Compressed Super-Resolution I: Maximal Rank Sum-of-Squares

Abstract

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.

Compressed Super-Resolution of Positive Sources

“…Inspired by [41], [42], we propose a novel approach to reduce the computational complexity of (ANM) by projecting the positivity constraint T (x) 0 to a lower dimension. Consider a matrix M ∈ C M ×N with M ≤ N and full rank, i.e.…”

“…Finally, numerical experiments are provided to demonstrate the effectiveness of the proposed algorithm. Our work is related to the compressed ANM proposed in [41], [42], but focuses on the positive case where we provide guarantees without imposing any separation condition on the sources.…”