In the recovery of sparse vectors from quadratic measurements, the presence of linear terms breaks the square root bottleneck

Motivated by recent results in the statistical physics of spin glasses, we study the recovery of a sparse vector $\mathbf{x}_0\in \mathbb{S}^{n-1}$, $\|\mathbf{x}_0\|_{\ell_0} = k<n$, from $m$ quadratic measurements of the form $(1-\lambda)\langle \mathbf{A}_i, \mathbf{x}_0\mathbf{x}_0^T \rangle + \lambda \langle\mathbf{c}_i,\mathbf{x}_0 \rangle$ where $\mathbf{A}_{i}, \mathbf{c}_{i}$ have i.i.d Gaussian entries. This can be related to a constrained version of the 2-spin Hamiltonian with external field for which it was recently shown (in the absence of any structural constraint and in the asymptotic regime) that the geometry of the energy landscape becomes trivial above a certain threshold $\lambda > \lambda_c\in (0,1)$. Building on this idea we study the evolution of the so-called square root bottleneck for $\lambda\in [0,1]$ in the setting of the sparse rank one matrix recovery/sensing problem. We show that recovery of the vector $\mathbf{x}_0$ can be guaranteed as soon as $m\gtrsim k^2 (1-\lambda)^2/\lambda^2$, $\lambda \gtrsim k^{-1/2}$ provided that this vector satisfies a sufficiently strong incoherence condition, thus retrieving the linear regime for an external field $(1-\lambda)/\lambda \lesssim k^{-1/2}$. Our proof relies on an interpolation between the linear and quadratic settings, as well as on standard convex geometry arguments

A short note on rank-2 relaxation for waveform inversion

This note is a first attempt to perform waveform inversion by utilizing recent developments in semidefinite relaxations for polynomial equations to mitigate non-convexity. The approach consists in reformulating the inverse problem as a set of constraints on a low-rank moment matrix in a higher-dimensional space. While this idea has mostly been a theoretical curiosity so far, the novelty of this note is the suggestion that a modified adjoint-state method enables algorithmic scalability of the relaxed formulation to standard 2D community models in geophysical imaging. Numerical experiments show that the new formulation leads to a modest increase in the basin of attraction of least-squares waveform inversion.TOTAL (Firm)Belgian National Foundation for Scientific ResearchMIT International Science and Technology InitiativesUnited States. Air Force. Office of Scientific ResearchUnited States. Office of Naval ResearchNational Science Foundation (U.S.

Semidefinite programming relaxations for matrix completion, inverse scattering and blind deconvolution

The thesis studies semidefinite programming relaxations for three instances of the general affine rank minimization problem. The first instance, rank one matrix completion, was known to be solved by non-linear propagation algorithms without stability guarantees. The thesis closes the line of work on this problem by introducing a stable algorithm based on two levels of semidefinite programming relaxation. For this algorithm, recovery of the unknown matrix is first certified in the absence of noise, at the information limit, through the construction of a dual (sum of squares) polynomial. In passing, this dual polynomial also provides a rationale for the use of the trace norm in semidefinite programming. The dual polynomial is then used to derive a stability estimate for the noisy version of the problem. For the second instance, inverse scattering, the thesis introduces a fast algorithm which leverages the traditional Adjoint State method by lifting the search space. The algorithm is based on a first level of semidefinite programming relaxation encoded through a low rank factorization which guarantees its scalability. Numerical experiments on 2D community models show a modest increase with respect to the basin of attraction of traditional least squares waveform inversion. A geometric intuition for this improvement is provided. Finally, the thesis studies nuclear norm relaxation for the blind deconvolution problem when multiple input signals are considered. Recovery is certified through the construction of a dual certificate. A candidate certificate is first constructed through the golfing scheme. Relying on the recent line of work on blind deconvolution, this candidate certificate is then shown to satisfy the conditions from duality theory for the recovery of the rank one matrix encoding the impulse response and input signals through the Orlicz version of the Bernstein concentration bound. ​(FSA - Sciences de l'ingénieur) -- UCL, 201

