655 research outputs found
A Unique "Nonnegative" Solution to an Underdetermined System: from Vectors to Matrices
This paper investigates the uniqueness of a nonnegative vector solution and
the uniqueness of a positive semidefinite matrix solution to underdetermined
linear systems. A vector solution is the unique solution to an underdetermined
linear system only if the measurement matrix has a row-span intersecting the
positive orthant. Focusing on two types of binary measurement matrices,
Bernoulli 0-1 matrices and adjacency matrices of general expander graphs, we
show that, in both cases, the support size of a unique nonnegative solution can
grow linearly, namely O(n), with the problem dimension n. We also provide
closed-form characterizations of the ratio of this support size to the signal
dimension. For the matrix case, we show that under a necessary and sufficient
condition for the linear compressed observations operator, there will be a
unique positive semidefinite matrix solution to the compressed linear
observations. We further show that a randomly generated Gaussian linear
compressed observations operator will satisfy this condition with
overwhelmingly high probability
Subspace Expanders and Matrix Rank Minimization
Matrix rank minimization (RM) problems recently gained extensive attention
due to numerous applications in machine learning, system identification and
graphical models. In RM problem, one aims to find the matrix with the lowest
rank that satisfies a set of linear constraints. The existing algorithms
include nuclear norm minimization (NNM) and singular value thresholding. Thus
far, most of the attention has been on i.i.d. Gaussian measurement operators.
In this work, we introduce a new class of measurement operators, and a novel
recovery algorithm, which is notably faster than NNM. The proposed operators
are based on what we refer to as subspace expanders, which are inspired by the
well known expander graphs based measurement matrices in compressed sensing. We
show that given an PSD matrix of rank , it can be uniquely
recovered from a minimal sampling of measurements using the proposed
structures, and the recovery algorithm can be cast as matrix inversion after a
few initial processing steps
Multireference Alignment using Semidefinite Programming
The multireference alignment problem consists of estimating a signal from
multiple noisy shifted observations. Inspired by existing Unique-Games
approximation algorithms, we provide a semidefinite program (SDP) based
relaxation which approximates the maximum likelihood estimator (MLE) for the
multireference alignment problem. Although we show that the MLE problem is
Unique-Games hard to approximate within any constant, we observe that our
poly-time approximation algorithm for the MLE appears to perform quite well in
typical instances, outperforming existing methods. In an attempt to explain
this behavior we provide stability guarantees for our SDP under a random noise
model on the observations. This case is more challenging to analyze than
traditional semi-random instances of Unique-Games: the noise model is on
vertices of a graph and translates into dependent noise on the edges.
Interestingly, we show that if certain positivity constraints in the SDP are
dropped, its solution becomes equivalent to performing phase correlation, a
popular method used for pairwise alignment in imaging applications. Finally, we
show how symmetry reduction techniques from matrix representation theory can
simplify the analysis and computation of the SDP, greatly decreasing its
computational cost
Frequency-Selective Vandermonde Decomposition of Toeplitz Matrices with Applications
The classical result of Vandermonde decomposition of positive semidefinite
Toeplitz matrices, which dates back to the early twentieth century, forms the
basis of modern subspace and recent atomic norm methods for frequency
estimation. In this paper, we study the Vandermonde decomposition in which the
frequencies are restricted to lie in a given interval, referred to as
frequency-selective Vandermonde decomposition. The existence and uniqueness of
the decomposition are studied under explicit conditions on the Toeplitz matrix.
The new result is connected by duality to the positive real lemma for
trigonometric polynomials nonnegative on the same frequency interval. Its
applications in the theory of moments and line spectral estimation are
illustrated. In particular, it provides a solution to the truncated
trigonometric -moment problem. It is used to derive a primal semidefinite
program formulation of the frequency-selective atomic norm in which the
frequencies are known {\em a priori} to lie in certain frequency bands.
Numerical examples are also provided.Comment: 23 pages, accepted by Signal Processin
Euclidean Distance Matrices: Essential Theory, Algorithms and Applications
Euclidean distance matrices (EDM) are matrices of squared distances between
points. The definition is deceivingly simple: thanks to their many useful
properties they have found applications in psychometrics, crystallography,
machine learning, wireless sensor networks, acoustics, and more. Despite the
usefulness of EDMs, they seem to be insufficiently known in the signal
processing community. Our goal is to rectify this mishap in a concise tutorial.
We review the fundamental properties of EDMs, such as rank or
(non)definiteness. We show how various EDM properties can be used to design
algorithms for completing and denoising distance data. Along the way, we
demonstrate applications to microphone position calibration, ultrasound
tomography, room reconstruction from echoes and phase retrieval. By spelling
out the essential algorithms, we hope to fast-track the readers in applying
EDMs to their own problems. Matlab code for all the described algorithms, and
to generate the figures in the paper, is available online. Finally, we suggest
directions for further research.Comment: - 17 pages, 12 figures, to appear in IEEE Signal Processing Magazine
- change of title in the last revisio
Regularization-free estimation in trace regression with symmetric positive semidefinite matrices
Over the past few years, trace regression models have received considerable
attention in the context of matrix completion, quantum state tomography, and
compressed sensing. Estimation of the underlying matrix from
regularization-based approaches promoting low-rankedness, notably nuclear norm
regularization, have enjoyed great popularity. In the present paper, we argue
that such regularization may no longer be necessary if the underlying matrix is
symmetric positive semidefinite (\textsf{spd}) and the design satisfies certain
conditions. In this situation, simple least squares estimation subject to an
\textsf{spd} constraint may perform as well as regularization-based approaches
with a proper choice of the regularization parameter, which entails knowledge
of the noise level and/or tuning. By contrast, constrained least squares
estimation comes without any tuning parameter and may hence be preferred due to
its simplicity
- …