18 research outputs found
A new ADMM algorithm for the Euclidean median and its application to robust patch regression
The Euclidean Median (EM) of a set of points in an Euclidean space
is the point x minimizing the (weighted) sum of the Euclidean distances of x to
the points in . While there exits no closed-form expression for the EM,
it can nevertheless be computed using iterative methods such as the Wieszfeld
algorithm. The EM has classically been used as a robust estimator of centrality
for multivariate data. It was recently demonstrated that the EM can be used to
perform robust patch-based denoising of images by generalizing the popular
Non-Local Means algorithm. In this paper, we propose a novel algorithm for
computing the EM (and its box-constrained counterpart) using variable splitting
and the method of augmented Lagrangian. The attractive feature of this approach
is that the subproblems involved in the ADMM-based optimization of the
augmented Lagrangian can be resolved using simple closed-form projections. The
proposed ADMM solver is used for robust patch-based image denoising and is
shown to exhibit faster convergence compared to an existing solver.Comment: 5 pages, 3 figures, 1 table. To appear in Proc. IEEE International
Conference on Acoustics, Speech, and Signal Processing, April 19-24, 201
A parallel implementation of 3D Zernike moment analysis
Zernike polynomials are a well known set of functions that find many applications in image or pattern characterization because they allow to construct shape descriptors that are invariant against translations, rotations or scale changes. The concepts behind them can be extended to higher dimension spaces, making them also fit to describe volumetric data. They have been less used than their properties might suggest due to their high computational cost. We present a parallel implementation of 3D Zernike moments analysis, written in C with CUDA extensions, which makes it practical to employ Zernike descriptors in interactive applications, yielding a performance of several frames per second in voxel datasets about 2003 in size. In our contribution, we describe the challenges of implementing 3D Zernike analysis in a general-purpose GPU. These include how to deal with numerical inaccuracies, due to the high precision demands of the algorithm, or how to deal with the high volume of input data so that it does not become a bottleneck for the system
Computing medians and means in Hadamard spaces
The geometric median as well as the Frechet mean of points in an Hadamard
space are important in both theory and applications. Surprisingly, no
algorithms for their computation are hitherto known. To address this issue, we
use a split version of the proximal point algorithm for minimizing a sum of
convex functions and prove that this algorithm produces a sequence converging
to a minimizer of the objective function, which extends a recent result of D.
Bertsekas (2001) into Hadamard spaces. The method is quite robust and not only
does it yield algorithms for the median and the mean, but it also applies to
various other optimization problems. We moreover show that another algorithm
for computing the Frechet mean can be derived from the law of large numbers due
to K.-T. Sturm (2002). In applications, computing medians and means is probably
most needed in tree space, which is an instance of an Hadamard space, invented
by Billera, Holmes, and Vogtmann (2001) as a tool for averaging phylogenetic
trees. It turns out, however, that it can be also used to model numerous other
tree-like structures. Since there now exists a polynomial-time algorithm for
computing geodesics in tree space due to M. Owen and S. Provan (2011), we
obtain efficient algorithms for computing medians and means, which can be
directly used in practice.Comment: Corrected version. Accepted in SIAM Journal on Optimizatio
Network Lasso: Clustering and Optimization in Large Graphs
Convex optimization is an essential tool for modern data analysis, as it
provides a framework to formulate and solve many problems in machine learning
and data mining. However, general convex optimization solvers do not scale
well, and scalable solvers are often specialized to only work on a narrow class
of problems. Therefore, there is a need for simple, scalable algorithms that
can solve many common optimization problems. In this paper, we introduce the
\emph{network lasso}, a generalization of the group lasso to a network setting
that allows for simultaneous clustering and optimization on graphs. We develop
an algorithm based on the Alternating Direction Method of Multipliers (ADMM) to
solve this problem in a distributed and scalable manner, which allows for
guaranteed global convergence even on large graphs. We also examine a
non-convex extension of this approach. We then demonstrate that many types of
problems can be expressed in our framework. We focus on three in particular -
binary classification, predicting housing prices, and event detection in time
series data - comparing the network lasso to baseline approaches and showing
that it is both a fast and accurate method of solving large optimization
problems