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In this article, we develop a distributed variable screening method for generalized linear models. This method is designed to handle situations where both the sample size and the number of covariates are large. Specifically, the proposed…
Expectation Maximization (EM) is among the most popular algorithms for estimating parameters of statistical models. However, EM, which is an iterative algorithm based on the maximum likelihood principle, is generally only guaranteed to find…
This work performs a non-asymptotic analysis of the generalized Lasso under the assumption of sub-exponential data. Our main results continue recent research on the benchmark case of (sub-)Gaussian sample distributions and thereby explore…
This note examines the behavior of generalization capabilities - as defined by out-of-sample mean squared error (MSE) - of Linear Gaussian (with a fixed design matrix) and Linear Least Squares regression. Particularly, we consider a…
Existing interpretation algorithms have found that, even deep models make the same and right predictions on the same image, they might rely on different sets of input features for classification. However, among these sets of features, some…
Linear structural equation models, which relate random variables via linear interdependencies and Gaussian noise, are a popular tool for modeling multivariate joint distributions. These models correspond to mixed graphs that include both…
We characterize generalized derivatives of the solution operator of the obstacle problem. This precise characterization requires the usage of the theory of so-called capacitary measures and the associated solution operators of relaxed…
Convolutional neural networks (CNNs) provide flexible function approximations for a wide variety of applications when the input variables are in the form of images or spatial data. Although CNNs often outperform traditional statistical…
The necessary and sufficient conditions for existence of a generalized representer theorem are presented for learning Hilbert space-valued functions. Representer theorems involving explicit basis functions and Reproducing Kernels are a…
Systems with both quantitative and qualitative responses are widely encountered in many applications. Design of experiment methods are needed when experiments are conducted to study such systems. Classic experimental design methods are…
We consider interpolation learning in high-dimensional linear regression with Gaussian data, and prove a generic uniform convergence guarantee on the generalization error of interpolators in an arbitrary hypothesis class in terms of the…
Equivariance of linear neural network layers is well studied. In this work, we relax the equivariance condition to only be true in a projective sense. We propose a way to construct a projectively equivariant neural network through building…
We propose a unified framework to draw inferences for regression coefficients in a generalized linear model (GLM) following Lasso-based variable selection. We adapt to non-Gaussian GLMs a recently developed parametric programming strategy…
This paper focuses on understanding how the generalization error scales with the amount of the training data for deep neural networks (DNNs). Existing techniques in statistical learning require computation of capacity measures, such as VC…
Cumulative link models have been widely used for ordered categorical responses. Uniform allocation of experimental units is commonly used in practice, but often suffers from a lack of efficiency. We consider D-optimal designs with ordered…
We consider universal inference in variance components models, focusing on settings where the parameter is near or at the boundary of the parameter set. Two cases, which are not handled by existing state-of-the-art methods, are of…
Algorithms often have tunable parameters that impact performance metrics such as runtime and solution quality. For many algorithms used in practice, no parameter settings admit meaningful worst-case bounds, so the parameters are made…
In this paper, we study the support vector machine and introduced the notion of generalized support vector machine for classification of data. We show that the problem of generalized support vector machine is equivalent to the problem of…
While deep learning is successful in a number of applications, it is not yet well understood theoretically. A satisfactory theoretical characterization of deep learning however, is beginning to emerge. It covers the following questions: 1)…
The generalization mystery in deep learning is the following: Why do over-parameterized neural networks trained with gradient descent (GD) generalize well on real datasets even though they are capable of fitting random datasets of…