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Dmitry Batenkov

Publications and source records attributed to Dmitry Batenkov.

At least 37 records · Page 2Linked to original sources
arXiv2016

Stability and super-resolution of generalized spike recovery

We consider the problem of recovering a linear combination of Dirac delta functions and derivatives from a finite number of Fourier samples corrupted by noise. This is a generalized version of the well-known spike recovery problem, which is receiving much attention recently. We analyze the numerical conditioning of this problem in two different settings depending on the order of magnitude of the quantity $Nη$, where $N$ is the number of Fourier samples and $η$ is the minimal distance between the generalized spikes. In the "well-conditioned" regime $Nη\gg1$, we provide upper bounds for first-order perturbation of the solution to the corresponding least-squares problem. In the near-colliding, or "super-resolution" regime $Nη\to0$ with a single cluster, we propose a natural regularization scheme based on decimating the samples \textendash{} essentially increasing the separation $η$ \textendash{} and demonstrate the effectiveness and near-optimality of this scheme in practice.

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arXiv2015

Uniform upper bounds for the cyclicity of the zero solution of the Abel differential equation

Given two polynomials $P,q$ we consider the following question: "how large can the index of the first non-zero moment $\tilde{m}_k=\int_a^b P^k q$ be, assuming the sequence is not identically zero?". The answer $K$ to this question is known as the moment Bautin index, and we provide the first general upper bound: $K\leqslant 2+\mathrm{deg} q+3(\mathrm{deg} P-1)^2$. The proof is based on qualitative analysis of linear ODEs, applied to Cauchy-type integrals of certain algebraic functions. The moment Bautin index plays an important role in the study of bifurcations of periodic solution in the polynomial Abel equation $y'=py^2+\varepsilon qy^3$ for $p,q$ polynomials and $\varepsilon \ll 1$. In particular, our result implies that for $p$ satisfying a well-known generic condition, the number of periodic solutions near the zero solution does not exceed $5+\mathrm{deg} q+3\mathrm{deg}^2 p$. This is the first such bound depending solely on the degrees of the Abel equation.

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arXiv2015

Accuracy of spike-train Fourier reconstruction for colliding nodes

We consider Fourier reconstruction problem for signals F, which are linear combinations of shifted delta-functions. We assume the Fourier transform of F to be known on the frequency interval [-N,N], with an absolute error not exceeding e > 0. We give an absolute lower bound (which is valid with any reconstruction method) for the "worst case" reconstruction error of F in situations where the nodes (i.e. the positions of the shifted delta-functions in F) are known to form an l elements cluster of a size h << 1. Using "decimation" reconstruction algorithm we provide an upper bound for the reconstruction error, essentially of the same form as the lower one. Roughly, our main result states that for N*h of order of (2l-1)-st root of e the worst case reconstruction error of the cluster nodes is of the same order as h, and hence the inside configuration of the cluster nodes (in the worst case scenario) cannot be reconstructed at all. On the other hand, decimation algorithm reconstructs F with the accuracy of order of 2l-st root of e.

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arXiv2014

Taylor Domination, Difference Equations, and Bautin Ideals

We compare three approaches to studying the behavior of an analytic function $f(z)=\sum_{k=0}^\infty a_kz^k$ from its Taylor coefficients. The first is "Taylor domination" property for $f(z)$ in the complex disk $D_R$, which is an inequality of the form \[ |a_{k}|R^{k}\leq C\ \max_{i=0,\dots,N}\ |a_{i}|R^{i}, \ k \geq N+1. \] The second approach is based on a possibility to generate $a_k$ via recurrence relations. Specifically, we consider linear non-stationary recurrences of the form \[ a_{k}=\sum_{j=1}^{d}c_{j}(k)\cdot a_{k-j},\ \ k=d,d+1,\dots, \] with uniformly bounded coefficients. In the third approach we assume that $a_k=a_k(λ)$ are polynomials in a finite-dimensional parameter $λ\in {\mathbb C}^n.$ We study "Bautin ideals" $I_k$ generated by $a_{1}(λ),\ldots,a_{k}(λ)$ in the ring ${\mathbb C}[λ]$ of polynomials in $λ$. \smallskip These three approaches turn out to be closely related. We present some results and questions in this direction.

