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

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Geometry of error amplification in solving the Prony system with near-colliding nodes
Mathematics of Computation ( IF 2.2 ) Pub Date : 2020-09-09 , DOI: 10.1090/mcom/3571
Andrey Akinshin , Gil Goldman , Yosef Yomdin

Abstract:We consider a reconstruction problem for ``spike-train'' signals

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

, $k=0,1,\ldots ,2d-1$ , 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

from

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

, 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.

[1] Andrey Akinshin, Dmitry Batenkov, and Yosef Yomdin,
Accuracy of spike-train Fourier reconstruction for colliding nodes,
2015 International Conference on Sampling Theory and Applications (SampTA), pages 617-621. IEEE, 2015.
[2]

中文翻译：

节点冲突的Prony系统求解中误差放大的几何

摘要：我们考虑``穗列车'重建问题的信号

先验已知形式的从他们的时刻，我们假设的时刻，都会被加上绝对误差不超过已知的。这个问题从本质上讲等于解决Prony系统。在节点接近碰撞（即形成大小簇）的情况下，我们从重建中研究了``误差放大的几何'' 。我们表明，在这种情况下，误差放大受信号参数空间中某些代数形式的支配。 $F（x）= \ sum _ {j = 1} ^ {d} a_ {j} \ delta \ left（x-x_ {j} \ right），$ $m_k（F）= \ int x ^ kF（x）dx。$

$k = 0,1，\ ldots，2d-1$ $\ epsilon> 0$ $\ sum _ {j = 1} ^ d a_jx_j ^ k = m_k（F），\ k = 0,1，\ ldots，2d-1。$

$x_1，\ ldots，x_d$ $h \ ll 1$

，我们称之为``Prony品种''。基于此，我们在最坏情况下的重构误差上产生了相同数量级的上下限。此外，我们在幅度和节点的重构上分别得出上下边界。最后，我们讨论了在给定其他先验信息的情况下，如何使用Prony品种的几何形状来提高重建精度。