236 research outputs found
Hierarchical isometry properties of hierarchical measurements
Compressed sensing studies linear recovery problems under structure assumptions. We introduce a new class of measurement operators, coined hierarchical measurement operators, and prove results guaranteeing the efficient, stable and robust recovery of hierarchically structured signals from such measurements. We derive bounds on their hierarchical restricted isometry properties based on the restricted isometry constants of their constituent matrices, generalizing and extending prior work on Kronecker-product measurements. As an exemplary application, we apply the theory to two communication scenarios. The fast and scalable HiHTP algorithm is shown to be suitable for solving these types of problems and its performance is evaluated numerically in terms of sparse signal recovery and block detection capability
Hierarchical compressed sensing
Compressed sensing is a paradigm within signal processing that provides the
means for recovering structured signals from linear measurements in a highly
efficient manner. Originally devised for the recovery of sparse signals, it has
become clear that a similar methodology would also carry over to a wealth of
other classes of structured signals. In this work, we provide an overview over
the theory of compressed sensing for a particularly rich family of such
signals, namely those of hierarchically structured signals. Examples of such
signals are constituted by blocked vectors, with only few non-vanishing sparse
blocks. We present recovery algorithms based on efficient hierarchical
hard-thresholding. The algorithms are guaranteed to converge, in a stable
fashion both with respect to measurement noise as well as to model mismatches,
to the correct solution provided the measurement map acts isometrically
restricted to the signal class. We then provide a series of results
establishing the required condition for large classes of measurement ensembles.
Building upon this machinery, we sketch practical applications of this
framework in machine-type communications and quantum tomography.Comment: This book chapter is a report on findings within the DFG-funded
priority program `Compressed Sensing in Information Processing' (CoSIP
A Kronecker-Based Sparse Compressive Sensing Matrix for Millimeter Wave Beam Alignment
Millimeter wave beam alignment (BA) is a challenging problem especially for
large number of antennas. Compressed sensing (CS) tools have been exploited due
to the sparse nature of such channels. This paper presents a novel
deterministic CS approach for BA. Our proposed sensing matrix which has a
Kronecker-based structure is sparse, which means it is computationally
efficient. We show that our proposed sensing matrix satisfies the restricted
isometry property (RIP) condition, which guarantees the reconstruction of the
sparse vector. Our approach outperforms existing random beamforming techniques
in practical low signal to noise ratio (SNR) scenarios.Comment: Accepted to 13th International Conference on Signal Processing and
Communication Systems (ICSPCS'2019
Low-rank Tensor Recovery
Low-rank tensor recovery is an interesting subject from both the theoretical and application point of view. On one side, it is a natural extension of the sparse vector and low-rank matrix recovery problem. On the other side, estimating a low-rank tensor has applications in many different areas such as machine learning, video compression, and seismic data interpolation. In this thesis, two approaches are introduced. The first approach is a convex optimization approach and could be considered as a tractable extension of -minimization for sparse vector and nuclear norm minimization for matrix recovery to tensor scenario. It is based on theta bodies – a recently introduced tool from real algebraic geometry. In particular, theta bodies of appropriately defined polynomial ideal correspond to the unit-theta norm balls. These unit-theta norm balls are relaxations of the unit-tensor-nuclear norm ball. Thus, in this case, we consider a canonical tensor format. The method requires computing the reduced Groebner basis (with respect to the graded reverse lexicographic ordering) of the appropriately defined polynomial ideal. Numerical results for third-order tensor recovery via -norm are provided. The second approach is a generalization of iterative hard thresholding algorithm for sparse vector and low-rank matrix recovery to tensor scenario (tensor IHT or TIHT algorithm). Here, we consider the Tucker format, the tensor train decomposition, and the hierarchical Tucker decomposition. The analysis of the algorithm is based on a version of the restricted isometry property (tensor RIP or TRIP) adapted to the tensor decomposition at hand. We show that subgaussian measurement ensembles satisfy TRIP with high probability under an almost optimal condition on the number of measurements. Additionally, we show that partial Fourier maps combined with random sign flips of the tensor entries satisfy TRIP with high probability. Under the assumption that the linear operator satisfies TRIP and under an additional assumption on the thresholding operator, we provide a linear convergence result for the TIHT algorithm. Finally, we present numerical results on low-Tucker-rank third-order tensors via partial Fourier maps combined with random sign flips of tensor entries, tensor completion, and Gaussian measurement ensembles
Semi-device-dependent blind quantum tomography
Extracting tomographic information about quantum states is a crucial task in
the quest towards devising high-precision quantum devices. Current schemes
typically require measurement devices for tomography that are a priori
calibrated to a high precision. Ironically, the accuracy of the measurement
calibration is fundamentally limited by the accuracy of state preparation,
establishing a vicious cycle. Here, we prove that this cycle can be broken and
the fundamental dependence on the measurement devices significantly relaxed. We
show that exploiting the natural low-rank structure of quantum states of
interest suffices to arrive at a highly scalable blind tomography scheme with a
classically efficient post-processing algorithm. We further improve the
efficiency of our scheme by making use of the sparse structure of the
calibrations. This is achieved by relaxing the blind quantum tomography problem
to the task of de-mixing a sparse sum of low-rank quantum states. Building on
techniques from model-based compressed sensing, we prove that the proposed
algorithm recovers a low-rank quantum state and the calibration provided that
the measurement model exhibits a restricted isometry property. For generic
measurements, we show that our algorithm requires a close-to-optimal number
measurement settings for solving the blind tomography task. Complementing these
conceptual and mathematical insights, we numerically demonstrate that blind
quantum tomography is possible by exploiting low-rank assumptions in a
practical setting inspired by an implementation of trapped ions using
constrained alternating optimization.Comment: 22 pages, 8 Figure
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