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

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Andrey Akinshin, Gil Goldman and Yosef Yomdin
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## 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.

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## Additional Information

**Andrey Akinshin**- Affiliation: Department of Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel
- MR Author ID: 1061721
- ORCID: 0000-0003-3553-9367
- Email: andrey.akinshin@gmail.com
**Gil Goldman**- Affiliation: Department of Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel
- MR Author ID: 1233252
- ORCID: 0000-0002-4087-7320
- Email: gilgoldm@gmail.com
**Yosef Yomdin**- Affiliation: Department of Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel
- MR Author ID: 185690
- Email: yosef.yomdin@weizmann.ac.il
- Received by editor(s): March 20, 2018
- Received by editor(s) in revised form: April 22, 2018, May 8, 2019, and October 31, 2019
- Published electronically: September 9, 2020
- Additional Notes: The research of the second and third authors was supported in part by the Minerva Foundation.
- © Copyright 2020 American Mathematical Society
- Journal: Math. Comp.
**90**(2021), 267-302 - MSC (2010): Primary 65H10, 94A12
- DOI: https://doi.org/10.1090/mcom/3571
- MathSciNet review: 4166461