Inverse problems in machine learning: An application to brain activity interpretation

Estimating vocal tract geometry from acoustic impedance using deep neural network

A data-driven approach using artificial neural networks is proposed to address the classic inverse area function problem, i.e., to determine the vocal tract geometry (modelled as a tube of nonuniform cylindrical cross-sections) from the vocal tract acoustic impedance spectrum. The predicted cylindrical radii and the actual radii were found to have high correlation in the three- and four-cylinder model (Pearson coefficient (ρ) and Lin concordance coefficient (ρ_c) exceeded 95%); however, for the six-cylinder model, the correlation was low (ρ around 75% and ρ_c around 69%). Upon standardizing the impedance value, the correlation improved significantly for all cases (ρ and ρ_c exceeded 90%).

Robust parameter estimation based on the generalized log-likelihood in the context of Sharma-Taneja-Mittal measure

The problem of obtaining physical parameters that cannot be directly measured from observed data arises in several scientific fields. In the classic approach, the well-known maximum likelihood estimation associated with a Gaussian distribution is employed to obtain the model parameters of a complex system. Although this approach is quite popular in statistical physics, only a handful of spurious observations (outliers) make this approach ineffective, violating the Gauss-Markov theorem. In this work, starting from the generalized logarithmic function associated to the Sharma-Taneja-Mittal (STM) information measure, we propose an outlier-resistant approach based on the generalized log-likelihood estimation. In particular, our proposal deforms the Gaussian distribution based on a two-parameter generalization of the ordinary logarithmic function. We have tested the effectiveness of our proposal considering a classic geophysical inverse problem with a very noisy data set. The results show that the task of obtaining physical parameters based on the STM measure from noisy data with several outliers outperforms the classic approach, and therefore, our proposal is a useful tool for statistical physics, information theory, and statistical inference problems.

Tsallis Entropy, Likelihood, and the Robust Seismic Inversion

The nonextensive statistical mechanics proposed by Tsallis have been successfully used to model and analyze many complex phenomena. Here, we study the role of the generalized Tsallis statistics on the inverse problem theory. Most inverse problems are formulated as an optimisation problem that aims to estimate the physical parameters of a system from indirect and partial observations. In the conventional approach, the misfit function that is to be minimized is based on the least-squares distance between the observed data and the modelled data (residuals or errors), in which the residuals are assumed to follow a Gaussian distribution. However, in many real situations, the error is typically non-Gaussian, and therefore this technique tends to fail. This problem has motivated us to study misfit functions based on non-Gaussian statistics. In this work, we derive a misfit function based on the q-Gaussian distribution associated with the maximum entropy principle in the Tsallis formalism. We tested our method in a typical geophysical data inverse problem, called post-stack inversion (PSI), in which the physical parameters to be estimated are the Earth’s reflectivity. Our results show that the PSI based on Tsallis statistics outperforms the conventional PSI, especially in the non-Gaussian noisy-data case.

A regularization algorithm for decoding perceptual temporal profiles from fMRI data

In several biomedical fields, researchers are faced with regression problems that can be stated as Statistical Learning problems. One example is given by decoding brain states from functional magnetic resonance imaging (fMRI) data. Recently, it has been shown that the general Statistical Learning problem can be restated as a linear inverse problem. Hence, new algorithms were proposed to solve this inverse problem in the context of Reproducing Kernel Hilbert Spaces. In this paper, we detail one iterative learning algorithm belonging to this class, called ν-method, and test its effectiveness in a between-subjects regression framework. Specifically, our goal was to predict the perceived pain intensity based on fMRI signals, during an experimental model of acute prolonged noxious stimulation. We found that, using a linear kernel, the psychophysical time profile was well reconstructed, while pain intensity was in some cases significantly over/underestimated. No substantial differences in terms of accuracy were found between the proposed approach and one of the state-of-the-art learning methods, the Support Vector Machines. Nonetheless, adopting the ν-method yielded a significant reduction in computational time, an advantage that became more evident when a relevant feature selection procedure was implemented. The ν-method can be easily extended and included in typical approaches for binary or multiple classification problems, and therefore it seems well-suited to build effective brain activity estimators.