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arXiv2014

Taylor Domination, Turán lemma, and Poincaré-Perron Sequences

We consider "Taylor domination" property for an analytic function $f(z)=\sum_{k=0}^{\infty}a_{k}z^{k},$ in the complex disk $D_R$, which is an inequality of the form \[ |a_{k}|R^{k}\leq C\ \max_{i=0,\dots,N}\ |a_{i}|R^{i}, \ k \geq N+1. \] This property is closely related to the classical notion of "valency" of $f$ in $D_R$. For $f$ - rational function we show that Taylor domination is essentially equivalent to a well-known and widely used Turán's inequality on the sums of powers. Next we consider linear recurrence relations of the Poincaré type \[ a_{k}=\sum_{j=1}^{d}[c_{j}+ψ_{j}(k)]a_{k-j},\ \ k=d,d+1,\dots,\quad\text{with }\lim_{k\rightarrow\infty}ψ_{j}(k)=0. \] We show that the generating functions of their solutions possess Taylor domination with explicitly specified parameters. As the main example we consider moment generating functions, i.e. the Stieltjes transforms \[ S_{g}\left(z\right)=\int\frac{g\left(x\right)dx}{1-zx}. \] We show Taylor domination property for such $S_{g}$ when $g$ is a piecewise D-finite function, satisfying on each continuity segment a linear ODE with polynomial coefficients.

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arXiv2014

Accuracy of Algebraic Fourier Reconstruction for Shifts of Several Signals

We consider the problem of "algebraic reconstruction" of linear combinations of shifts of several known signals $f_1,\ldots,f_k$ from the Fourier samples. Following \cite{Bat.Sar.Yom2}, for each $j=1,\ldots,k$ we choose sampling set $S_j$ to be a subset of the common set of zeroes of the Fourier transforms ${\cal F}(f_\ell), \ \ell \ne j$, on which ${\cal F}(f_j)\ne 0$. It was shown in \cite{Bat.Sar.Yom2} that in this way the reconstruction system is "decoupled" into $k$ separate systems, each including only one of the signals $f_j$. The resulting systems are of a "generalized Prony" form. However, the sampling sets as above may be non-uniform/not "dense enough" to allow for a unique reconstruction of the shifts and amplitudes. In the present paper we study uniqueness and robustness of non-uniform Fourier sampling of signals as above, investigating sampling of exponential polynomials with purely imaginary exponents. As the main tool we apply a well-known result in Harmonic Analysis: the Turán-Nazarov inequality (\cite{Naz}), and its generalization to discrete sets, obtained in \cite{Fri.Yom}. We illustrate our general approach with examples, and provide some simulation results.

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arXiv2014

Local and global geometry of Prony systems and Fourier reconstruction of piecewise-smooth functions

Many reconstruction problems in signal processing require solution of a certain kind of nonlinear systems of algebraic equations, which we call Prony systems. We study these systems from a general perspective, addressing questions of global solvability and stable inversion. Of special interest are the so-called "near-singular" situations, such as a collision of two closely spaced nodes. We also discuss the problem of reconstructing piecewise-smooth functions from their Fourier coefficients, which is easily reduced by a well-known method of K.Eckhoff to solving a particular Prony system. As we show in the paper, it turns out that a modification of this highly nonlinear method can reconstruct the jump locations and magnitudes of such functions, as well as the pointwise values between the jumps, with the maximal possible accuracy.

