160 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
Klee sets and Chebyshev centers for the right Bregman distance
We systematically investigate the farthest distance function, farthest
points, Klee sets, and Chebyshev centers, with respect to Bregman distances
induced by Legendre functions. These objects are of considerable interest in
Information Geometry and Machine Learning; when the Legendre function is
specialized to the energy, one obtains classical notions from Approximation
Theory and Convex Analysis.
The contribution of this paper is twofold. First, we provide an affirmative
answer to a recently-posed question on whether or not every Klee set with
respect to the right Bregman distance is a singleton. Second, we prove
uniqueness of the Chebyshev center and we present a characterization that
relates to previous works by Garkavi, by Klee, and by Nielsen and Nock.Comment: 23 pages, 2 figures, 14 image
Right Bregman nonexpansive operators in Banach spaces
We introduce and study new classes of Bregman nonexpansive operators
in reflexive Banach spaces. These classes of operators are associated with the Bregman distance induced by a convex function. In particular, we characterize sunny right quasi-Bregman nonexpansive retractions, and as a consequence we show that the fixed point set of any right quasi-Bregman nonexpansive operator is a sunny right quasi-Bregman nonexpansive retract of the ambient Banach space.Dirección General de Enseñanza SuperiorJunta de AndalucÃaIsrael Science FoundationGraduate School of the TechnionFund for the Promotion of Research at the TechnionTechnion VPR Fun
Graph Spectral Image Processing
Recent advent of graph signal processing (GSP) has spurred intensive studies
of signals that live naturally on irregular data kernels described by graphs
(e.g., social networks, wireless sensor networks). Though a digital image
contains pixels that reside on a regularly sampled 2D grid, if one can design
an appropriate underlying graph connecting pixels with weights that reflect the
image structure, then one can interpret the image (or image patch) as a signal
on a graph, and apply GSP tools for processing and analysis of the signal in
graph spectral domain. In this article, we overview recent graph spectral
techniques in GSP specifically for image / video processing. The topics covered
include image compression, image restoration, image filtering and image
segmentation
- …