843 research outputs found
Structured random measurements in signal processing
Compressed sensing and its extensions have recently triggered interest in
randomized signal acquisition. A key finding is that random measurements
provide sparse signal reconstruction guarantees for efficient and stable
algorithms with a minimal number of samples. While this was first shown for
(unstructured) Gaussian random measurement matrices, applications require
certain structure of the measurements leading to structured random measurement
matrices. Near optimal recovery guarantees for such structured measurements
have been developed over the past years in a variety of contexts. This article
surveys the theory in three scenarios: compressed sensing (sparse recovery),
low rank matrix recovery, and phaseless estimation. The random measurement
matrices to be considered include random partial Fourier matrices, partial
random circulant matrices (subsampled convolutions), matrix completion, and
phase estimation from magnitudes of Fourier type measurements. The article
concludes with a brief discussion of the mathematical techniques for the
analysis of such structured random measurements.Comment: 22 pages, 2 figure
A physics-driven CNN model for real-time sea waves 3D reconstruction
One of the most promising techniques for the analysis of Spatio-Temporal ocean wave fields is stereo vision. Indeed, the reconstruction accuracy and resolution typically outperform other approaches like radars, satellites, etc. However, it is computationally expensive so its application is typically restricted to the analysis of short pre-recorded sequences. What prevents such methodology from being truly real-time is the final 3D surface estimation from a scattered, non-equispaced point cloud. Recently, we studied a novel approach exploiting the temporal dependence of subsequent frames to iteratively update the wave spectrum over time. Albeit substantially faster, the unpre-dictable convergence time of the optimization involved still prevents its usage as a continuously running remote sensing infrastructure. In this work, we build upon the same idea, but investigat-ing the feasibility of a fully data-driven Machine Learning (ML) approach. We designed a novel Convolutional Neural Network that learns how to produce an accurate surface from the scattered elevation data of three subsequent frames. The key idea is to embed the linear dispersion relation into the model itself to physically relate the sparse points observed at different times. Assuming that the scattered data are uniformly distributed in the spatial domain, this has the same effect of increasing the sample density of each single frame. Experiments demonstrate how the proposed technique, even if trained with purely synthetic data, can produce accurate and physically consistent surfaces at five frames per second on a modern PC
Tensor Methods for Nonlinear Matrix Completion
In the low rank matrix completion (LRMC) problem, the low rank assumption
means that the columns (or rows) of the matrix to be completed are points on a
low-dimensional linear algebraic variety. This paper extends this thinking to
cases where the columns are points on a low-dimensional nonlinear algebraic
variety, a problem we call Low Algebraic Dimension Matrix Completion (LADMC).
Matrices whose columns belong to a union of subspaces (UoS) are an important
special case. We propose a LADMC algorithm that leverages existing LRMC methods
on a tensorized representation of the data. For example, a second-order
tensorization representation is formed by taking the outer product of each
column with itself, and we consider higher order tensorizations as well. This
approach will succeed in many cases where traditional LRMC is guaranteed to
fail because the data are low-rank in the tensorized representation but not in
the original representation. We also provide a formal mathematical
justification for the success of our method. In particular, we show bounds of
the rank of these data in the tensorized representation, and we prove sampling
requirements to guarantee uniqueness of the solution. Interestingly, the
sampling requirements of our LADMC algorithm nearly match the information
theoretic lower bounds for matrix completion under a UoS model. We also provide
experimental results showing that the new approach significantly outperforms
existing state-of-the-art methods for matrix completion in many situations
- …