspect Models, Probabilistic PCA, Exponential PCA, Non-Negative Matrix Factorization and others). In this thesis we address several issues related to learning with matrix factorizations: we study the asymptotic behavior and generalization ability of existing methods, suggest new optimization methods, and present a novel maximum-margin high-dimensional matrix factorization formulation.by Nathan Srebro.Ph.D

Dynamical Systems; Proceedings of an IIASA Workshop, Sopron, Hungary, September 9-13, 1985

The investigation of special topics in systems dynamics -- uncertain dynamic processes, viability theory, nonlinear dynamics in models for biomathematics, inverse problems in control systems theory -- has become a major issue at the System and Decision Sciences Research Program of IIASA. The above topics actually reflect two different perspectives in the investigation of dynamic processes. The first, motivated by control theory, is concerned with the properties of dynamic systems that are stable under variations in the systems' parameters. This allows us to specify classes of dynamic systems for which it is possible to construct and control a whole "tube" of trajectories assigned to a system with uncertain parameters and to resolve some inverse problems of control theory within numerically stable solution schemes. The second perspective is to investigate generic properties of dynamic systems that are due to nonlinearity (as bifurcations theory, chaotic behavior, stability properties, and related problems in the qualitative theory of differential systems). Special stress is given to the applications of nonlinear dynamic systems theory to biomathematics and ecology. The proceedings of a workshop on the "Mathematics of Dynamic Processes", dealing with these topics is presented in this volume

Function estimation of irregularly spaced data with long memory dependence

We examine the problem of estimating an underlying function from collected data. The methods considered include parametric regression, density estimation, kernel estimation, wavelet regression, and specific results from when our underlying function f\left(x\right) is a member of the Besov or the Triebel spaces. Then we consider the problem of long memory error in several settings, including data which is equally spaced, data which is unequally spaced, and data which is a member of the Holder class and several other spaces. Ultimately we focus on three different problems. The first involves using linear interpolation or local averaging to account for the problem of irregularly spaced data. The second involves using a function H to reorder the data in a more general space. The third involves solving the problem in the matrix setting and considers the use of penalty functions. This method leads to general equations which describe the Mean Square Error in terms of Oracle risk. All three of these problems attempt to bound the Mean Integrated Square Error when the data is subject to long memory error

Mean field modelling of human EEG: application to epilepsy

Aggregated electrical activity from brain regions recorded via an electroencephalogram (EEG), reveal that the brain is never at rest, producing a spectrum of ongoing oscillations that change as a result of different behavioural states and neurological conditions. In particular, this thesis focusses on pathological oscillations associated with absence seizures that typically affect 2–16 year old children. Investigation of the cellular and network mechanisms for absence seizures studies have implicated an abnormality in the cortical and thalamic activity in the generation of absence seizures, which have provided much insight to the potential cause of this disease. A number of competing hypotheses have been suggested, however the precise cause has yet to be determined. This work attempts to provide an explanation of these abnormal rhythms by considering a physiologically based, macroscopic continuum mean-field model of the brain's electrical activity. The methodology taken in this thesis is to assume that many of the physiological details of the involved brain structures can be aggregated into continuum state variables and parameters. The methodology has the advantage to indirectly encapsulate into state variables and parameters, many known physiological mechanisms underlying the genesis of epilepsy, which permits a reduction of the complexity of the problem. That is, a macroscopic description of the involved brain structures involved in epilepsy is taken and then by scanning the parameters of the model, identification of state changes in the system are made possible. Thus, this work demonstrates how changes in brain state as determined in EEG can be understood via dynamical state changes in the model providing an explanation of absence seizures. Furthermore, key observations from both the model and EEG data motivates a number of model reductions. These reductions provide approximate solutions of seizure oscillations and a better understanding of periodic oscillations arising from the involved brain regions. Local analysis of oscillations are performed by employing dynamical systems theory which provide necessary and sufficient conditions for their appearance. Finally local and global stability is then proved for the reduced model, for a reduced region in the parameter space. The results obtained in this thesis can be extended and suggestions are provided for future progress in this area

Changing Choices

Changing choices psychological relativity theory unifying theory transformation parameters psychology psygologie koornstra choice dynamics The book contains a unifying theory on how the common object space is metrically transformed by individuals with different transformation parameters, due to their other previous experiences, to individually different psychological spaces for judgment on the one hand and preference on the other hand. Individual experiences also change generally, whereby the psychological spaces also change generally for each individual. The theory, therefore, is a psychological relativity theory of perception, judgment, preference, and choice dynamics. This book is a must read for all behavioural, economic, and social scientists with theoretical interest and some understanding of multidimensional data analyses. It integrates more than twenty theories on perception, judgment, preference, and risk decisions into one mathematical theory. Knowledge of advanced mathematics and modern geometry is not needed, because the mathematical subsections can be skipped without loss of understanding, due to their explanation and illustration by figures in the text