1,410 research outputs found
On the Success Probability of the Box-Constrained Rounding and Babai Detectors
In communications, one frequently needs to detect a parameter vector \hbx
in a box from a linear model. The box-constrained rounding detector \x^\sBR
and Babai detector \x^\sBB are often used to detect \hbx due to their high
probability of correct detection, which is referred to as success probability,
and their high efficiency of implimentation. It is generally believed that the
success probability P^\sBR of \x^\sBR is not larger than the success
probability P^\sBB of \x^\sBB. In this paper, we first present formulas for
P^\sBR and P^\sBB for two different situations: \hbx is deterministic and
\hbx is uniformly distributed over the constraint box. Then, we give a simple
example to show that P^\sBR may be strictly larger than P^\sBB if \hbx is
deterministic, while we rigorously show that P^\sBR\leq P^\sBB always holds
if \hbx is uniformly distributed over the constraint box.Comment: to appear in ISIT 201
Least Squares Ranking on Graphs
Given a set of alternatives to be ranked, and some pairwise comparison data,
ranking is a least squares computation on a graph. The vertices are the
alternatives, and the edge values comprise the comparison data. The basic idea
is very simple and old: come up with values on vertices such that their
differences match the given edge data. Since an exact match will usually be
impossible, one settles for matching in a least squares sense. This formulation
was first described by Leake in 1976 for rankingfootball teams and appears as
an example in Professor Gilbert Strang's classic linear algebra textbook. If
one is willing to look into the residual a little further, then the problem
really comes alive, as shown effectively by the remarkable recent paper of
Jiang et al. With or without this twist, the humble least squares problem on
graphs has far-reaching connections with many current areas ofresearch. These
connections are to theoretical computer science (spectral graph theory, and
multilevel methods for graph Laplacian systems); numerical analysis (algebraic
multigrid, and finite element exterior calculus); other mathematics (Hodge
decomposition, and random clique complexes); and applications (arbitrage, and
ranking of sports teams). Not all of these connections are explored in this
paper, but many are. The underlying ideas are easy to explain, requiring only
the four fundamental subspaces from elementary linear algebra. One of our aims
is to explain these basic ideas and connections, to get researchers in many
fields interested in this topic. Another aim is to use our numerical
experiments for guidance on selecting methods and exposing the need for further
development.Comment: Added missing references, comparison of linear solvers overhauled,
conclusion section added, some new figures adde
Forward-backward truncated Newton methods for convex composite optimization
This paper proposes two proximal Newton-CG methods for convex nonsmooth
optimization problems in composite form. The algorithms are based on a a
reformulation of the original nonsmooth problem as the unconstrained
minimization of a continuously differentiable function, namely the
forward-backward envelope (FBE). The first algorithm is based on a standard
line search strategy, whereas the second one combines the global efficiency
estimates of the corresponding first-order methods, while achieving fast
asymptotic convergence rates. Furthermore, they are computationally attractive
since each Newton iteration requires the approximate solution of a linear
system of usually small dimension
Conic Optimization Theory: Convexification Techniques and Numerical Algorithms
Optimization is at the core of control theory and appears in several areas of
this field, such as optimal control, distributed control, system
identification, robust control, state estimation, model predictive control and
dynamic programming. The recent advances in various topics of modern
optimization have also been revamping the area of machine learning. Motivated
by the crucial role of optimization theory in the design, analysis, control and
operation of real-world systems, this tutorial paper offers a detailed overview
of some major advances in this area, namely conic optimization and its emerging
applications. First, we discuss the importance of conic optimization in
different areas. Then, we explain seminal results on the design of hierarchies
of convex relaxations for a wide range of nonconvex problems. Finally, we study
different numerical algorithms for large-scale conic optimization problems.Comment: 18 page
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