15,652 research outputs found
Convexity in source separation: Models, geometry, and algorithms
Source separation or demixing is the process of extracting multiple
components entangled within a signal. Contemporary signal processing presents a
host of difficult source separation problems, from interference cancellation to
background subtraction, blind deconvolution, and even dictionary learning.
Despite the recent progress in each of these applications, advances in
high-throughput sensor technology place demixing algorithms under pressure to
accommodate extremely high-dimensional signals, separate an ever larger number
of sources, and cope with more sophisticated signal and mixing models. These
difficulties are exacerbated by the need for real-time action in automated
decision-making systems.
Recent advances in convex optimization provide a simple framework for
efficiently solving numerous difficult demixing problems. This article provides
an overview of the emerging field, explains the theory that governs the
underlying procedures, and surveys algorithms that solve them efficiently. We
aim to equip practitioners with a toolkit for constructing their own demixing
algorithms that work, as well as concrete intuition for why they work
Contrast in Multipath Interference and Quantum Coherence
We develop a rigorous connection between statistical properties of an
interference pattern and the coherence properties of the underlying quantum
state. With explicit examples, we demonstrate that even for inaccurate
reconstructions of interference patterns properly defined statistical moments
permit a reliable characterization of quantum coherence.Comment: 10 page
Sum-Rate Maximization in Two-Way AF MIMO Relaying: Polynomial Time Solutions to a Class of DC Programming Problems
Sum-rate maximization in two-way amplify-and-forward (AF) multiple-input
multiple-output (MIMO) relaying belongs to the class of difference-of-convex
functions (DC) programming problems. DC programming problems occur as well in
other signal processing applications and are typically solved using different
modifications of the branch-and-bound method. This method, however, does not
have any polynomial time complexity guarantees. In this paper, we show that a
class of DC programming problems, to which the sum-rate maximization in two-way
MIMO relaying belongs, can be solved very efficiently in polynomial time, and
develop two algorithms. The objective function of the problem is represented as
a product of quadratic ratios and parameterized so that its convex part (versus
the concave part) contains only one (or two) optimization variables. One of the
algorithms is called POlynomial-Time DC (POTDC) and is based on semi-definite
programming (SDP) relaxation, linearization, and an iterative search over a
single parameter. The other algorithm is called RAte-maximization via
Generalized EigenvectorS (RAGES) and is based on the generalized eigenvectors
method and an iterative search over two (or one, in its approximate version)
optimization variables. We also derive an upper-bound for the optimal values of
the corresponding optimization problem and show by simulations that this
upper-bound can be achieved by both algorithms. The proposed methods for
maximizing the sum-rate in the two-way AF MIMO relaying system are shown to be
superior to other state-of-the-art algorithms.Comment: 35 pages, 10 figures, Submitted to the IEEE Trans. Signal Processing
in Nov. 201
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
Spatial networks with wireless applications
Many networks have nodes located in physical space, with links more common
between closely spaced pairs of nodes. For example, the nodes could be wireless
devices and links communication channels in a wireless mesh network. We
describe recent work involving such networks, considering effects due to the
geometry (convex,non-convex, and fractal), node distribution,
distance-dependent link probability, mobility, directivity and interference.Comment: Review article- an amended version with a new title from the origina
Finding a low-rank basis in a matrix subspace
For a given matrix subspace, how can we find a basis that consists of
low-rank matrices? This is a generalization of the sparse vector problem. It
turns out that when the subspace is spanned by rank-1 matrices, the matrices
can be obtained by the tensor CP decomposition. For the higher rank case, the
situation is not as straightforward. In this work we present an algorithm based
on a greedy process applicable to higher rank problems. Our algorithm first
estimates the minimum rank by applying soft singular value thresholding to a
nuclear norm relaxation, and then computes a matrix with that rank using the
method of alternating projections. We provide local convergence results, and
compare our algorithm with several alternative approaches. Applications include
data compression beyond the classical truncated SVD, computing accurate
eigenvectors of a near-multiple eigenvalue, image separation and graph
Laplacian eigenproblems
A Sparse Multi-Scale Algorithm for Dense Optimal Transport
Discrete optimal transport solvers do not scale well on dense large problems
since they do not explicitly exploit the geometric structure of the cost
function. In analogy to continuous optimal transport we provide a framework to
verify global optimality of a discrete transport plan locally. This allows
construction of an algorithm to solve large dense problems by considering a
sequence of sparse problems instead. The algorithm lends itself to being
combined with a hierarchical multi-scale scheme. Any existing discrete solver
can be used as internal black-box.Several cost functions, including the noisy
squared Euclidean distance, are explicitly detailed. We observe a significant
reduction of run-time and memory requirements.Comment: Published "online first" in Journal of Mathematical Imaging and
Vision, see DO
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