40 research outputs found
Recovery of Low-Rank Matrices under Affine Constraints via a Smoothed Rank Function
In this paper, the problem of matrix rank minimization under affine
constraints is addressed. The state-of-the-art algorithms can recover matrices
with a rank much less than what is sufficient for the uniqueness of the
solution of this optimization problem. We propose an algorithm based on a
smooth approximation of the rank function, which practically improves recovery
limits on the rank of the solution. This approximation leads to a non-convex
program; thus, to avoid getting trapped in local solutions, we use the
following scheme. Initially, a rough approximation of the rank function subject
to the affine constraints is optimized. As the algorithm proceeds, finer
approximations of the rank are optimized and the solver is initialized with the
solution of the previous approximation until reaching the desired accuracy.
On the theoretical side, benefiting from the spherical section property, we
will show that the sequence of the solutions of the approximating function
converges to the minimum rank solution. On the experimental side, it will be
shown that the proposed algorithm, termed SRF standing for Smoothed Rank
Function, can recover matrices which are unique solutions of the rank
minimization problem and yet not recoverable by nuclear norm minimization.
Furthermore, it will be demonstrated that, in completing partially observed
matrices, the accuracy of SRF is considerably and consistently better than some
famous algorithms when the number of revealed entries is close to the minimum
number of parameters that uniquely represent a low-rank matrix.Comment: Accepted in IEEE TSP on December 4th, 201
Low-rank Matrix Sensing With Dithered One-Bit Quantization
We explore the impact of coarse quantization on low-rank matrix sensing in
the extreme scenario of dithered one-bit sampling, where the high-resolution
measurements are compared with random time-varying threshold levels. To recover
the low-rank matrix of interest from the highly-quantized collected data, we
offer an enhanced randomized Kaczmarz algorithm that efficiently solves the
emerging highly-overdetermined feasibility problem. Additionally, we provide
theoretical guarantees in terms of the convergence and sample size
requirements. Our numerical results demonstrate the effectiveness of the
proposed methodology.Comment: arXiv admin note: substantial text overlap with arXiv:2308.0069
Calibration Using Matrix Completion with Application to Ultrasound Tomography
We study the calibration process in circular ultrasound tomography devices
where the sensor positions deviate from the circumference of a perfect circle.
This problem arises in a variety of applications in signal processing ranging
from breast imaging to sensor network localization. We introduce a novel method
of calibration/localization based on the time-of-flight (ToF) measurements
between sensors when the enclosed medium is homogeneous. In the presence of all
the pairwise ToFs, one can easily estimate the sensor positions using
multi-dimensional scaling (MDS) method. In practice however, due to the
transitional behaviour of the sensors and the beam form of the transducers, the
ToF measurements for close-by sensors are unavailable. Further, random
malfunctioning of the sensors leads to random missing ToF measurements. On top
of the missing entries, in practice an unknown time delay is also added to the
measurements. In this work, we incorporate the fact that a matrix defined from
all the ToF measurements is of rank at most four. In order to estimate the
missing ToFs, we apply a state-of-the-art low-rank matrix completion algorithm,
OPTSPACE . To find the correct positions of the sensors (our ultimate goal) we
then apply MDS. We show analytic bounds on the overall error of the whole
process in the presence of noise and hence deduce its robustness. Finally, we
confirm the functionality of our method in practice by simulations mimicking
the measurements of a circular ultrasound tomography device.Comment: submitted to IEEE Transaction on Signal Processin