On algebraic properties of low rank approximations of Prony systems

We consider the reconstruction of spike train signals of the form $$F(x) = \sum_{i=1}^d a_i \delta(x-x_i),$$ from their moments measurements $m_k(F)=\int x^k F(x) dx = \sum_{i=1}^d a_ix^k$. When some of the nodes $x_i$ near collide the inversion becomes unstable. Given noisy moments measurements, a typical consequence is that reconstruction algorithms estimate the signal $F$ with a signal having fewer nodes, $\tilde{F}$. We derive lower bounds for the moments difference between a signal $F$ with $d$ nodes and a signal $\tilde{F}$ with strictly less nodes, $l$. Next we consider the geometry of the non generic case of $d$ nodes signals $F$, for which there exists an $l<d$ nodes signal $\tilde{F}$, with moments \begin{align*} m_0(\tilde{F})=m_{0}(F),\ldots,m_{p}(\tilde{F})=m_{p}(F),&& p>2l-1 . \end{align*} We give a complete description for the case of a general $d$, $l=1$ and $p=2$. We give a reference for the case $p=2l-1$ which can be inferred from earlier work.