10,454 research outputs found
Weighted norms in subspace-based methods for time series analysis
Many modern approaches of time series analysis belong to the class of methods based on approximating high-dimensional spaces by low-dimensional subspaces. A typical method would embed a given time series into a structured matrix and find a low-dimensional approximation to this structured matrix. The purpose of this paper is twofold: (i) to establish a correspondence between a class of SVD-compatible matrix norms on the space of Hankel matrices and weighted vector norms (and provide methods to construct this correspondence) and (ii) to motivate the importance of this for problems in time series analysis. Examples are provided to demonstrate the merits of judiciously selecting weights on imputing missing data and forecasting in time series. Copyright © 2016 John Wiley & Sons, Ltd
Robust computation of linear models by convex relaxation
Consider a dataset of vector-valued observations that consists of noisy
inliers, which are explained well by a low-dimensional subspace, along with
some number of outliers. This work describes a convex optimization problem,
called REAPER, that can reliably fit a low-dimensional model to this type of
data. This approach parameterizes linear subspaces using orthogonal projectors,
and it uses a relaxation of the set of orthogonal projectors to reach the
convex formulation. The paper provides an efficient algorithm for solving the
REAPER problem, and it documents numerical experiments which confirm that
REAPER can dependably find linear structure in synthetic and natural data. In
addition, when the inliers lie near a low-dimensional subspace, there is a
rigorous theory that describes when REAPER can approximate this subspace.Comment: Formerly titled "Robust computation of linear models, or How to find
a needle in a haystack
Weighted projections and Riesz frames
Let be a (separable) Hilbert space and a
fixed orthonormal basis of . Motivated by many papers on scaled
projections, angles of subspaces and oblique projections, we define and study
the notion of compatibility between a subspace and the abelian algebra of
diagonal operators in the given basis. This is used to refine previous work on
scaled projections, and to obtain a new characterization of Riesz frames.Comment: 23 pages, to appear in Linear Algebra and its Application
A Compact Formulation for the Mixed-Norm Minimization Problem
Parameter estimation from multiple measurement vectors (MMVs) is a
fundamental problem in many signal processing applications, e.g., spectral
analysis and direction-of- arrival estimation. Recently, this problem has been
address using prior information in form of a jointly sparse signal structure. A
prominent approach for exploiting joint sparsity considers mixed-norm
minimization in which, however, the problem size grows with the number of
measurements and the desired resolution, respectively. In this work we derive
an equivalent, compact reformulation of the mixed-norm
minimization problem which provides new insights on the relation between
different existing approaches for jointly sparse signal reconstruction. The
reformulation builds upon a compact parameterization, which models the
row-norms of the sparse signal representation as parameters of interest,
resulting in a significant reduction of the MMV problem size. Given the sparse
vector of row-norms, the jointly sparse signal can be computed from the MMVs in
closed form. For the special case of uniform linear sampling, we present an
extension of the compact formulation for gridless parameter estimation by means
of semidefinite programming. Furthermore, we derive in this case from our
compact problem formulation the exact equivalence between the
mixed-norm minimization and the atomic-norm minimization. Additionally, for the
case of irregular sampling or a large number of samples, we present a low
complexity, grid-based implementation based on the coordinate descent method
- …