For (3), we relax our proposed test using techniques from robust statistics and imprecise probabilities.

We prove in this work that the well-known lasso problem can be solved exactly without homotopy using novel differential inclusions techniques. Specifically, we show that a selection principle from the theory of differential inclusions transforms the dual lasso problem into the problem of calculating the trajectory of a projected dynamical system that we prove is integrable. Our analysis yields an exact algorithm for the lasso problem, numerically up to machine precision, that is amenable to computing regularization paths and is very fast. Moreover, we show the continuation of solutions to the integrable projected dynamical system in terms of the hyperparameter naturally yields a rigorous homotopy algorithm. Numerical experiments confirm that our algorithm outperforms the state-of-the-art algorithms in both efficiency and accuracy. Beyond this work, we expect our results and analysis can be adapted to compute exact or approximate solutions to a broader class of polyhedral-constrained optimization problems.

For (3), we relax our proposed test using techniques from robust statistics and imprecise probabilities.

Why is the same i used here?

Predictive modeling using structural magnetic resonance imaging (MRI) data is a prominent approach to study brain-aging. Machine learning algorithms and feature extraction methods have been employed to improve predictions and explore healthy and accelerated aging e.g. neurodegenerative and psychiatric disorders. The high-dimensional MRI data pose challenges to building generalizable and interpretable models as well as for data privacy. Common practices are resampling or averaging voxels within predefined parcels, which reduces anatomical specificity and biological interpretability as voxels within a region may differently relate to aging. Effectively, naive fusion by averaging can result in information loss and reduced accuracy. We present a conceptually novel two-level stacking ensemble (SE) approach. The first level comprises regional models for predicting individuals' age based on voxel-wise information, fused by a second-level model yielding final predictions. Eight data fusion scenarios were explored using as input Gray matter volume (GMV) estimates from four datasets covering the adult lifespan. Performance, measured using mean absolute error (MAE), R2, correlation and prediction bias, showed that SE outperformed the region-wise averages. The best performance was obtained when first-level regional predictions were obtained as out-of-sample predictions on the application site with second-level models trained on independent and site-specific data (MAE=4.75 vs baseline regional mean GMV MAE=5.68). Performance improved as more datasets were used for training. First-level predictions showed improved and more robust aging signal providing new biological insights and enhanced data privacy. Overall, the SE improves accuracy compared to the baseline while preserving or enhancing data privacy.

MEG are non invasive neuroimaging techniques with excellent temporal and spatial resolution, crucial for studying brain function in dementia and Alzheimer Disease. They identify changes in brain activity at various Alzheimer stages, including preclinical and prodromal phases. MEG may detect pathological changes before clinical symptoms, offering potential biomarkers for intervention. This study evaluates classification techniques using MEG features to distinguish between healthy controls and mild cognitive impairment participants from the BioFIND study. We compare MEG based biomarkers with MRI based anatomical features, both independently and combined. We used 3 Tesla MRI and MEG data from 324 BioFIND participants;158 MCI and 166 HC. Analyses were performed using MATLAB with SPM12 and OSL toolboxes. Machine learning analyses, including 100 Monte Carlo replications of 10 fold cross validation, were conducted on sensor and source spaces. Combining MRI with MEG features achieved the best performance; 0.76 accuracy and AUC of 0.82 for GLMNET using LCMV source based MEG. MEG only analyses using LCMV and eLORETA also performed well, suggesting that combining uncorrected MEG with z-score-corrected MRI features is optimal.

Given the vast number of classifiers that have been (and continue to be) proposed, reliable methods for comparing them are becoming increasingly important. The desire for reliability is broken down into three main aspects: (1) Comparisons should allow for different quality metrics simultaneously. (2) Comparisons should take into account the statistical uncertainty induced by the choice of benchmark suite. (3) The robustness of the comparisons under small deviations in the underlying assumptions should be verifiable. To address (1), we propose to compare classifiers using a generalized stochastic dominance ordering (GSD) and present the GSD-front as an information-efficient alternative to the classical Pareto-front. For (2), we propose a consistent statistical estimator for the GSD-front and construct a statistical test for whether a (potentially new) classifier lies in the GSD-front of a set of state-of-the-art classifiers. For (3), we relax our proposed test using techniques from robust statistics and imprecise probabilities. We illustrate our concepts on the benchmark suite PMLB and on the platform OpenML.

We propose a fast method for solving compressed sensing, Lasso regression, and Logistic Lasso regression problems that iteratively runs an appropriate solver using an active set approach. We design a strategy to update the active set that achieves a large speedup over a single call of several solvers, including gradient projection for sparse reconstruction (GPSR), lassoglm of Matlab, and glmnet. For compressed sensing, the hybrid of our method and GPSR is 31.41 times faster than GPSR on average for Gaussian ensembles and 25.64 faster on average for binary ensembles. For Lasso regression, the hybrid of our method and GPSR achieves a 30.67-fold average speedup in our experiments. In our experiments on Logistic Lasso regression, the hybrid of our method and lassoglm gives an 11.95-fold average speedup, and the hybrid of our method and glmnet gives a 1.40-fold average speedup.

Ridge regression is a method we can use to fit a regression model when multicollinearity is present in the data. This second term in the equation is known as a shrinkage penalty. In ridge regression, we select a value for λ that produces the lowest possible test MSE (mean squared error). This tutorial provides a step-by-step example of how to perform ridge regression in R. For this example, we'll use the R built-in dataset called mtcars. To perform ridge regression, we'll use functions from the glmnet package.

This post shows how to use the R packages for estimating an exclusive lasso and a group lasso. These lasso variants have a given grouping order in common but differ in how this grouping constraint is functioning when a variable selection is performed. Lasso, Group Lasso, and Exclusive Lasso While LASSO (least absolute shrinkage and selection operator) has many variants and extensions, our focus is on two lasso models: Group Lasso and Exclusive Lasso. Before we dive into the specifics, let's go over the similarities and differences of these two lasso variants from the following figure. In the above figure, 15 variables are categorized into 5 groups.