TY - JOUR

T1 - On Algebraic Properties of Low Rank Approximations of Prony Systems

AU - Goldman, Gil

AU - Yomdin, Yosef

N1 - Publisher Copyright:
© 2018, Springer Nature Switzerland AG.

PY - 2019/9/1

Y1 - 2019/9/1

N2 - 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 xi 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.

AB - 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 xi 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.

KW - Moments inversion

KW - Non-linear models

KW - Signal acquisition

KW - Singularities

UR - http://www.scopus.com/inward/record.url?scp=85051113359&partnerID=8YFLogxK

U2 - 10.1007/s11785-018-0829-y

DO - 10.1007/s11785-018-0829-y

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AN - SCOPUS:85051113359

SN - 1661-8254

VL - 13

SP - 2799

EP - 2811

JO - Complex Analysis and Operator Theory

JF - Complex Analysis and Operator Theory

IS - 6

ER -