A method for the solution of certain non – linear problems in least squares
01 Jul 1944-Quarterly of Applied Mathematics (QUARTERLY OF APPLIED MATHEMATICS)-Vol. 2, Iss: 2, pp 164-168
TL;DR: In this article, the problem of least square problems with non-linear normal equations is solved by an extension of the standard method which insures improvement of the initial solution, which can also be considered an extension to Newton's method.
Abstract: The standard method for solving least squares problems which lead to non-linear normal equations depends upon a reduction of the residuals to linear form by first order Taylor approximations taken about an initial or trial solution for the parameters.2 If the usual least squares procedure, performed with these linear approximations, yields new values for the parameters which are not sufficiently close to the initial values, the neglect of second and higher order terms may invalidate the process, and may actually give rise to a larger value of the sum of the squares of the residuals than that corresponding to the initial solution. This failure of the standard method to improve the initial solution has received some notice in statistical applications of least squares3 and has been encountered rather frequently in connection with certain engineering applications involving the approximate representation of one function by another. The purpose of this article is to show how the problem may be solved by an extension of the standard method which insures improvement of the initial solution.4 The process can also be used for solving non-linear simultaneous equations, in which case it may be considered an extension of Newton's method. Let the function to be approximated be h{x, y, z, • • • ), and let the approximating function be H{oc, y, z, • • ■ ; a, j3, y, ■ • ■ ), where a, /3, 7, • ■ ■ are the unknown parameters. Then the residuals at the points, yit zit • • • ), i = 1, 2, ■ • • , n, are
Citations
More filters
Book•
[...]
TL;DR: Deep learning as mentioned in this paper is a form of machine learning that enables computers to learn from experience and understand the world in terms of a hierarchy of concepts, and it is used in many applications such as natural language processing, speech recognition, computer vision, online recommendation systems, bioinformatics, and videogames.
Abstract: Deep learning is a form of machine learning that enables computers to learn from experience and understand the world in terms of a hierarchy of concepts. Because the computer gathers knowledge from experience, there is no need for a human computer operator to formally specify all the knowledge that the computer needs. The hierarchy of concepts allows the computer to learn complicated concepts by building them out of simpler ones; a graph of these hierarchies would be many layers deep. This book introduces a broad range of topics in deep learning. The text offers mathematical and conceptual background, covering relevant concepts in linear algebra, probability theory and information theory, numerical computation, and machine learning. It describes deep learning techniques used by practitioners in industry, including deep feedforward networks, regularization, optimization algorithms, convolutional networks, sequence modeling, and practical methodology; and it surveys such applications as natural language processing, speech recognition, computer vision, online recommendation systems, bioinformatics, and videogames. Finally, the book offers research perspectives, covering such theoretical topics as linear factor models, autoencoders, representation learning, structured probabilistic models, Monte Carlo methods, the partition function, approximate inference, and deep generative models. Deep Learning can be used by undergraduate or graduate students planning careers in either industry or research, and by software engineers who want to begin using deep learning in their products or platforms. A website offers supplementary material for both readers and instructors.
38,208 citations
Book•
01 Jan 1995TL;DR: This is the first comprehensive treatment of feed-forward neural networks from the perspective of statistical pattern recognition, and is designed as a text, with over 100 exercises, to benefit anyone involved in the fields of neural computation and pattern recognition.
Abstract: From the Publisher:
This is the first comprehensive treatment of feed-forward neural networks from the perspective of statistical pattern recognition. After introducing the basic concepts, the book examines techniques for modelling probability density functions and the properties and merits of the multi-layer perceptron and radial basis function network models. Also covered are various forms of error functions, principal algorithms for error function minimalization, learning and generalization in neural networks, and Bayesian techniques and their applications. Designed as a text, with over 100 exercises, this fully up-to-date work will benefit anyone involved in the fields of neural computation and pattern recognition.
19,056 citations
Cites methods from "A method for the solution of certai..."
...61) we arrive at the Levenberg-Marquardt approximation (Levenberg, 1944; Marquardt, 1963) or outer product approximation (since the Hessian matrix is built up from a sum of outer products of vectors), given by...
[...]
...In the Levenberg-Marquardt algorithm (Levenberg, 1944; Marquardt, 1963), this problem is addressed by seeking to minimize the error function while at the same time trying to keep the step size small so as to ensure that the linear approximation remains valid....
[...]
Book•
01 Nov 2008TL;DR: Numerical Optimization presents a comprehensive and up-to-date description of the most effective methods in continuous optimization, responding to the growing interest in optimization in engineering, science, and business by focusing on the methods that are best suited to practical problems.
Abstract: Numerical Optimization presents a comprehensive and up-to-date description of the most effective methods in continuous optimization. It responds to the growing interest in optimization in engineering, science, and business by focusing on the methods that are best suited to practical problems. For this new edition the book has been thoroughly updated throughout. There are new chapters on nonlinear interior methods and derivative-free methods for optimization, both of which are used widely in practice and the focus of much current research. Because of the emphasis on practical methods, as well as the extensive illustrations and exercises, the book is accessible to a wide audience. It can be used as a graduate text in engineering, operations research, mathematics, computer science, and business. It also serves as a handbook for researchers and practitioners in the field. The authors have strived to produce a text that is pleasant to read, informative, and rigorous - one that reveals both the beautiful nature of the discipline and its practical side.
17,420 citations
TL;DR: This historical survey compactly summarizes relevant work, much of it from the previous millennium, review deep supervised learning, unsupervised learning, reinforcement learning & evolutionary computation, and indirect search for short programs encoding deep and large networks.
14,635 citations
Additional excerpts
...Least-squares methods (Gauss– Newton, Levenberg–Marquardt) (Gauss, 1809; Levenberg, 1944; Marquardt, 1963; Newton, 1687; Schaback & Werner, 1992) and quasi-Newtonmethods (Broyden–Fletcher–Goldfarb–Shanno, BFGS) (Broyden et al., 1965; Fletcher & Powell, 1963; Goldfarb, 1970; Shanno, 1970) are…...
[...]
TL;DR: A number of theorems are proved to show that it always converges and that it converges rapidly, and this method has been used to solve a system of one hundred non-linear simultaneous equations.
Abstract: © The British Computer Society Issue Section: Articles Download all figures A powerful iterative descent method for finding a local minimum of a function of several variables is described. A number of theorems are proved to show that it always converges and that it converges rapidly. Numerical tests on a variety of functions confirm these theorems. The method has been used to solve a system of one hundred non-linear simultaneous equations. Related articles in Web of Science
4,305 citations