Estimating L1-Norm Best-Fit Lines for Data

The general formulation for finding the L1-norm best-fit subspace for a point set in $m$-dimensions is a nonlinear, nonconvex, nonsmooth optimization problem. In this paper we present a procedure to estimate the L1-norm best-fit one-dimensional subspace (a line through the origin) to data in $\Re^m$ based on an optimization criterion involving linear programming but which … Read more

A TVSCAD approach for image deblurring with impulsive noise

We consider image deblurring problem in the presence of impulsive noise. It is known that \emph{total variation} (TV) regularization with L1-norm penalized data fitting (TVL1 for short) works reasonably well only when the level of impulsive noise is relatively low. For high level impulsive noise, TVL1 works poorly. The reason is that all data, both … Read more

pcaL1: An Implementation in R of Three Methods for L1-Norm Principal Component Analysis

pcaL1 is a package for the R environment for finding principal components using methods based on the L1 norm. The principal components derived using traditional principal component analysis (PCA) can be interpreted as an optimal solution to several optimization problems involving the L2 norm. Using the L1 norm in these problems provides an alternative that … Read more

Sparse and Low-Rank Matrix Decomposition Via Alternating Direction Methods

The problem of recovering the sparse and low-rank components of a matrix captures a broad spectrum of applications. Authors in [4] proposed the concept of “rank-sparsity incoherence” to characterize the fundamental identifiability of the recovery, and derived practical sufficient conditions to ensure the high possibility of recovery. This exact recovery is achieved via solving a … Read more

The L1-Norm Best-Fit Hyperplane Problem

We formalize an algorithm for solving the L1-norm best-fit hyperplane problem derived using first principles and geometric insights about L1 projection and L1 regression. The procedure follows from a new proof of global optimality and relies on the solution of a small number of linear programs. The procedure is implemented for validation and testing. This … Read more