Statistics Seminar - 01/29/21

Abstract: Efficient recovery of a low-dimensional structure from high-dimensional data has been pursued in various settings including wavelet denoising, generalized linear models and low-rank matrix estimation. It is also present in the sparse regularization of the active neurons for each hidden layer in deep learning architectures. By thresholding some parameters to zero, thresholding estimators (such as lasso) perform variable selection and can be employed whether the model includes more parameters than observations (high dimensional data) or not. Based on thresholding estimators, we derive a new class of statistical tests for high dimensional generalized linear models. We show through simulations that our tests have better control of the nominal level and higher power than existing tests. Second, we propose the Quantile Universal Threshold (QUT) to choose the thresholding parameter that governs the amount of regularization in thresholding estimators. QUT is at the detection edge and numerical results show the effectiveness of our approach in terms of model selection and prediction. We also use our methodologies to solve a cosmology problem that consists in recovering the 3D gas emissivity of a galaxy cluster from a 2D image taken by a telescope. We show how our methodology outperforms the current state-of-the-art approach in terms of mean squared errors, and how it has good coverage probability. Finally, we show ongoing projects with applications in geology, traffic prediction, disease control and deep learning autoencoders, using our results.