Identifying Linear Combinations of Ridge Functions

Identifying Linear Combinations of Ridge Functions

This paper is about an inverse problem. We assume we are given a function f(x) which is some sum of ridge functions of the form ∑m i=1 gi(a i · x) and we just know an upper bound on m. We seek to identify the functions gi and also the directions a i from such limited information. Several ways to solve this nonlinear problem are discussed in this work. §

Convex Domains and Linear Combinations of Continuous Functions

Fundamentality of Ridge Functions

For a given integer d, 1 ≤ d ≤ n − 1, let Ω be a subset of the set of all d × n real matrices. Define the subspace M(Ω) = span{g(Ax) : A ∈ Ω, g ∈ C(IR, IR)} . We give necessary and sufficient conditions on Ω so that M(Ω) is dense in C(IR, IR) in the topology of uniform convergence on compact subsets. This generalizes work of Vostrecov and Kreines. We also consider some related problems. §

Linear combinations of orders

This note explores some properties of a notion of linear combination of partial order relations over a given set. A bilinear form is used to represent compatibility of orders, and we study combinations up to equivalence through this form. A precise description of the quotient space is provided according to the nature of the semiring of coefficients.

On the Linear Combinations of Slanted Half-Plane Harmonic Mappings

‎In this paper,  the sufficient conditions for the linear combinations of slanted half-plane harmonic mappings to be univalent and convex in the direction of $(-gamma) $ are studied. Our result improves some recent works. Furthermore, a illustrative example and imagine domains of the linear combinations satisfying the desired conditions are enumerated.