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

Scaling Law for Recovering the Sparsest Element in a Subspace

We address the problem of recovering a sparse $n$-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 $k$ random vectors. We prove that the sparse vector will be the output to one of $n$ linear programs with high probability if its support size $s$ satisfies $s \lesssim n/\sqrt{k \log n}$. The scaling law still holds when the desired vector is approximately sparse. To get a single estimate for the sparse vector from the $n$ 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 $\ell_1/\ell_\infty$ sense, then the scaling law can not be better than $s \lesssim n/\sqrt{k}$. Computer simulations show that selecting the sparsest output in the $\ell_1/\ell_2$ or thresholded-$\ell_0$ senses can lead to a larger parameter range for successful recovery than that given by the $\ell_1/\ell_\infty$ sense

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 $$\begin{equation*}\beta_{\ast} = 0.913\,958\,905 \pm 1e-8.\end{equation*}$$ 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/

Does socio-economic disadvantage lead to acting out? A reinvigoration of an old question

Research into socio-economic determinants of school deviance is inconclusive. Recently, scholars argued that economic deprivation, rather than SES background, affects delinquency. Using multilevel analyses on data of 9,174 students across 111 schools in 4 European cities (2013-2014), we investigate the association of SES and economic deprivation with school-deviant behavior. Furthermore, we study the role of academic self-efficacy. Lower-SES and deprived students might perceive goal blockage with regard to study-related goals, leading to deviant coping – that is self-efficacy as mediator – or self-efficacy might condition SES and deprivation effects – that is self-efficacy as moderator. Results showed that deprivation relates to school-deviant behavior. This association was not mediated, nor moderated, by academic self-efficacy. The relationship with SES was moderated by academic self-efficacy. We conclude that deprived and lower SES-students are prone to break school rules, the latter more so when feeling less competent at reaching academic goals

Convex recovery from interferometric measurements

This note formulates a deterministic recovery result for vectors $x$ from quadratic measurements of the form $(Ax)_i \overline{(Ax)_j}$ for some left-invertible $A$. Recovery is exact, or stable in the noisy case, when the couples $(i,j)$ 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