56 research outputs found
Efficient estimation of nearly sparse many-body quantum Hamiltonians
We develop an efficient and robust approach to Hamiltonian identification for
multipartite quantum systems based on the method of compressed sensing. This
work demonstrates that with only O(s log(d)) experimental configurations,
consisting of random local preparations and measurements, one can estimate the
Hamiltonian of a d-dimensional system, provided that the Hamiltonian is nearly
s-sparse in a known basis. We numerically simulate the performance of this
algorithm for three- and four-body interactions in spin-coupled quantum dots
and atoms in optical lattices. Furthermore, we apply the algorithm to
characterize Hamiltonian fine structure and unknown system-bath interactions.Comment: 8 pages, 2 figures. Title is changed. Detailed error analysis is
added. Figures are updated with additional clarifying discussion
High-dimensional wave atoms and compression of seismic datasets
Wave atoms are a low-redundancy alternative to curvelets, suitable for high-dimensional seismic data processing. This abstract extends the wave atom orthobasis construction to 3D, 4D, and 5D Cartesian arrays, and parallelizes it in a shared-memory environment. An implementation of the algorithm for NVIDIA CUDA capable graphics processing units (GPU) is also developed to accelerate computation for 2D and 3D data. The new transforms are benchmarked against the Fourier transform for compression of data generated from synthetic 2D and 3D acoustic models.National Science Foundation (U.S.); Alfred P. Sloan Foundatio
Beyond convergence rates: Exact recovery with Tikhonov regularization with sparsity constraints
The Tikhonov regularization of linear ill-posed problems with an
penalty is considered. We recall results for linear convergence rates and
results on exact recovery of the support. Moreover, we derive conditions for
exact support recovery which are especially applicable in the case of ill-posed
problems, where other conditions, e.g. based on the so-called coherence or the
restricted isometry property are usually not applicable. The obtained results
also show that the regularized solutions do not only converge in the
-norm but also in the vector space (when considered as the
strict inductive limit of the spaces as tends to infinity).
Additionally, the relations between different conditions for exact support
recovery and linear convergence rates are investigated.
With an imaging example from digital holography the applicability of the
obtained results is illustrated, i.e. that one may check a priori if the
experimental setup guarantees exact recovery with Tikhonov regularization with
sparsity constraints
Sparsity without the Complexity: Loss Localisation using Tree Measurements
We study network loss tomography based on observing average loss rates over a
set of paths forming a tree -- a severely underdetermined linear problem for
the unknown link loss probabilities. We examine in detail the role of sparsity
as a regularising principle, pointing out that the problem is technically
distinct from others in the compressed sensing literature. While sparsity has
been applied in the context of tomography, key questions regarding uniqueness
and recovery remain unanswered. Our work exploits the tree structure of path
measurements to derive sufficient conditions for sparse solutions to be unique
and the condition that minimization recovers the true underlying
solution. We present a fast single-pass linear algorithm for
minimization and prove that a minimum solution is both unique and
sparsest for tree topologies. By considering the placement of lossy links
within trees, we show that sparse solutions remain unique more often than is
commonly supposed. We prove similar results for a noisy version of the problem
Matrix-free interior point method for compressed sensing problems
We consider a class of optimization problems for sparse signal reconstruction
which arise in the field of Compressed Sensing (CS). A plethora of approaches
and solvers exist for such problems, for example GPSR, FPC AS, SPGL1, NestA,
\ell_{1}_\ell_{s}, PDCO to mention a few. Compressed Sensing applications
lead to very well conditioned optimization problems and therefore can be solved
easily by simple first-order methods. Interior point methods (IPMs) rely on the
Newton method hence they use the second-order information. They have numerous
advantageous features and one clear drawback: being the second-order approach
they need to solve linear equations and this operation has (in the general
dense case) an computational complexity. Attempts have been made to
specialize IPMs to sparse reconstruction problems and they have led to
interesting developments implemented in and PDCO softwares. We
go a few steps further. First, we use the matrix-free interior point method, an
approach which redesigns IPM to avoid the need to explicitly formulate (and
store) the Newton equation systems. Secondly, we exploit the special features
of the signal processing matrices within the matrix-free IPM. Two such features
are of particular interest: an excellent conditioning of these matrices and the
ability to perform inexpensive (low complexity) matrix-vector multiplications
with them. Computational experience with large scale one-dimensional signals
confirms that the new approach is efficient and offers an attractive
alternative to other state-of-the-art solvers
Compressed Sensing of Speech Signals Based on Linear Prediction Coefficients and Difference Transformation
A New Algorithm Based on Linearized Bregman Iteration with Generalized Inverse for Compressed Sensing
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