Stable Rank-One Matrix Completion is Solved by the Level 2 Lasserre Relaxation

Abstract This paper studies the problem of deterministic rank-one matrix completion. It is known that the simplest semidefinite programming relaxation, involving minimization of the nuclear norm, does not in general return the solution for this problem. In this paper, we show that in every instance where the problem has a unique solution, one can provably recover the original matrix through the level 2 Lasserre relaxation with minimization of the trace norm. We further show that the solution of the proposed semidefinite program is Lipschitz stable with respect to perturbations of the observed entries, unlike more basic algorithms such as nonlinear propagation or ridge regression. Our proof is based on recursively building a certificate of optimality corresponding to a dual sum-of-squares (SoS) polynomial. This SoS polynomial is built from the polynomial ideal generated by the completion constraints and the monomials provided by the minimization of the trace. The proposed relaxation fits in the framework of the Lasserre hierarchy, albeit with the key addition of the trace objective function. Finally, we show how to represent and manipulate the moment tensor in favorable complexity by means of a hierarchical low-rank factorization

Rank-one matrix completion is solved by the sum-of-squares relaxation of order two

This note studies the problem of nonsymmetric rank-one matrix completion. We show that in every instance where the problem has a unique solution, one can recover the original matrix through the second round of the sum-of-squares/Lasserre hierarchy with minimization of the trace of the moments matrix. Our proof system is based on iteratively building a sum of N - 1 linearly independent squares, where N is the number of monomials of degree at most two, corresponding to the canonical basis (z[superscript α] - z[subscript 0][superscript α])[superscript 2]. Those squares are constructed from the ideal I generated by the constraints and the monomials provided by the minimization of the trace.United States. Office of Naval ResearchNational Science Foundation (U.S.)TOTAL (Firm

A convex approach to blind deconvolution with diverse inputs

This note considers the problem of blind identification of a linear, time-invariant (LTI) system when the input signals are unknown, but belong to sufficiently diverse, known subspaces. This problem can be recast as the recovery of a rank-1 matrix, and is effectively relaxed using a semidefinite program (SDP). We show that exact recovery of both the unknown impulse response, and the unknown inputs, occurs when the following conditions are met: (1) the impulse response function is spread in the Fourier domain, and (2) the N input vectors belong to generic, known subspaces of dimension K in ℝL. Recent results in the well-understood area of low-rank recovery from underdetermined linear measurements can be adapted to show that exact recovery occurs with high probablility (on the genericity of the subspaces) provided that K,L, and N obey the information-theoretic scalings, namely L ≳ K and N ≳ 1 up to log factors.Fonds national de la recherche scientifique (Belgium)MIT International Science and Technology InitiativesUnited States. Air Force. Office of Scientific ResearchUnited States. Office of Naval ResearchNational Science Foundation (U.S.)Total S

Diffeomorphic surface-based registration for MR-US fusion in prostate brachytherapy

In nowadays prostate interstitial brachytherapy, transrectal ultrasound guidance is typically used to localize the prostate and the rectum. However, the ability of ultrasound to distinct normal from cancer tissues is weak. In most treatments, the organ as a whole is thus irradiated, leading to numerous side effects such as urinary or sexual dysfunction. The quality of magnetic resonance images (MRI), on the other hand, has improved over the last few years and allows for an accurate delineation of the tumor [1]. This paper proposes a novel framework for MRI-US surface registration of the prostate and the rectum. The preoperative MRI is first segmented using a multiresolution graph-cut method. The intraoperative US image is manually segmented by the surgeon. The registration is then performed in two steps. In a first step, a bi-affine registration is performed on the surfaces of the rectum and the prostate using an expectation maximization iterative closest point method (EM-ICP). In the second step, non-rigid registration is applied to the distance maps resulting from the pre-registered surfaces. Our approach has been applied on 5 MR/US pairs and shows a relative independence between prostate and rectum motions