44 research outputs found
Bregman distances and Chebyshev sets
A closed set of a Euclidean space is said to be Chebyshev if every point in
the space has one and only one closest point in the set. Although the situation
is not settled in infinite-dimensional Hilbert spaces, in 1932 Bunt showed that
in Euclidean spaces a closed set is Chebyshev if and only if the set is convex.
In this paper, from the more general perspective of Bregman distances, we show
that if every point in the space has a unique nearest point in a closed set,
then the set is convex. We provide two approaches: one is by nonsmooth
analysis; the other by maximal monotone operator theory. Subdifferentiability
properties of Bregman nearest distance functions are also given
Variable Metric Forward-Backward Splitting with Applications to Monotone Inclusions in Duality
We propose a variable metric forward-backward splitting algorithm and prove
its convergence in real Hilbert spaces. We then use this framework to derive
primal-dual splitting algorithms for solving various classes of monotone
inclusions in duality. Some of these algorithms are new even when specialized
to the fixed metric case. Various applications are discussed
Generalised Br\`{e}gman relative entropies: a brief introduction
We present some basic elements of the theory of generalised Br\`{e}gman
relative entropies over nonreflexive Banach spaces. Using nonlinear embeddings
of Banach spaces together with the Euler--Legendre functions, this approach
unifies two former approaches to Br\`{e}gman relative entropy: one based on
reflexive Banach spaces, another based on differential geometry. This
construction allows to extend Br\`{e}gman relative entropies, and related
geometric and operator structures, to arbitrary-dimensional state spaces of
probability, quantum, and postquantum theory. We give several examples, not
considered previously in the literature.Comment: arXiv admin note: text overlap with arXiv:1710.01837 (author's note:
this overlap will be removed in the upcoming v5 of arXiv:1710.01837 and the
upcoming v4 of this paper); v3: minor change (grant numbers