52,205 research outputs found
Diagonal and Low-Rank Matrix Decompositions, Correlation Matrices, and Ellipsoid Fitting
In this paper we establish links between, and new results for, three problems
that are not usually considered together. The first is a matrix decomposition
problem that arises in areas such as statistical modeling and signal
processing: given a matrix formed as the sum of an unknown diagonal matrix
and an unknown low rank positive semidefinite matrix, decompose into these
constituents. The second problem we consider is to determine the facial
structure of the set of correlation matrices, a convex set also known as the
elliptope. This convex body, and particularly its facial structure, plays a
role in applications from combinatorial optimization to mathematical finance.
The third problem is a basic geometric question: given points
(where ) determine whether there is a centered
ellipsoid passing \emph{exactly} through all of the points.
We show that in a precise sense these three problems are equivalent.
Furthermore we establish a simple sufficient condition on a subspace that
ensures any positive semidefinite matrix with column space can be
recovered from for any diagonal matrix using a convex
optimization-based heuristic known as minimum trace factor analysis. This
result leads to a new understanding of the structure of rank-deficient
correlation matrices and a simple condition on a set of points that ensures
there is a centered ellipsoid passing through them.Comment: 20 page
A condition number for the tensor rank decomposition
The tensor rank decomposition problem consists of recovering the unique set
of parameters representing a robustly identifiable low-rank tensor when the
coordinate representation of the tensor is presented as input. A condition
number for this problem measuring the sensitivity of the parameters to an
infinitesimal change to the tensor is introduced and analyzed. It is
demonstrated that the absolute condition number coincides with the inverse of
the least singular value of Terracini's matrix. Several basic properties of
this condition number are investigated.Comment: 45 pages, 4 figure
Dynamic mode decomposition with control
We develop a new method which extends Dynamic Mode Decomposition (DMD) to
incorporate the effect of control to extract low-order models from
high-dimensional, complex systems. DMD finds spatial-temporal coherent modes,
connects local-linear analysis to nonlinear operator theory, and provides an
equation-free architecture which is compatible with compressive sensing. In
actuated systems, DMD is incapable of producing an input-output model;
moreover, the dynamics and the modes will be corrupted by external forcing. Our
new method, Dynamic Mode Decomposition with control (DMDc), capitalizes on all
of the advantages of DMD and provides the additional innovation of being able
to disambiguate between the underlying dynamics and the effects of actuation,
resulting in accurate input-output models. The method is data-driven in that it
does not require knowledge of the underlying governing equations, only
snapshots of state and actuation data from historical, experimental, or
black-box simulations. We demonstrate the method on high-dimensional dynamical
systems, including a model with relevance to the analysis of infectious disease
data with mass vaccination (actuation).Comment: 10 pages, 4 figure
Decomposition tables for experiments I. A chain of randomizations
One aspect of evaluating the design for an experiment is the discovery of the
relationships between subspaces of the data space. Initially we establish the
notation and methods for evaluating an experiment with a single randomization.
Starting with two structures, or orthogonal decompositions of the data space,
we describe how to combine them to form the overall decomposition for a
single-randomization experiment that is ``structure balanced.'' The
relationships between the two structures are characterized using efficiency
factors. The decomposition is encapsulated in a decomposition table. Then, for
experiments that involve multiple randomizations forming a chain, we take
several structures that pairwise are structure balanced and combine them to
establish the form of the orthogonal decomposition for the experiment. In
particular, it is proven that the properties of the design for such an
experiment are derived in a straightforward manner from those of the individual
designs. We show how to formulate an extended decomposition table giving the
sources of variation, their relationships and their degrees of freedom, so that
competing designs can be evaluated.Comment: Published in at http://dx.doi.org/10.1214/09-AOS717 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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