On Algebraic Properties of Low Rank Approximations of Prony Systems

We consider the reconstruction of spike train signals of the form F(x)=∑i=1daiδ(x-xi),from their moments measurements mk(F)=∫xkF(x)dx=∑i=1daixk. 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, F~. We derive lower bounds for the moments difference between a signal F with d nodes and a signal 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 F~ , with moments m0(F~)=m0(F),…,mp(F~)=mp(F),p>2l-1.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= 2 l- 1 which can be inferred from earlier work.