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

Abstract

Let $\mu(t) = \sum_{\tau\in S} \alpha_\tau \delta(t-\tau)$ denote an $|S|$-atomic measure defined on $[0,1]$, satisfying $\min_{\tau\neq \tau'}|\tau - \tau'|\geq |S|\cdot n^{-1}$. Let $\eta(\theta) = \sum_{\tau\in S} a_\tau D_n(\theta - \tau) + b_\tau D'_n(\theta - \tau)$, denote the polynomial obtained from the Dirichlet kernel $D_n(\theta) = \frac{1}{n+1}\sum_{|k|\leq n} e^{2\pi i k \theta}$ and its derivative by solving the system $\left\{\eta(\tau) = 1, \eta'(\tau) = 0,\; \forall \tau \in S\right\}$. We provide evidence that for sufficiently large $n$, $\Delta> |S|^2 n^{-1}$, the non negative polynomial $1 - |\eta(\theta)|^2$ which vanishes at the atoms $\tau \in S$, 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 Fejer-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 $100$.