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Register a new userRobust and Efficient Optimization Using a Marquardt-Levenberg Algorithm with R Package marqLevAlg
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Added by
Viviane Philipps
Publication date
2020
fields
Mathematical Statistics
and research's language is
English
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Implementations in R of classical general-purpose algorithms generally have two major limitations which make them unusable in complex problems: too loose convergence criteria and too long calculation time. By relying on a Marquardt-Levenberg algorithm (MLA), a Newton-like method particularly robust for solving local optimization problems, we provide with marqLevAlg package an efficient and general-purpose local optimizer which (i) prevents convergence to saddle points by using a stringent convergence criterion based on the relative distance to minimum/maximum in addition to the stability of the parameters and of the objective function; and (ii) reduces the computation time in complex settings by allowing parallel calculations at each iteration. We demonstrate through a variety of cases from the literature that our implementation reliably and consistently reaches the optimum (even when other optimizers fail), and also largely reduces computational time in complex settings through the example of maximum likelihood estimation of different sophisticated statistical models.
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The phase retrieval problem, where one aims to recover a complex-valued image from far-field intensity measurements, is a classic problem encountered in a range of imaging applications. Modern phase retrieval approaches usually rely on gradient descent methods in a nonlinear minimization framework. Calculating closed-form gradients for use in these methods is tedious work, and formulating second order derivatives is even more laborious. Additionally, second order techniques often require the storage and inversion of large matrices of partial derivatives, with memory requirements that can be prohibitive for data-rich imaging modalities. We use a reverse-mode automatic differentiation (AD) framework to implement an efficient matrix-free version of the Levenberg-Marquardt (LM) algorithm, a longstanding method that finds popular use in nonlinear least-square minimization problems but which has seen little use in phase retrieval. Furthermore, we extend the basic LM algorithm so that it can be applied for general constrained optimization problems beyond just the least-square applications. Since we use AD, we only need to specify the physics-based forward model for a specific imaging application; the derivative terms are calculated automatically through matrix-vector products, without explicitly forming any large Jacobian or Gauss-Newton matrices. We demonstrate that this algorithm can be used to solve both the unconstrained ptychographic object retrieval problem and the constrained blind ptychographic object and probe retrieval problems, under both the Gaussian and Poisson noise models, and that this method outperforms best-in-class first-order ptychographic reconstruction methods: it provides excellent convergence guarantees with (in many cases) a superlinear rate of convergence, all with a computational cost comparable to, or lower than, the tested first-order algorithms.
Although various forms of linkage map construction software are widely available, there is a distinct lack of packages for use in the R statistical computing environment. This article introduces the ASMap linkage map construction R package which contains functions that use the efficient MSTmap algorithm for clustering and optimally ordering large sets of markers. Additional to the construction functions, the package also contains a suite of tools to assist in the rapid diagnosis and repair of a constructed linkage map. The package functions can also be used for post linkage map construction techniques such as fine mapping or combining maps of the same population. To showcase the efficiency and functionality of ASMap, the complete linkage map construction process is demonstrated with a high density barley backcross marker data set.
The inverse problem in Acousto-Electric tomography concerns the reconstruction of the electric conductivity in a domain from knowledge of the power density function in the interior of the body. This interior power density results from currents prescribed at boundary electrodes (and can be obtained through electro-static boundary measurements together with auxiliary acoustic measurement. In Electrical Impedance Tomography, the complete electrode model is known to be the most accurate model for the forward modelling. In this paper, the reconstruction problem of Acousto-Electric tomography is posed using the (smooth) complete electrode model, and a Levenberg-Marquardt iteration is formulated in appropriate function spaces. This results in a system of partial differential equations to be solved in each iteration. To increase the computational efficiency and stability, a strategy based on both the complete electrode model and the continuum model with Dirichlet boundary condition is proposed. The system of equations is implemented numerically for a two dimensional scenario and the algorithm is tested on two different numerical phantoms, a heart and lung model and a human brain model. Several numerical experiments are carried out confirming the feasibility, accuracy and stability of the methods.
We introduce the UPG package for highly efficient Bayesian inference in probit, logit, multinomial logit and binomial logit models. UPG offers a convenient estimation framework for balanced and imbalanced data settings where sampling efficiency is ensured through Markov chain Monte Carlo boosting methods. All sampling algorithms are implemented in C++, allowing for rapid parameter estimation. In addition, UPG provides several methods for fast production of output tables and summary plots that are easily accessible to a broad range of users.
Recurrent event analyses have found a wide range of applications in biomedicine, public health, and engineering, among others, where study subjects may experience a sequence of event of interest during follow-up. The R package reReg (Chiou and Huang 2021) offers a comprehensive collection of practical and easy-to-use tools for regression analysis of recurrent events, possibly with the presence of an informative terminal event. The regression framework is a general scale-change model which encompasses the popular Cox-type model, the accelerated rate model, and the accelerated mean model as special cases. Informative censoring is accommodated through a subject-specific frailty without no need for parametric specification. Different regression models are allowed for the recurrent event process and the terminal event. Also included are visualization and simulation tools.
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