10,258 research outputs found
Dynamical Functional Theory for Compressed Sensing
We introduce a theoretical approach for designing generalizations of the
approximate message passing (AMP) algorithm for compressed sensing which are
valid for large observation matrices that are drawn from an invariant random
matrix ensemble. By design, the fixed points of the algorithm obey the
Thouless-Anderson-Palmer (TAP) equations corresponding to the ensemble. Using a
dynamical functional approach we are able to derive an effective stochastic
process for the marginal statistics of a single component of the dynamics. This
allows us to design memory terms in the algorithm in such a way that the
resulting fields become Gaussian random variables allowing for an explicit
analysis. The asymptotic statistics of these fields are consistent with the
replica ansatz of the compressed sensing problem.Comment: 5 pages, accepted for ISIT 201
Belief propagation decoding of quantum channels by passing quantum messages
Belief propagation is a powerful tool in statistical physics, machine
learning, and modern coding theory. As a decoding method, it is ubiquitous in
classical error correction and has also been applied to stabilizer-based
quantum error correction. The algorithm works by passing messages between nodes
of the factor graph associated with the code and enables efficient decoding, in
some cases even up to the Shannon capacity of the channel. Here we construct a
belief propagation algorithm which passes quantum messages on the factor graph
and is capable of decoding the classical-quantum channel with pure state
outputs. This gives explicit decoding circuits whose number of gates is
quadratic in the blocklength of the code. We also show that this decoder can be
modified to work with polar codes for the pure state channel and as part of a
polar decoder for transmitting quantum information over the amplitude damping
channel. These represent the first explicit capacity-achieving decoders for
non-Pauli channels.Comment: v3: final version for publication; v2: improved discussion of the
algorithm; 7 pages & 2 figures. v1: 6 pages, 1 figur
Unitary Approximate Message Passing for Sparse Bayesian Learning and Bilinear Recovery
Over the past several years, the approximate message passing (AMP) algorithm has been applied to a broad range of problems, including compressed sensing (CS), robust regression, Bayesian estimation, etc. AMP was originally developed for compressed sensing based on the loopy belief propagation (BP). Compared to convex optimization based algorithms, AMP has low complexity and its performance can be rigorously characterized by a scalar state evolution (SE) in the case of a large independent and identically distributed (i.i.d.) (sub-) Gaussian matrix. AMP was then extended to solve general estimation problems with a generalized linear observation model. However, AMP performs poorly on a generic matrix such as non-zero mean, rank-deficient, correlated, or ill-conditioned matrix, resulting in divergence and degraded performance. It was discovered later that applying AMP to a unitary transform of the original model can remarkably enhance the robustness to difficult matrices. This variant is named unitary AMP (UAMP), or formally called UTAMP. In this thesis, leveraging UAMP, we propose UAMP-SBL for sparse signal recovery and Bi-UAMP for bilinear recovery, both of which inherit the low complexity and robustness of UAMP.
Sparse Bayesian learning (SBL) is a powerful tool for recovering a sparse signal from noisy measurements, which finds numerous applications in various areas. As a traditional implementation of SBL, e.g., Tippingâs method, involves matrix inversion in each iteration, the computational complexity can be prohibitive for large scale problems. To circumvent this, AMP and its variants have been used as low-complexity solutions. Unfortunately, they will diverge for âdifficultâ measurement matrices as previously mentioned. In this thesis, leveraging UAMP, a novel SBL algorithm called UAMP-SBL is proposed where UAMP is incorporated into the structured variational message passing (SVMP) to handle the most computationally intensive part of message computations. It is shown that, compared to state-of-the-art AMP based SBL algorithms, the proposed UAMP-SBL is more robust and efficient, leading to remarkably better performance.
The bilinear recovery problem has many applications such as dictionary learning, selfcalibration, compressed sensing with matrix uncertainty, etc. Compared to existing nonmessage passing alternates, several AMP based algorithms have been developed to solve bilinear problems. By using UAMP, a more robust and faster approximate inference algorithm for bilinear recovery is proposed in this thesis, which is called Bi-UAMP. With the lifting approach, the original bilinear problem is reformulated as a linear one. Then, variational inference (VI), expectation propagation (EP) and BP are combined with UAMP to implement the approximate inference algorithm Bi-UAMP, where UAMP is adopted for the most computationally intensive part. It is shown that, compared to state-of-the-art bilinear recovery algorithms, the proposed Bi-UAMP is much more robust and faster, and delivers significantly better performance.
Recently, UAMP has also been employed for many other applications such as inverse synthetic aperture radar (ISAR) imaging, low-complexity direction of arrival (DOA) estimation, iterative detection for orthogonal time frequency space modulation (OTFS), channel estimation for RIS-Aided MIMO communications, etc. Promising performance was achieved in all of the applications, and more applications of UAMP are expected in the future
Orthogonal Transform Multiplexing with Memoryless Nonlinearity: a Possible Alternative to Traditional Coded-Modulation Schemes
In this paper, we propose a novel joint coding-modulation technique based on
serial concatenation of orthogonal linear transform, such as discrete Fourier
transform (DFT) or Walsh-Hadamard transform (WHT), with memoryless
nonlinearity. We demonstrate that such a simple signal construction may exhibit
properties of a random code ensemble, as a result approaching channel capacity.
Our computer simulations confirm that if the decoder relies on a modified
approximate message passing algorithm, the proposed modulation technique
exhibits performance on par with state-of-the-art coded modulation schemes that
use capacity-approaching component codes. The proposed modulation scheme could
be used directly or as a pre-coder for a conventional orthogonal frequency
division multiplexing (OFDM) transmitter, resulting in a system possessing all
benefits of OFDM along with reduced peak-to-average power ratio (PAPR)
Reexamination of Quantum Bit Commitment: the Possible and the Impossible
Bit commitment protocols whose security is based on the laws of quantum
mechanics alone are generally held to be impossible. In this paper we give a
strengthened and explicit proof of this result. We extend its scope to a much
larger variety of protocols, which may have an arbitrary number of rounds, in
which both classical and quantum information is exchanged, and which may
include aborts and resets. Moreover, we do not consider the receiver to be
bound to a fixed "honest" strategy, so that "anonymous state protocols", which
were recently suggested as a possible way to beat the known no-go results are
also covered. We show that any concealing protocol allows the sender to find a
cheating strategy, which is universal in the sense that it works against any
strategy of the receiver. Moreover, if the concealing property holds only
approximately, the cheat goes undetected with a high probability, which we
explicitly estimate. The proof uses an explicit formalization of general two
party protocols, which is applicable to more general situations, and a new
estimate about the continuity of the Stinespring dilation of a general quantum
channel. The result also provides a natural characterization of protocols that
fall outside the standard setting of unlimited available technology, and thus
may allow secure bit commitment. We present a new such protocol whose security,
perhaps surprisingly, relies on decoherence in the receiver's lab.Comment: v1: 26 pages, 4 eps figures. v2: 31 pages, 5 eps figures; replaced
with published version; title changed to comply with puzzling Phys. Rev.
regulations; impossibility proof extended to protocols with infinitely many
rounds or a continuous communication tree; security proof of decoherence
monster protocol expanded; presentation clarifie
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