arXiv subjects

Deirel Paz-Linares

Publications and source records attributed to Deirel Paz-Linares.

6 recordsLinked to original sources

arXiv2023

Multivariate Intrinsic Local Polynomial Regression on Isometric Riemannian Manifolds: Applications to Positive Definite Data

The paper introduces a novel non-parametric Riemannian regression method using Isometric Riemannian Manifolds (IRMs). The proposed technique, Intrinsic Local Polynomial Regression on IRMs (ILPR-IRMs), enables global data mapping between Riemannian manifolds while preserving underlying geometries. The ILPR method is generalized to handle multivariate covariates on any Riemannian manifold and isometry. Specifically, for manifolds equipped with Euclidean Pullback Metrics (EPMs), a closed analytical formula is derived for the multivariate ILPR (ILPR-EPM). Asymptotic statistical properties of the ILPR-EPM for the multivariate local linear case are established, including a formula for the asymptotic bias, establishing estimator consistency. The paper showcases possible applications of the method by focusing on a group of Riemannian metrics on the Symmetric Positive Definite (SPD) manifold, which arises in machine learning and neuroscience. It is demonstrated that several metrics on the SPD manifold are EPMs, resulting in a closed analytical expression for the multivariate ILPR estimator on the SPD manifold. The paper evaluates the ILPR estimator's performance under two specific EPMs, Log-Cholesky and Log-Euclidean, on simulated data on the SPD manifold and compares it with extrinsic LPR using the Affine-Invariant when scaling the manifold and covariate dimension. The results show that the ILPR using the Log-Cholesky metric is computationally faster and provides a better trade-off between error and time than other metrics. Finally, the Log-Cholesky metric on the SPD manifold is employed to implement an efficient and intrinsic version of Rie-SNE for visualizing high-dimensional SPD data. The code for implementing ILPR-EPMs and other relevant calculations is available on the GitHub page.

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arXiv2022

Identification of oscillatory brain networks with Hidden Gaussian Graphical Spectral models of EEG/MEG

Identifying the functional networks underpinning indirectly observed processes poses an inverse problem for neurosciences or other fields. A solution of such inverse problems estimates as a first step the activity emerging within functional networks from EEG or MEG data. These EEG or MEG estimates are a direct reflect functional brain network activity with a temporal resolution that no other in vivo neuroimage may provide. A second step estimating functional connectivity from such activity pseudodata unveil the oscillatory brain networks that strongly correlate with all cognition and behavior. Simulations of such MEG or EEG inverse problem also reveal estimation errors of the functional connectivity determined by any of the state-of-the-art inverse solutions. We disclose a significant cause of estimation errors originating from misspecification of the functional network model incorporated into either inverse solution steps. We introduce the Bayesian identification of a Hidden Gaussian Graphical Spectral (HIGGS) model specifying such oscillatory brain networks model. In human EEG alpha rhythm simulations estimation errors measured as ROC performance do not surpass 2% in our HIGGS inverse solution and reach 20% in state-of-the-art methods. Macaque simultaneous EEG/ECoG recordings provide experimental confirmation for our inverse-solution with 1/3 more congruence according to Riemannian distances than state-of-the-art methods.

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arXiv2022

FKreg: A MATLAB toolbox for fast Multivariate Kernel Regression

Kernel smooth is the most fundamental technique for data density and regression estimation. However, time-consuming is the biggest obstacle for the application that the direct evaluation of kernel smooth for $N$ samples needs ${O}\left( {{N}^{2}} \right)$ operations. People have developed fast smooth algorithms using the idea of binning with FFT. Unfortunately, the accuracy is not controllable, and the implementation for multivariable and its bandwidth selection for the fast method is not available. Hence, we introduce a new MATLAB toolbox for fast multivariate kernel regression with the idea of non-uniform FFT (NUFFT), which implemented the algorithm for $M$ gridding points with ${O}\left( N+M\log M \right)$ complexity and accuracy controllability. The bandwidth selection problem utilizes the Fast Monte-Carlo algorithm to estimate the degree of freedom (DF), saving enormous cross-validation time even better when data share the same grid space for multiple regression. Up to now, this is the first toolbox for fast-binning high-dimensional kernel regression. Moreover, the estimation for local polynomial regression, the conditional variance for the heteroscedastic model, and the complex-valued datasets are also implemented in this toolbox. The performance is demonstrated with simulations and an application on the quantitive EEG.

