In this paper we establish links between, and new results for, three problems that are not usually considered together. The first is a matrix decomposition problem that arises in areas such as statistical modeling and signal processing: given a matrix X formed as the sum of an unknown diagonal matrix and an unknown low rank positive semidefinite matrix, decompose X into these constituents. The ...

In this paper, a new definition of locally convex L-topological vector spaces is given. The relationship between this new definition and the previous definition of locally convex L-topological vector spaces given by Yan and Fang in 1999 is investigated. Moreover, the concept of generalized L-fuzzy semi-norm is introduced. By using a family of generalized L-fuzzy semi-norms, a characterization o...

The concepts of L-convex function and M-convex function have recently been introduced by Murota as generalizations of submodular function and base polyhedron, respectively, and discrete separation theorems are established for L-convex/concave functions and for M-convex/concave functions as generalizations of Frank's discrete separation theorem for submodular/supermodular set functions and Edmon...

This paper shows the equivalence between Murota’s L-convex functions and Favati and Tardella’s submodular integrally convex functions: For a submodular integrally convex function g(p1, . . . , pn), the function g̃ defined by g̃(p0, p1, . . . , pn) = g(p1 − p0, . . . , pn − p0) is an L-convex function, and vice versa. This fact implies, in combination with known results for L-convex functions, tha...

We consider a broad class of first-order optimization algorithms which are oblivious, in the sense that their step sizes are scheduled regardless of the function under consideration, except for limited side-information such as smoothness or strong convexity parameters. With the knowledge of these two parameters, we show that any such algorithm attains an iteration complexity lower bound of Ω( √...

In this paper, the concepts of L-concave structures, concave Linterior operators and concave L-neighborhood systems are introduced. It is shown that the category of L-concave spaces and the category of concave Linterior spaces are isomorphic, and they are both isomorphic to the category of concave L-neighborhood systems whenever L is a completely distributive lattice. Also, it is proved that th...

This paper introduces an estimation scheme aimed at improving the performance of adaptive filters. The basic idea consists of a convex combination of three adaptive filters, all based on the affine projection algorithm with different numbers of normalization hyperplanes. The three filters are adjusted separately and then combined to minimize the MSE and to increase the speed of convergence of t...