Triangular norms, or t-norms for short, play an important role for the semantics of fuzzy logics. Although an enormous number of examples and a remarkable number of construction methods for this kind of operation has been established, a uniform approach is still outstanding. This paper is devoted to a specific algebraic-geometrical framework within which t-norms, up to isomorphism, can be descr...

The aim of this paper is to present the application of an approach to study contraction theory recently developed for piecewise smooth and switched systems. The approach that can be used to analyze incremental stability properties of so-called Filippov systems (or variable structure systems) is based on the use of regularization, a procedure to make the vector field of interest differentiable b...

We present a new method for solving total variation (TV) minimization problems in image restoration. The main idea is to remove some of the singularity caused by the nondifferentiability of the quantity |∇u| in the definition of the TV-norm before we apply a linearization technique such as Newton’s method. This is accomplished by introducing an additional variable for the flux quantity appearin...

We consider the problem of minimizing a sum of Euclidean norms, f(x) = ∑m i=1 ‖bi− Ai x‖. This problem is a nonsmooth problem because f is not differentiable at a point x when one of the norms is zero. In this paper we present a smoothing Newton method for this problem by applying the smoothing Newton method proposed by Qi, Sun, and Zhou [Math. Programming, 87 (2000), pp. 1–35] directly to a sy...

Fast indexing in time sequence databases for similarity searching has attracted a lot of research recently. Most of the proposals, however, typically centered around the Euclidean distance and its derivatives. We examine the problem of multi-modal similarity search in which users can choose the best one from multiple similarity models for their needs. In this paper, we present a novel and fast ...

The measurement of distance is one of the key steps in the unsupervised learning process, as it is through these distance measurements that patterns and correlations are discovered. We examined the characteristics of both non-Euclidean norms and data normalisation within the unsupervised learning environment. We empirically assessed the performance of the K-means, Neural Gas, Growing Neural Gas...

The inequality in (a) is naturally the best possible. Theorem l(a) is connected with questions such as the Beck-Fiala theorem or the Koml6s conjecture (see [2J, [3J and [7]). Combinatorial motivations are presented exhaustively in [8J (see also [6]). In a slightly different form, Theorem l(a) was used in [IJ in the proof that nuclear Frechet spaces satisfy the Levy-Steinitz theorem on rearrange...

In this paper, we consider composite convex minimization problems. We advocate the merit of considering Generalized Proximal gradient Methods (GPM) where the norm employed is not Euclidean. To that end, we show the tractability of the general proximity operator for a broad class of structure priors by proposing a polynomial-time approach to approximately compute it. We also identify a special c...

This paper introduces the family of CVaR norms in Rn , based on the CVaR concept. The CVaR norm is defined in two variations: scaled and non-scaled. The well-known L1 and L∞ norms are limiting cases of the new family of norms. The D-norm, used in robust optimization, is equivalent to the non-scaled CVaR norm. We present two relatively simple definitions of the CVaR norm: (i) as the average or t...