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arXiv2014

Complete Algebraic Reconstruction of Piecewise-Smooth Functions from Fourier Data

In this paper we provide a reconstruction algorithm for piecewise-smooth functions with a-priori known smoothness and number of discontinuities, from their Fourier coefficients, posessing the maximal possible asymptotic rate of convergence -- including the positions of the discontinuities and the pointwise values of the function. This algorithm is a modification of our earlier method, which is in turn based on the algebraic method of K.Eckhoff proposed in the 1990s. The key ingredient of the new algorithm is to use a different set of Eckhoff's equations for reconstructing the location of each discontinuity. Instead of consecutive Fourier samples, we propose to use a "decimated" set which is evenly spread throughout the spectrum.

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arXiv2013

Decimated generalized Prony systems

We continue studying robustness of solving algebraic systems of Prony type (also known as the exponential fitting systems), which appear prominently in many areas of mathematics, in particular modern "sub-Nyquist" sampling theories. We show that by considering these systems at arithmetic progressions (or "decimating" them), one can achieve better performance in the presence of noise. We also show that the corresponding lower bounds are closely related to well-known estimates, obtained for similar problems but in different contexts.

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arXiv2013

Algebraic signal sampling, Gibbs phenomenon and Prony-type systems

Systems of Prony type appear in various signal reconstruction problems such as finite rate of innovation, superresolution and Fourier inversion of piecewise smooth functions. We propose a novel approach for solving Prony-type systems, which requires sampling the signal at arithmetic progressions. By keeping the number of equations small and fixed, we demonstrate that such "decimation" can lead to practical improvements in the reconstruction accuracy. As an application, we provide a solution to the so-called Eckhoff's conjecture, which asked for reconstructing jump positions and magnitudes of a piecewise-smooth function from its Fourier coefficients with maximal possible asymptotic accuracy -- thus eliminating the Gibbs phenomenon.

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arXiv2013

Decoupling of Fourier Reconstruction System for Shifts of Several Signals

We consider the problem of ``algebraic reconstruction'' of linear combinations of shifts of several signals $f_1,\ldots,f_k$ from the Fourier samples. For each $r=1,\ldots,k$ we choose sampling set $S_r$ to be a subset of the common set of zeroes of the Fourier transforms ${\cal F}(f_ł), \ ł\ne r$, on which ${\cal F}(f_r)\ne 0$. We show that in this way the reconstruction system is reduced to $k$ separate systems, each including only one of the signals $f_r$. Each of the resulting systems is of a ``generalized Prony'' form. We discuss the problem of unique solvability of such systems, and provide some examples.

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arXiv2013

Moment vanishing of piecewise solutions of linear ODEs

We consider the "moment vanishing problem" for a general class of piecewise-analytic functions which satisfy on each continuity interval a linear ODE with polynomial coefficients. This problem, which essentially asks how many zero first moments can such a (nonzero) function have, turns out to be related to several difficult questions in analytic theory of ODEs (Poincare's Center-Focus problem) as well as in Approximation Theory and Signal Processing ("Algebraic Sampling"). While the solution space of any particular ODE admits such a bound, it will in the most general situation depend on the coefficients of this ODE. We believe that a good understanding of this dependence may provide a clue for attacking the problems mentioned above. In this paper we undertake an approach to the moment vanishing problem which utilizes the fact that the moment sequences under consideration satisfy a recurrence relation of fixed length, whose coefficients are polynomials in the index. For any given operator, we prove a general bound for its moment vanishing index. We also provide uniform bounds for several operator families.

