1,956 research outputs found
Convergence of the -Means Minimization Problem using -Convergence
The -means method is an iterative clustering algorithm which associates
each observation with one of clusters. It traditionally employs cluster
centers in the same space as the observed data. By relaxing this requirement,
it is possible to apply the -means method to infinite dimensional problems,
for example multiple target tracking and smoothing problems in the presence of
unknown data association. Via a -convergence argument, the associated
optimization problem is shown to converge in the sense that both the -means
minimum and minimizers converge in the large data limit to quantities which
depend upon the observed data only through its distribution. The theory is
supplemented with two examples to demonstrate the range of problems now
accessible by the -means method. The first example combines a non-parametric
smoothing problem with unknown data association. The second addresses tracking
using sparse data from a network of passive sensors
Chebushev Greedy Algorithm in convex optimization
Chebyshev Greedy Algorithm is a generalization of the well known Orthogonal
Matching Pursuit defined in a Hilbert space to the case of Banach spaces. We
apply this algorithm for constructing sparse approximate solutions (with
respect to a given dictionary) to convex optimization problems. Rate of
convergence results in a style of the Lebesgue-type inequalities are proved
A forward-backward splitting algorithm for the minimization of non-smooth convex functionals in Banach space
We consider the task of computing an approximate minimizer of the sum of a
smooth and non-smooth convex functional, respectively, in Banach space.
Motivated by the classical forward-backward splitting method for the
subgradients in Hilbert space, we propose a generalization which involves the
iterative solution of simpler subproblems. Descent and convergence properties
of this new algorithm are studied. Furthermore, the results are applied to the
minimization of Tikhonov-functionals associated with linear inverse problems
and semi-norm penalization in Banach spaces. With the help of
Bregman-Taylor-distance estimates, rates of convergence for the
forward-backward splitting procedure are obtained. Examples which demonstrate
the applicability are given, in particular, a generalization of the iterative
soft-thresholding method by Daubechies, Defrise and De Mol to Banach spaces as
well as total-variation based image restoration in higher dimensions are
presented
Universal Scalable Robust Solvers from Computational Information Games and fast eigenspace adapted Multiresolution Analysis
We show how the discovery of robust scalable numerical solvers for arbitrary
bounded linear operators can be automated as a Game Theory problem by
reformulating the process of computing with partial information and limited
resources as that of playing underlying hierarchies of adversarial information
games. When the solution space is a Banach space endowed with a quadratic
norm , the optimal measure (mixed strategy) for such games (e.g. the
adversarial recovery of , given partial measurements with
, using relative error in -norm as a loss) is a
centered Gaussian field solely determined by the norm , whose
conditioning (on measurements) produces optimal bets. When measurements are
hierarchical, the process of conditioning this Gaussian field produces a
hierarchy of elementary bets (gamblets). These gamblets generalize the notion
of Wavelets and Wannier functions in the sense that they are adapted to the
norm and induce a multi-resolution decomposition of that is
adapted to the eigensubspaces of the operator defining the norm .
When the operator is localized, we show that the resulting gamblets are
localized both in space and frequency and introduce the Fast Gamblet Transform
(FGT) with rigorous accuracy and (near-linear) complexity estimates. As the FFT
can be used to solve and diagonalize arbitrary PDEs with constant coefficients,
the FGT can be used to decompose a wide range of continuous linear operators
(including arbitrary continuous linear bijections from to or
to ) into a sequence of independent linear systems with uniformly bounded
condition numbers and leads to
solvers and eigenspace adapted Multiresolution Analysis (resulting in near
linear complexity approximation of all eigensubspaces).Comment: 142 pages. 14 Figures. Presented at AFOSR (Aug 2016), DARPA (Sep
2016), IPAM (Apr 3, 2017), Hausdorff (April 13, 2017) and ICERM (June 5,
2017
- β¦