Related papers: $L_1$-norm regularized $L_1$-norm best-fit line pr…
We study the robustness properties of $\ell_1$ norm minimization for the classical linear regression problem with a given design matrix and contamination restricted to the dependent variable. We perform a fine error analysis of the $\ell_1$…
Optimization problems with $L^1$-control cost functional subject to an elliptic partial differential equation (PDE) are considered. However, different from the finite dimensional $l^1$-regularization optimization, the resulting discretized…
We describe a probabilistic, {\it sublinear} runtime, measurement-optimal system for model-based sparse recovery problems through dimensionality reducing, {\em dense} random matrices. Specifically, we obtain a linear sketch $u\in \R^M$ of a…
We consider a discrete optimization formulation for learning sparse classifiers, where the outcome depends upon a linear combination of a small subset of features. Recent work has shown that mixed integer programming (MIP) can be used to…
Problems in signal processing and medical imaging often lead to calculating sparse solutions to under-determined linear systems. Methodologies for solving this problem are presented as background to the method used in this work where the…
In this work, we consider a class of differentiable criteria for sparse image computing problems, where a nonconvex regularization is applied to an arbitrary linear transform of the target image. As special cases, it includes…
The $L_1$-regularized models are widely used for sparse regression or classification tasks. In this paper, we propose the orthant-wise passive descent algorithm (OPDA) for optimizing $L_1$-regularized models, as an improved substitute of…
Sparse coding algorithms are about finding a linear basis in which signals can be represented by a small number of active (non-zero) coefficients. Such coding has many applications in science and engineering and is believed to play an…
Principal component analysis (PCA) is a well-known linear dimension-reduction method that has been widely used in data analysis and modeling. It is an unsupervised learning technique that identifies a suitable linear subspace for the input…
In this paper, we present a novel nonlinear programming-based approach to fine-tune pre-trained neural networks to improve robustness against adversarial attacks while maintaining high accuracy on clean data. Our method introduces…
We consider the problem of learning a linear subspace from data corrupted by outliers. Classical approaches are typically designed for the case in which the subspace dimension is small relative to the ambient dimension. Our approach works…
We study principal component analysis (PCA), where given a dataset in $\mathbb{R}^d$ from a distribution, the task is to find a unit vector $v$ that approximately maximizes the variance of the distribution after being projected along $v$.…
We propose an unrolled algorithm approach for learning spatially adaptive parameter maps in the framework of convolutional synthesis-based $\ell_1$ regularization. More precisely, we consider a family of pre-trained convolutional filters…
Principal Component Analysis (PCA) is a well known procedure to reduce intrinsic complexity of a dataset, essentially through simplifying the covariance structure or the correlation structure. We introduce a novel algebraic, model-based…
Sparse principal component analysis (PCA) is a popular dimensionality reduction technique for obtaining principal components which are linear combinations of a small subset of the original features. Existing approaches cannot supply…
The problems of Lasso regression and optimal design of experiments share a critical property: their optimal solutions are typically \emph{sparse}, i.e., only a small fraction of the optimal variables are non-zero. Therefore, the…
Neural networks are usually not the tool of choice for nonparametric high-dimensional problems where the number of input features is much larger than the number of observations. Though neural networks can approximate complex multivariate…
For the problem of high-dimensional sparse linear regression, it is known that an $\ell_0$-based estimator can achieve a $1/n$ "fast" rate on the prediction error without any conditions on the design matrix, whereas in absence of…
We propose a data aggregation-based algorithm with monotonic convergence to a global optimum for a generalized version of the L1-norm error fitting model with an assumption of the fitting function. The proposed algorithm generalizes the…
It was recently established that for convex optimization problems with sparse optimal solutions (be it entry-wise sparsity or matrix rank-wise sparsity) it is possible to design first-order methods with linear convergence rates that depend…