Geometry of error amplification in solving the Prony system with near-colliding nodes

Abstract:

We consider a reconstruction problem for “spike-train” signals $F$ of an a priori known form $F(x)=\sum _{j=1}^{d}a_{j}\delta \left (x-x_{j}\right ),$ from their moments $m_k(F)=\int x^kF(x)dx.$ We assume that the moments $m_k(F)$, $k=0,1,\ldots ,2d-1$, are known with an absolute error not exceeding $\epsilon > 0$. This problem is essentially equivalent to solving the Prony system $\sum _{j=1}^d a_jx_j^k=m_k(F), \ k=0,1,\ldots ,2d-1.$

We study the “geometry of error amplification” in reconstruction of $F$ from $m_k(F),$ in situations where the nodes $x_1,\ldots ,x_d$ near-collide, i.e., form a cluster of size $h \ll 1$. We show that in this case, error amplification is governed by certain algebraic varieties in the parameter space of signals $F$, which we call the “Prony varieties”.

Based on this we produce lower and upper bounds, of the same order, on the worst case reconstruction error. In addition we derive separate lower and upper bounds on the reconstruction of the amplitudes and the nodes.

Finally we discuss how to use the geometry of the Prony varieties to improve the reconstruction accuracy given additional a priori information.