220 research outputs found
A small frame and a certificate of its injectivity
We present a complex frame of eleven vectors in 4-space and prove that it
defines injective measurements. That is, any rank-one Hermitian
matrix is uniquely determined by its values as a Hermitian form on this
collection of eleven vectors. This disproves a recent conjecture of Bandeira,
Cahill, Mixon, and Nelson. We use algebraic computations and certificates in
order to prove injectivity.Comment: 4 pages, 3 figure
Phase retrieval from very few measurements
In many applications, signals are measured according to a linear process, but
the phases of these measurements are often unreliable or not available. To
reconstruct the signal, one must perform a process known as phase retrieval.
This paper focuses on completely determining signals with as few intensity
measurements as possible, and on efficient phase retrieval algorithms from such
measurements. For the case of complex M-dimensional signals, we construct a
measurement ensemble of size 4M-4 which yields injective intensity
measurements; this is conjectured to be the smallest such ensemble. For the
case of real signals, we devise a theory of "almost" injective intensity
measurements, and we characterize such ensembles. Later, we show that phase
retrieval from M+1 almost injective intensity measurements is NP-hard,
indicating that computationally efficient phase retrieval must come at the
price of measurement redundancy.Comment: 18 pages, 1 figur
Low Complexity Regularization of Linear Inverse Problems
Inverse problems and regularization theory is a central theme in contemporary
signal processing, where the goal is to reconstruct an unknown signal from
partial indirect, and possibly noisy, measurements of it. A now standard method
for recovering the unknown signal is to solve a convex optimization problem
that enforces some prior knowledge about its structure. This has proved
efficient in many problems routinely encountered in imaging sciences,
statistics and machine learning. This chapter delivers a review of recent
advances in the field where the regularization prior promotes solutions
conforming to some notion of simplicity/low-complexity. These priors encompass
as popular examples sparsity and group sparsity (to capture the compressibility
of natural signals and images), total variation and analysis sparsity (to
promote piecewise regularity), and low-rank (as natural extension of sparsity
to matrix-valued data). Our aim is to provide a unified treatment of all these
regularizations under a single umbrella, namely the theory of partial
smoothness. This framework is very general and accommodates all low-complexity
regularizers just mentioned, as well as many others. Partial smoothness turns
out to be the canonical way to encode low-dimensional models that can be linear
spaces or more general smooth manifolds. This review is intended to serve as a
one stop shop toward the understanding of the theoretical properties of the
so-regularized solutions. It covers a large spectrum including: (i) recovery
guarantees and stability to noise, both in terms of -stability and
model (manifold) identification; (ii) sensitivity analysis to perturbations of
the parameters involved (in particular the observations), with applications to
unbiased risk estimation ; (iii) convergence properties of the forward-backward
proximal splitting scheme, that is particularly well suited to solve the
corresponding large-scale regularized optimization problem
Phase retrieval from low-rate samples
The paper considers the phase retrieval problem in N-dimensional complex
vector spaces. It provides two sets of deterministic measurement vectors which
guarantee signal recovery for all signals, excluding only a specific subspace
and a union of subspaces, respectively. A stable analytic reconstruction
procedure of low complexity is given. Additionally it is proven that signal
recovery from these measurements can be solved exactly via a semidefinite
program. A practical implementation with 4 deterministic diffraction patterns
is provided and some numerical experiments with noisy measurements complement
the analytic approach.Comment: Preprint accepted for publication in Sampling Theory in Signal and
Image Processing -- Special issue on SampTa 201
Phase retrieval by hyperplanes
We show that a scalable frame does phase retrieval if and only if the
hyperplanes of its orthogonal complements do phase retrieval. We then show this
result fails in general by giving an example of a frame for which
does phase retrieval but its induced hyperplanes fail phase retrieval.
Moreover, we show that such frames always exist in for any
dimension . We also give an example of a frame in which fails
phase retrieval but its perps do phase retrieval. We will also see that a
family of hyperplanes doing phase retrieval in must contain at
least hyperplanes. Finally, we provide an example of six hyperplanes in
which do phase retrieval
Stable optimizationless recovery from phaseless linear measurements
We address the problem of recovering an n-vector from m linear measurements
lacking sign or phase information. We show that lifting and semidefinite
relaxation suffice by themselves for stable recovery in the setting of m = O(n
log n) random sensing vectors, with high probability. The recovery method is
optimizationless in the sense that trace minimization in the PhaseLift
procedure is unnecessary. That is, PhaseLift reduces to a feasibility problem.
The optimizationless perspective allows for a Douglas-Rachford numerical
algorithm that is unavailable for PhaseLift. This method exhibits linear
convergence with a favorable convergence rate and without any parameter tuning
- …