The notion of a Bregman retraction of a closed convex set in Euclidean space is introduced. Bregman retractions include backward Bregman projections, forward Bregman projections, as well as their convex combinations, and are thus quite flexible. The main result on iterating Bregman retractions unifies several convergence results on projection methods for solving convex feasibility problems. It ...

Nonconvex quadratic constraints can be linearized to obtain relaxations in a wellunderstood manner. We propose to tighten the relaxation by using second order cone constraints, resulting in a convex quadratic relaxation. Our quadratic approximation to the bilinear term is compared to the linear McCormick bounds. The second order cone constraints are based on linear combinations of pairs of vari...

We derive new bounds on covering numbers for hypothesis classes generated by convex combinations of basis functions. These are useful in bounding the generalization performance of algorithms such as RBF-networks, boosting and a new class of linear programming machines similar to SV machines. We show that p-convex combinations with p > 1 lead to diverging bounds, whereas for p = 1 good bounds in...

No-regret algorithms are powerful tools for learning in online convex problems that have received increased attention in recent years. Considering affine and external regret, we investigate what happens when a set of no-regret learners (voters) merge their respective strategies in each learning iteration to a single, common one in form of a convex combination. We show that an agent who executes...

We study the convex combinations of $(d+1)$ generalized Pauli dynamical maps in a Hilbert space dimension $d$. For certain choices decoherence function, are noninvertible and they remain under as well. case characterized by function $(1-e^{-ct})/n$ with parameter $n$ decay factor $c$, we evaluate fraction invertible obtained upon mixing, which is found to increase superexponentially

The paper is an updated survey of our work on the approximation of univariate setvalued functions by samples-based linear approximation operators, beyond the results reported in our previous overview. Our approach is to adapt operators for real-valued functions to set-valued functions, by replacing operations between numbers by operations between sets. For set-valued functions with compact conv...

As appropriate generalizations of convex combinations with uncountably many terms, we introduce the so-called Choquet combinations, decomposable and as well their corresponding hull operators acting on power sets Lebesgue-Bochner spaces. We show that coincides in finite-dimensional setting, yet tends to be larger infinite dimensions. also provide a quantitative characterization hull, without an...

We consider the space of w-mixtures that are the set of finite statistical mixtures sharing the same prescribed component distributions. The geometry induced by the Kullback-Leibler (KL) divergence on this family of w-mixtures is a dually flat space in information geometry called the mixture family manifold. It follows that the KL divergence between two w-mixtures is equivalent to a Bregman Div...

Properties of compositions and convex combinations of averaged nonexpansive operators are investigated and applied to the design of new fixed point algorithms in Hilbert spaces. An extended version of the forward-backward splitting algorithm for finding a zero of the sum of two monotone operators is obtained.