Abstract-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 R L . 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.

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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.

Abstract-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-ofsquares/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 α − z α 0 ) 2 . Those squares are constructed from the ideal I generated by the constraints and the monomials provided by the minimization of the trace. I. NON SYMMETRIC MATRIX COMPLETIONThis paper introduces a deterministic recovery result for non symmetric rank-1 matrix completion by using the Lasserre hierarchy of semidefinite programming relaxations. To our knowledge, the closest result in the current literature is [1] where the authors use the same hierarchy and certify recovery in the case of tensor decomposition. Our paper also shares its deterministic nature with [2] where the authors derive recovery from the spectral properties of a graph Laplacian. Finally, this paper can also be related to [3] in which the authors study noisy tensor completion and use the sixth round of the Lasserre hierarchy to derive probabilistic recovery guarantees.We will use M(r; m × n) to denote the set of matrices of rank r. This set is an algebraic determinantal variety that can be completely characterized through the vanishing of the (r + 1)-minors. This determinantal variety has dimension (m+n−r)r.The general nonsymmetric rank-1 matrix completion problem consists in recovering an unknown matrix X ∈ M(1; m× n) such that X = xy T , given a fixed subset of its entries [4], find X subject to rank(X) = 1As a slight abuse, we also speak of constraints X ij = A ij as belonging to Ω. In relation to problem (1), we introduce the mapping R Ω : R m×n → R |Ω| that corresponds to extracting the observed entries of the matrix. We let R 1 Ω denote the restriction of R Ω to matrices of rank-1, i.e R 1 Ω : M(1; m × n) → R |Ω| . Invertibility of this restriction R 1 Ω is a natural question. In other words, when can one uniquely recover the matrix X from the knowledge of R Ω (X) and the fact that X has rank 1 ?In particular, this paper considers the completion problem on M * (1, m×n), where M * (1, m×n) denotes the restriction of M(1; m × n) to matrices for which none of the entries are zero. The reason for this is that if a rank-1 matrix has a zero element, then the corresponding row or column will be zero, and it is easy to see that the completion problem will generically lack injectivity.Respectively denote by V 1 , V 2 the row and column indices of X. We consider the bipartite graph G(V 1 , V 2 , E) associated to problem (1), where the set of edges in the graph is defined by (i, j) ∈ E iff (i, j) ∈ Ω. The conditions for the recovery of the matrix X from the set Ω are related to the properties of this bipartite graph. In particular, we have the f...

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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.

This paper discusses the recovery of an unknown signal x ∈ R L through the result of its convolution with an unknown filter h ∈ R L . This problem, also known as blind deconvolution, has been studied extensively by the signal processing and applied mathematics communities, leading to a diversity of proofs and algorithms based on various assumptions on the filter and its input. Sparsity of this filter, or in contrast, non vanishing of its Fourier transform are instances of such assumptions. The main result of this paper shows that blind deconvolution can be solved through nuclear norm relaxation in the case of a fully unknown channel, as soon as this channel is probed through a few N µ 2 m K 1/2 input signals xn = Cnmn, n = 1, . . . , N, that are living in known K-dimensional subspaces Cn of R L . This result holds with high probability on the genericity of the subspaces Cn as soon as L K 3/2 and N K 1/2 up to log factors. Our proof system relies on the construction of a certificate of optimality for the underlying convex program. This certificate expands as a Neumann series and is shown to satisfy the conditions for the recovery of the matrix encoding the unknowns by controlling the terms in this series. We apply specific concentration bounds to the first two terms in the series, in order to reduce the sample complexities, and bound the remaining terms through a more general argument. The first term in the series is bounded through the subexponential Bernstein inequality. The second term is decomposed into two contributions, each corresponding to a fourth order gaussian chaos. The first contribution, containing the univariate fourth order monomials in the random vectors defining the subspaces Cn, is bounded through a matrix version of the Rosenthal-Pinelis inequality. The second contribution, containing the cross terms, is bounded by using a decoupling argument for U-Statistics. An incidental consequence of the result of this paper, following from the lack of assumptions on the filter, is that nuclear norm relaxation can be extended from blind deconvolution to blind super-resolution, as soon as the unknown ideal low pass filter has a sufficiently large support compared to the ambient dimension L. Numerical experiments supporting the theory as well as its application to blind super-resolution are provided.Acknowledgement. AC was supported by the FNRS, FSMP, BAEF and Francqui Foundations. AC thanks MIT Math, Harvard IACS and The University of Chicago, for hosting him during this work. AC is grateful to Laurent Demanet and Ali Ahmed for interesting discussions as well as Joel Tropp for pointing out the matrix version of the Rosenthal-Pinelis inequality.

Let µ(t) = ∑ τ ∈S ατ δ(t − τ ) denote an S -atomic measure defined on [0, 1], satisfying min, denote the polynomial obtained from the Dirichlet kernel Dn(θ) = 1 n+1 ∑ k ≤n e 2πikθ and its derivative by solving the system {η(τ ) = 1, η ′ (τ ) = 0, ∀τ ∈ S}. We provide evidence that for sufficiently large n, ∆ ≳ S 2 n −1 , the non negative polynomial 1 − η(θ) 2 which vanishes at the atoms τ ∈ S, and is bounded by 1 everywhere else on the [0, 1] interval, can be written as a sum-of-squares with associated Gram matrix of rank n − S . Unlike previous work, our approach does not rely on the Fejér-Riesz Theorem, which prevents developing intuition on the Gram matrix, but requires instead a lower bound on the singular values of a (truncated) large (O(1e10)) matrix. Despite the memory requirements which currently prevent dealing with such a matrix efficiently, we show how such lower bounds can be derived through Power iterations and convolutions with special functions for sizes up to O(1e7). We also provide numerical simulations suggesting that the spectrum remains approximately constant with the truncation size as soon as this size is larger than 100.Acknowledgement. This work was funded by the Fondation Sciences Mathématiques de Paris, the CNRS and the Air Force Office of Scientific research by means of AFOSR Grant FA9550-18-1-7007. AC is grateful to Gabriel Peyré and Irène Waldspurger for their help and valuable comments. AC also acknowledges funding from the FNRS and the Francqui Foundation.