Geometry and singularities of Prony varieties

Goldman, Gideon; Salman, Yehonatan; Yomdin, Yosef

doi:10.4310/maa.2018.v25.n3.a5

Cited by 3 publications

(3 citation statements)

References 9 publications

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“…This issue is related to multidimensional variants of Prony's method. Indeed, the Hankel matrix (14) which connects polytopal densities and its node points on R 1 with the axial moments is analogous to that for the classical Prony system [16]. Extending known results about the Prony system to our setting in R d may lead to applications in signal processing.…”

Section: Discussionmentioning

confidence: 84%

“…However, these polynomials are still simpler than the rational functions we obtain for moments of polytopes other than simplices. For instance, consider the subalgebra of R[X] generated by all moments m I (X) in (16) where I runs over N d . We shall argue in Section 7 that this is the algebra of multisymmetric polynomials [11].…”

Section: Simplicesmentioning

confidence: 99%

“…Remark 4.6. Both the moments (16) and the cumulants (22) are multisymmetric functions in x 1 , x 2 , . .…”

Section: Simplicesmentioning

confidence: 99%

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Moment varieties of measures on polytopes

Kohn¹,

Shapiro²,

Sturmfels³

2020

Annali Scuola Normale Superiore - Classe Di Scienze

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The uniform probability measure on a convex polytope induces piecewise polynomial densities on its projections. For a fixed combinatorial type of simplicial polytopes, the moments of these measures are rational functions in the vertex coordinates. We study projective varieties that are parametrized by finite collections of such rational functions. Our focus lies on determining the prime ideals of these moment varieties. Special cases include Hankel determinantal ideals for polytopal splines on line segments, and the relations among multisymmetric functions given by the cumulants of a simplex. In general, our moment varieties are more complicated than in these two special cases. They offer challenges for both numerical and symbolic computing in algebraic geometry.

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Section: Discussionmentioning

confidence: 84%

Section: Simplicesmentioning

confidence: 99%

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Moment varieties of measures on polytopes

Kohn¹,

Shapiro²,

Sturmfels³

2020

Annali Scuola Normale Superiore - Classe Di Scienze

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

2020

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We consider a reconstruction problem for “spike-train” signals F F of an a priori known form F ( x ) = ∑ j = 1 d a j δ ( x − x j ) , F(x)=\sum _{j=1}^{d}a_{j}\delta \left (x-x_{j}\right ), from their moments m k ( F ) = ∫ x k F ( x ) d x . m_k(F)=\int x^kF(x)dx. We assume that the moments m k ( F ) m_k(F) , k = 0 , 1 , … , 2 d − 1 k=0,1,\ldots ,2d-1 , are known with an absolute error not exceeding ϵ > 0 \epsilon > 0 . This problem is essentially equivalent to solving the Prony system ∑ j = 1 d a j x j k = m k ( F ) , k = 0 , 1 , … , 2 d − 1. \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 F from m k ( F ) , m_k(F), in situations where the nodes x 1 , … , x d x_1,\ldots ,x_d near-collide, i.e., form a cluster of size h ≪ 1 h \ll 1 . We show that in this case, error amplification is governed by certain algebraic varieties in the parameter space of signals F 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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Super-resolution of near-colliding point sources

Batenkov

Goldman

Yomdin

2020

Information and Inference: A Journal of the IMA

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Abstract We consider the problem of stable recovery of sparse signals of the form $$\begin{equation*}F(x)=\sum_{j=1}^d a_j\delta(x-x_j),\quad x_j\in\mathbb{R},\;a_j\in\mathbb{C}, \end{equation*}$$from their spectral measurements, known in a bandwidth $\varOmega $ with absolute error not exceeding $\epsilon>0$. We consider the case when at most $p\leqslant d$ nodes $\{x_j\}$ of $F$ form a cluster whose extent is smaller than the Rayleigh limit ${1\over \varOmega }$, while the rest of the nodes is well separated. Provided that $\epsilon \lessapprox \operatorname{SRF}^{-2p+1}$, where $\operatorname{SRF}=(\varOmega \varDelta )^{-1}$ and $\varDelta $ is the minimal separation between the nodes, we show that the minimax error rate for reconstruction of the cluster nodes is of order ${1\over \varOmega }\operatorname{SRF}^{2p-1}\epsilon $, while for recovering the corresponding amplitudes $\{a_j\}$ the rate is of the order $\operatorname{SRF}^{2p-1}\epsilon $. Moreover, the corresponding minimax rates for the recovery of the non-clustered nodes and amplitudes are ${\epsilon \over \varOmega }$ and $\epsilon $, respectively. These results suggest that stable super-resolution is possible in much more general situations than previously thought. Our numerical experiments show that the well-known matrix pencil method achieves the above accuracy bounds.

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Geometry and singularities of Prony varieties

Cited by 3 publications

References 9 publications

Moment varieties of measures on polytopes

Moment varieties of measures on polytopes

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

Super-resolution of near-colliding point sources

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