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Concept# Levenberg–Marquardt algorithm

Summary

In mathematics and computing, the Levenberg–Marquardt algorithm (LMA or just LM), also known as the damped least-squares (DLS) method, is used to solve non-linear least squares problems. These minimization problems arise especially in least squares curve fitting. The LMA interpolates between the Gauss–Newton algorithm (GNA) and the method of gradient descent. The LMA is more robust than the GNA, which means that in many cases it finds a solution even if it starts very far off the final minimum. For well-behaved functions and reasonable starting parameters, the LMA tends to be slower than the GNA. LMA can also be viewed as Gauss–Newton using a trust region approach.
The algorithm was first published in 1944 by Kenneth Levenberg, while working at the Frankford Army Arsenal. It was rediscovered in 1963 by Donald Marquardt, who worked as a statistician at DuPont, and independently by Girard, Wynne and

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Johanni Michael Brea, Wulfram Gerstner, Alireza Modirshanechi, Shuqi Wang

Fitting network models to neural activity is an important tool in neuroscience. A popular approach is to model a brain area with a probabilistic recurrent spiking network whose parameters maximize the likelihood of the recorded activity. Although this is widely used, we show that the resulting model does not produce realistic neural activity. To correct for this, we suggest to augment the log-likelihood with terms that measure the dissimilarity between simulated and recorded activity. This dissimilarity is defined via summary statistics commonly used in neuroscience and the optimization is efficient because it relies on back-propagation through the stochastically simulated spike trains. We analyze this method theoretically and show empirically that it generates more realistic activity statistics. We find that it improves upon other fitting algorithms for spiking network models like GLMs (Generalized Linear Models) which do not usually rely on back-propagation. This new fitting algorithm also enables the consideration of hidden neurons which is otherwise notoriously hard, and we show that it can be crucial when trying to infer the network connectivity from spike recordings.

2021Jan Sickmann Hesthaven, Stefano Ubbiali

We develop a non-intrusive reduced basis (RB) method for parametrized steady-state partial differential equations (PDEs). The method extracts a reduced basis from a collection of high-fidelity solutions via a proper orthogonal decomposition (POD) and employs artificial neural networks (ANNs), particularly multi-layer perceptrons (MLPs), to accurately approxi- mate the coefficients of the reduced model. The search for the optimal number of neurons and the minimum amount of training samples to avoid overfitting is carried out in the offline phase through an automatic routine, relying upon a joint use of the latin hypercube sampling (LHS) and the Levenberg-Marquardt training algorithm. This guarantees a complete offline-online decoupling, leading to an efficient RB method - referred to as POD-NN - suitable also for general nonlinear problems with a non-affine parametric dependence. Numerical studies are presented for the nonlinear Poisson equation and for driven cavity viscous flows, modeled through the steady incompressible Navier-Stokes equations. Both physical and geometrical parametrizations are considered. Several results confirm the accuracy of the POD-NN method and show the substantial speed-up enabled at the online stage as compared to a traditional RB strategy.

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Dalia Salem Hassan Fahmy El Badawy

Inspired by the human ability to localize sounds, even with only one ear, as well as to recognize objects using active echolocation, we investigate the role of sound scattering and prior knowledge in regularizing ill-posed inverse problems in acoustics. In particular, we study direction of arrival estimation with one microphone, acoustic imaging with a small number of microphones, and microphone array localization. Not only are these problems ill-posed but also non-convex in the variables of interest when formulated as optimization problems. To restore well-posedness, we thus use sound scattering which we construe as a physical form of regularization. We additionally use standard regularization in the form of appropriate priors on the variables. The non-convexity is then handled with tools such as linearization or semidefinite relaxation.
We begin with direction of arrival estimation. While conventional approaches require at least two microphones, we show how to estimate the direction of one or more sound sources using only one. This is made possible thanks to regularization by sound scattering which we achieve by compact structures made from LEGO that scatter the sound in a direction-dependent manner. We also impose a prior on the source spectra where we assume they can be sparsely represented in a learned dictionary. Using algorithms based on non-negative matrix factorization, we show how to use the LEGO devices and a speaker-independent dictionary to successfully localize one or two simultaneous speakers.
Next, we study acoustic imaging of 2D shapes using a small number of microphones. Unlike in echolocation where the source is known, we show how to image an unknown object using an unknown source. In this case, we enforce a prior on the object using a total variation norm penalty but no priors on the source. We also show how to use microphones embedded in the ears of a dummy head to benefit from the diversity encoded in the head-related transfer function. We then propose an algorithm to jointly reconstruct the shape of the object and the sound source spectrum. We demonstrate the effectiveness of our approach using numerical and real experiments with speech and noise sources.
Finally, the need to know the microphone positions in acoustic imaging and a number of other applications led us to study microphone localization. We assume the positions of the loudspeakers are also unknown and that all devices are not synchronized. In this case, the times of arrival from the loudspeakers to the microphones are shifted by unknown source emission times and unknown sensor capture times. We thus propose an objective that is timing-invariant allowing us to localize the setup without first having to estimate the unknown timing information. We also propose an approach to handle missing data as well as show how to include side information such as knowledge of some of the distances between the devices. We derive a semidefinite relaxation of the objective which provides a good initialization to a subsequent refinement using the Levenberg-Marquardt algorithm. Using numerical and real experiments, we show we can localize unsynchronized devices even in near-minimal configurations.