119 research outputs found
Scaling Law for Recovering the Sparsest Element in a Subspace
We address the problem of recovering a sparse -vector within a given
subspace. This problem is a subtask of some approaches to dictionary learning
and sparse principal component analysis. Hence, if we can prove scaling laws
for recovery of sparse vectors, it will be easier to derive and prove recovery
results in these applications. In this paper, we present a scaling law for
recovering the sparse vector from a subspace that is spanned by the sparse
vector and random vectors. We prove that the sparse vector will be the
output to one of linear programs with high probability if its support size
satisfies . The scaling law still holds when
the desired vector is approximately sparse. To get a single estimate for the
sparse vector from the linear programs, we must select which output is the
sparsest. This selection process can be based on any proxy for sparsity, and
the specific proxy has the potential to improve or worsen the scaling law. If
sparsity is interpreted in an sense, then the scaling law
can not be better than . Computer simulations show that
selecting the sparsest output in the or thresholded-
senses can lead to a larger parameter range for successful recovery than that
given by the sense
Stable optimizationless recovery from phaseless linear measurements
We address the problem of recovering an n-vector from m linear measurements
lacking sign or phase information. We show that lifting and semidefinite
relaxation suffice by themselves for stable recovery in the setting of m = O(n
log n) random sensing vectors, with high probability. The recovery method is
optimizationless in the sense that trace minimization in the PhaseLift
procedure is unnecessary. That is, PhaseLift reduces to a feasibility problem.
The optimizationless perspective allows for a Douglas-Rachford numerical
algorithm that is unavailable for PhaseLift. This method exhibits linear
convergence with a favorable convergence rate and without any parameter tuning
Numerical verification of a gap condition for a linearized nonlinear Schrödinger equation
We make a detailed numerical study of the spectrum of two Schrödinger operators L± arising from the linearization of the supercritical nonlinear Schrödinger equation (NLS) about the standing wave, in three dimensions. This study was motivated by a recent result of the second author on the conditional asymptotic stability of solitary waves in the case of a cubic nonlinearity. Underlying the validity of this result is a spectral condition on the operators L±, namely that they have no eigenvalues nor resonances in the gap (a region of the positive real axis between zero and the continuous spectrum), which we call the gap property. The present numerical study verifies this spectral condition and shows further that the gap property holds for NLS exponents of the form 2 β + 1, as long as β* < β ≤ 1, where
Our strategy consists of rewriting the original eigenvalue problem via the Birman–Schwinger method. From a numerical analysis viewpoint, our main contribution is an efficient quadrature rule for the kernel 1/|x - y| in {\mathbb R}^3 , i.e. proved spectrally accurate. As a result, we are able to give similar accuracy estimates for all our eigenvalue computations. We also propose an improvement in Petviashvili's iteration for the computation of standing wave profiles which automatically chooses the radial solution.
All our numerical experiments are reproducible. The Matlab code can be downloaded from http://www.acm.caltech.edu/~demanet/NLS/
Convex recovery from interferometric measurements
This note formulates a deterministic recovery result for vectors from
quadratic measurements of the form for some
left-invertible . Recovery is exact, or stable in the noisy case, when the
couples are chosen as edges of a well-connected graph. One possible way
of obtaining the solution is as a feasible point of a simple semidefinite
program. Furthermore, we show how the proportionality constant in the error
estimate depends on the spectral gap of a data-weighted graph Laplacian. Such
quadratic measurements have found applications in phase retrieval, angular
synchronization, and more recently interferometric waveform inversion
Compressive Wave Computation
This paper considers large-scale simulations of wave propagation phenomena.
We argue that it is possible to accurately compute a wavefield by decomposing
it onto a largely incomplete set of eigenfunctions of the Helmholtz operator,
chosen at random, and that this provides a natural way of parallelizing wave
simulations for memory-intensive applications.
This paper shows that L1-Helmholtz recovery makes sense for wave computation,
and identifies a regime in which it is provably effective: the one-dimensional
wave equation with coefficients of small bounded variation. Under suitable
assumptions we show that the number of eigenfunctions needed to evolve a sparse
wavefield defined on N points, accurately with very high probability, is
bounded by C log(N) log(log(N)), where C is related to the desired accuracy and
can be made to grow at a much slower rate than N when the solution is sparse.
The PDE estimates that underlie this result are new to the authors' knowledge
and may be of independent mathematical interest; they include an L1 estimate
for the wave equation, an estimate of extension of eigenfunctions, and a bound
for eigenvalue gaps in Sturm-Liouville problems.
Numerical examples are presented in one spatial dimension and show that as
few as 10 percents of all eigenfunctions can suffice for accurate results.
Finally, we argue that the compressive viewpoint suggests a competitive
parallel algorithm for an adjoint-state inversion method in reflection
seismology.Comment: 45 pages, 4 figure
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