TY - JOUR

T1 - GEOMETRY OF ERROR AMPLIFICATION IN SOLVING THE PRONY SYSTEM WITH NEAR-COLLIDING NODES

AU - Akinshin, Andrey

AU - Goldman, Gil

AU - Yomdin, Yosef

N1 - Publisher Copyright:
© 2021. All Rights Reserved.

PY - 2021/1

Y1 - 2021/1

N2 - We consider a reconstruction problem for “spike-train” signals F of an a priori known form F(x) =∑dj=i ajδ (x — Xj), from their moments mk(F)= xkF(x)dx. We assume that the moments mk(F), k =0, 1,…,2d-1, are known with an absolute error not exceeding ε > 0. This problem is essentially equivalent to solving the Prony system ∑jd=1 ajxjk = mk(F),k= 0, 1,…,2d — 1. We study the “geometry of error amplification” in reconstruction of F from mk(F), in situations where the nodes x1,…,xd near-collide, i.e., form a cluster of size h 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.

AB - We consider a reconstruction problem for “spike-train” signals F of an a priori known form F(x) =∑dj=i ajδ (x — Xj), from their moments mk(F)= xkF(x)dx. We assume that the moments mk(F), k =0, 1,…,2d-1, are known with an absolute error not exceeding ε > 0. This problem is essentially equivalent to solving the Prony system ∑jd=1 ajxjk = mk(F),k= 0, 1,…,2d — 1. We study the “geometry of error amplification” in reconstruction of F from mk(F), in situations where the nodes x1,…,xd near-collide, i.e., form a cluster of size h 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.

UR - http://www.scopus.com/inward/record.url?scp=85100137264&partnerID=8YFLogxK

U2 - 10.1090/mcom/3571

DO - 10.1090/mcom/3571

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:85100137264

SN - 0025-5718

VL - 90

SP - 267

EP - 302

JO - Mathematics of Computation

JF - Mathematics of Computation

IS - 327

ER -