595 research outputs found
Ball hulls, ball intersections, and 2-center problems for gauges
The notions of ball hull and ball intersection of nite sets, important in Banach space theory, are extended from normed planes to generalized normed planes, i.e., to (asymmetric) convex distance functions which are also called gauges. In this more general setting we derive various new results about these notions and their relations to each other. Further on, we extend the known 2-center problem and a modified version of it from the Euclidean situation to norms and gauges or, in other words, from Euclidean circles to arbitrary closed convex curves. We derive algorithmical results on the construction of ball hulls and ball intersections, and computational approaches to the 2-center problem with constrained circles and, in case of strictly convex norms and gauges, for the fixed 2-center problem are also given
Designing structured tight frames via an alternating projection method
Tight frames, also known as general Welch-bound- equality sequences, generalize orthonormal systems. Numerous applications - including communications, coding, and sparse approximation- require finite-dimensional tight frames that possess additional structural properties. This paper proposes an alternating projection method that is versatile enough to solve a huge class of inverse eigenvalue problems (IEPs), which includes the frame design problem. To apply this method, one needs only to solve a matrix nearness problem that arises naturally from the design specifications. Therefore, it is the fast and easy to develop versions of the algorithm that target new design problems. Alternating projection will often succeed even if algebraic constructions are unavailable. To demonstrate that alternating projection is an effective tool for frame design, the paper studies some important structural properties in detail. First, it addresses the most basic design problem: constructing tight frames with prescribed vector norms. Then, it discusses equiangular tight frames, which are natural dictionaries for sparse approximation. Finally, it examines tight frames whose individual vectors have low peak-to-average-power ratio (PAR), which is a valuable property for code-division multiple-access (CDMA) applications. Numerical experiments show that the proposed algorithm succeeds in each of these three cases. The appendices investigate the convergence properties of the algorithm
How Many Dissimilarity/Kernel Self Organizing Map Variants Do We Need?
In numerous applicative contexts, data are too rich and too complex to be
represented by numerical vectors. A general approach to extend machine learning
and data mining techniques to such data is to really on a dissimilarity or on a
kernel that measures how different or similar two objects are. This approach
has been used to define several variants of the Self Organizing Map (SOM). This
paper reviews those variants in using a common set of notations in order to
outline differences and similarities between them. It discusses the advantages
and drawbacks of the variants, as well as the actual relevance of the
dissimilarity/kernel SOM for practical applications
Atomistic simulations of rare events using gentlest ascent dynamics
The dynamics of complex systems often involve thermally activated barrier
crossing events that allow these systems to move from one basin of attraction
on the high dimensional energy surface to another. Such events are ubiquitous,
but challenging to simulate using conventional simulation tools, such as
molecular dynamics. Recently, Weinan E et al. [Nonlinearity, 24(6),1831(2011)]
proposed a set of dynamic equations, the gentlest ascent dynamics (GAD), to
describe the escape of a system from a basin of attraction and proved that
solutions of GAD converge to index-1 saddle points of the underlying energy. In
this paper, we extend GAD to enable finite temperature simulations in which the
system hops between different saddle points on the energy surface. An effective
strategy to use GAD to sample an ensemble of low barrier saddle points located
in the vicinity of a locally stable configuration on the high dimensional
energy surface is proposed. The utility of the method is demonstrated by
studying the low barrier saddle points associated with point defect activity on
a surface. This is done for two representative systems, namely, (a) a surface
vacancy and ad-atom pair and (b) a heptamer island on the (111) surface of
copper.Comment: total 30 page
A geometric analysis of subspace clustering with outliers
This paper considers the problem of clustering a collection of unlabeled data
points assumed to lie near a union of lower-dimensional planes. As is common in
computer vision or unsupervised learning applications, we do not know in
advance how many subspaces there are nor do we have any information about their
dimensions. We develop a novel geometric analysis of an algorithm named sparse
subspace clustering (SSC) [In IEEE Conference on Computer Vision and Pattern
Recognition, 2009. CVPR 2009 (2009) 2790-2797. IEEE], which significantly
broadens the range of problems where it is provably effective. For instance, we
show that SSC can recover multiple subspaces, each of dimension comparable to
the ambient dimension. We also prove that SSC can correctly cluster data points
even when the subspaces of interest intersect. Further, we develop an extension
of SSC that succeeds when the data set is corrupted with possibly
overwhelmingly many outliers. Underlying our analysis are clear geometric
insights, which may bear on other sparse recovery problems. A numerical study
complements our theoretical analysis and demonstrates the effectiveness of
these methods.Comment: Published in at http://dx.doi.org/10.1214/12-AOS1034 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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