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.
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.
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