In this paper, we propose a variational multiphase image segmentation model based on fuzzy membership functions and L1-norm fidelity. Then we apply the alternating direction method of multipliers to solve an equivalent problem. All the subproblems can be solved efficiently. Specifically, we propose a fast method to calculate the fuzzy median. Experimental results and comparisons show that the L...

In the last lecture we defined metric spaces, normed spaces, and considered the distortion resulting from certain embeddings. In particular, we proved that l1 norms cannot always be embedded isometrically into l2 by considering a specific four-point l1 norm and showing that it requires at least √ 2 distortion. Today’s lecture further explores the 1 norm. We see a couple of interesting examples ...

Approaches to multiple kernel learning (MKL) employ l1-norm constraints on the mixing coefficients to promote sparse kernel combinations. When features encode orthogonal characterizations of a problem, sparseness may lead to discarding useful information and may thus result in poor generalization performance. We study non-sparse multiple kernel learning by imposing an l2-norm constraint on the ...

This paper introduces DEVIATION, a soft global constraint to obtain balanced solutions. A violation measure of the perfect balance can be defined as the Lp norm of the vector variables minus their mean. SPREAD constraints the sum of square deviations to the mean [5, 7] i.e. the L2 norm. The L1 norm is considered here. Neither criterion subsumes the other but the design of a propagator for L1 is...

We consider the empirical risk minimization problem for linear supervised learning, with regularization by structured sparsity-inducing norms. These are defined as sums of Euclidean norms on certain subsets of variables, extending the usual l1-norm and the group l1-norm by allowing the subsets to overlap. This leads to a specific set of allowed nonzero patterns for the solutions of such problem...

The proportionate normalized least-mean-square (PNLMS) algorithm was developed in the context of network echo cancellation. It has been proven to be efficient when the echo path is sparse, which is not always the case in realworld echo cancellation. The improved PNLMS (IPNLMS) algorithm is less sensitive to the sparseness character of the echo path. This algorithm uses the l1 norm to exploit sp...

We characterize the approximate monomial complexity, sign monomial complexity, and the approximate L1 norm of symmetric functions in terms of simple combinatorial measures of the functions. Our characterization of the approximate L1 norm solves the main conjecture in [AFH12]. As an application of the characterization of the sign monomial complexity, we prove a conjecture in [ZS09] and provide a...

L1-norm is better than L2-norm at dealing with noisy data and yielding blocky models, features crucial in many geophysical applications. In this report, we develop a hybrid-norm solver proposed by Claerbout (2009) to perform L1 regressions. The solver is tested on a 1-D field RMS velocity inversion, a 2-D regularized Kirchhoff migration inversion and a 2-D velocity analysis problem. The results...

In most compressive sensing problems l1 norm is used during the signal reconstruction process. In this article the use of entropy functional is proposed to approximate the l1 norm. A modified version of the entropy functional is continuous, differentiable and convex. Therefore, it is possible to construct globally convergent iterative algorithms using Bregman’s row action D-projection method fo...

This paper proposes a distributed algorithm for multi-agent networks to achieve a minimum l1-norm solution to a linear equation Ax = b where A has full row rank. When the underlying network is undirected and fixed, it is proved that the proposed algorithm drive all agents’ individual states to converge in finite-time to the same minimum l1-norm solution. Numerical simulations are also provided ...