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arXiv2019

Caulking the Leakage Effect in MEEG Source Connectivity Analysis

Simplistic estimation of neural connectivity in MEEG sensor space is impossible due to volume conduction. The only viable alternative is to carry out connectivity estimation in source space. Among the neuroscience community this is claimed to be impossible or misleading due to Leakage: linear mixing of the reconstructed sources. To address this problematic we propose a novel solution method that caulks the Leakage in MEEG source activity and connectivity estimates: BC-VARETA. It is based on a joint estimation of source activity and connectivity in the frequency domain representation of MEEG time series. To achieve this, we go beyond current methods that assume a fixed gaussian graphical model for source connectivity. In contrast we estimate this graphical model in a Bayesian framework by placing priors on it, which allows for highly optimized computations of the connectivity, via a new procedure based on the local quadratic approximation under quite general prior models. A further contribution of this paper is the rigorous definition of leakage via the Spatial Dispersion Measure and Earth Movers Distance based on the geodesic distances over the cortical manifold. Both measures are extended for the first time to quantify Connectivity Leakage by defining them on the cartesian product of cortical manifolds. Using these measures, we show that BC-VARETA outperforms most state of the art inverse solvers by several orders of magnitude.

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arXiv2019

Third Generation MEEG Source Connectivity Analysis Toolbox (BC-VARETA 1.0) and Validation Benchmark

This paper presents a new toolbox for MEEG source activity and connectivity estimation: Brain Connectivity Variable Resolution Tomographic Analysis version 1.0 (BC-VARETA 1.0). It relies on the third generation of nonlinear methods for the analysis of resting state MEEG Time Series. Into the state of the art of MEEG analysis, the methodology underlying our tool (BC-VARETA) brings out several assets. First: Constitutes a Bayesian Identification approach of Linear Dynamical Systems in the Frequency Domain, grounded in more consistent models (third generation). Second: Achieves Super-Resolution, through the iterative solution of a Sparse Hermitian Source Graphical Model. Third: Tackles efficiently in High Dimensional and Complex set up the estimation of connectivity. Fourth: Incorporates priors at the connectivity level by penalizing the groups of variables, corresponding to the Gray Matter anatomical segmentation, and including a probability mask of the anatomically plausible connections. Along with the implementation of our method, we include in this toolbox a benchmark for the validation of MEEG source analysis methods, that would serve for the evaluation of sophisticated methodologies (third generation). It incorporates two elements. First: A realistic simulation framework, for the generation of MEEG synthetic data, given an underlying source connectivity structure. Second: Sensitive quality measures that allow for a reliable evaluation of the source activity and connectivity reconstruction performance, based on the Spatial Dispersion and Earth Movers Distance, in both source and connectivity space.

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arXiv2016

Empirical bayes formulation of the elastic net and mixed-norm models: application to the eeg inverse problem

The estimation of EEG generating sources constitutes an Inverse Problem (IP) in Neuroscience. This is an ill-posed problem, due to the non-uniqueness of the solution, and many kinds of prior information have been used to constrain it. A combination of smoothness (L2 norm-based) and sparseness (L1 norm-based) constraints is a flexible approach that have been pursued by important examples such as the Elastic Net (ENET) and mixed-norm (MXN) models. The former is used to find solutions with a small number of smooth non-zero patches, while the latter imposes sparseness and smoothness simultaneously along different dimensions of the spatio-temporal matrix solutions. Both models have been addressed within the penalized regression approach, where the regularization parameters are selected heuristically, leading usually to non-optimal solutions. The existing Bayesian formulation of ENET allows hyperparameter learning, but using computationally intensive Monte Carlo/Expectation Maximization methods. In this work we attempt to solve the EEG IP using a Bayesian framework for models based on mixtures of L1/L2 norms penalization functions (Laplace/Normal priors) such as ENET and MXN. We propose a Sparse Bayesian Learning algorithm based on combining the Empirical Bayes and the iterative coordinate descent procedures to estimate both the parameters and hyperparameters. Using simple but realistic simulations we found that our methods are able to recover complicated source setups more accurately and with a more robust variable selection than the ENET and LASSO solutions using classical algorithms. We also solve the EEG IP using data coming from a visual attention experiment, finding more interpretable neurophysiological patterns with our methods, as compared with other known methods such as LORETA, ENET and LASSO FUSION using the classical regularization approach.

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