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arXiv2013

Geometry and Singularities of the Prony mapping

Prony mapping provides the global solution of the Prony system of equations \[ Σ_{i=1}^{n}A_{i}x_{i}^{k}=m_{k},\ k=0,1,...,2n-1. \] This system appears in numerous theoretical and applied problems arising in Signal Reconstruction. The simplest example is the problem of reconstruction of linear combination of $δ$-functions of the form $g(x)=\sum_{i=1}^{n}a_{i}δ(x-x_{i})$, with the unknown parameters $a_{i},\ x_{i},\ i=1,...,n,$ from the "moment measurements" $m_{k}=\int x^{k}g(x)dx.$ Global solution of the Prony system, i.e. inversion of the Prony mapping, encounters several types of singularities. One of the most important ones is a collision of some of the points $x_{i}.$ The investigation of this type of singularities has been started in \cite{yom2009Singularities} where the role of finite differences was demonstrated. In the present paper we study this and other types of singularities of the Prony mapping, and describe its global geometry. We show, in particular, close connections of the Prony mapping with the "Vieta mapping" expressing the coefficients of a polynomial through its roots, and with hyperbolic polynomials and "Vandermonde mapping" studied by V. Arnold.

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arXiv2012

On the accuracy of solving confluent Prony systems

In this paper we consider several nonlinear systems of algebraic equations which can be called "Prony-type". These systems arise in various reconstruction problems in several branches of theoretical and applied mathematics, such as frequency estimation and nonlinear Fourier inversion. Consequently, the question of stability of solution with respect to errors in the right-hand side becomes critical for the success of any particular application. We investigate the question of "maximal possible accuracy" of solving Prony-type systems, putting stress on the "local" behavior which approximates situations with low absolute measurement error. The accuracy estimates are formulated in very simple geometric terms, shedding some light on the structure of the problem. Numerical tests suggest that "global" solution techniques such as Prony's algorithm and ESPRIT method are suboptimal when compared to this theoretical "best local" behavior.

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arXiv2012

Reconstruction of Planar Domains from Partial Integral Measurements

We consider the problem of reconstruction of planar domains from their moments. Specifically, we consider domains with boundary which can be represented by a union of a finite number of pieces whose graphs are solutions of a linear differential equation with polynomial coefficients. This includes domains with piecewise-algebraic and, in particular, piecewise-polynomial boundaries. Our approach is based on one-dimensional reconstruction method of [Bat]* and a kind of "separation of variables" which reduces the planar problem to two one-dimensional problems, one of them parametric. Several explicit examples of reconstruction are given. Another main topic of the paper concerns "invisible sets" for various types of incomplete moment measurements. We suggest a certain point of view which stresses remarkable similarity between several apparently unrelated problems. In particular, we discuss zero quadrature domains (invisible for harmonic polynomials), invisibility for powers of a given polynomial, and invisibility for complex moments (Wermer's theorem and further developments). The common property we would like to stress is a "rigidity" and symmetry of the invisible objects. * D.Batenkov, Moment inversion of piecewise D-finite functions, Inverse Problems 25 (2009) 105001

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arXiv2011

Algebraic reconstruction of piecewise-smooth functions from integral measurements

This paper presents some results on a well-known problem in Algebraic Signal Sampling and in other areas of applied mathematics: reconstruction of piecewise-smooth functions from their integral measurements (like moments, Fourier coefficients, Radon transform, etc.). Our results concern reconstruction (from the moments or Fourier coefficients) of signals in two specific classes: linear combinations of shifts of a given function, and "piecewise $D$-finite functions" which satisfy on each continuity interval a linear differential equation with polynomial coefficients. In each case the problem is reduced to a solution of a certain type of non-linear algebraic system of equations ("Prony-type system"). We recall some known methods for explicitly solving such systems in one variable, and provide extensions to some multi-dimensional cases. Finally, we investigate the local stability of solving the Prony-type systems.

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arXiv2010

Algebraic Fourier reconstruction of piecewise smooth functions

Accurate reconstruction of piecewise-smooth functions from a finite number of Fourier coefficients is an important problem in various applications. The inherent inaccuracy, in particular the Gibbs phenomenon, is being intensively investigated during the last decades. Several nonlinear reconstruction methods have been proposed, and it is by now well-established that the "classical" convergence order can be completely restored up to the discontinuities. Still, the maximal accuracy of determining the positions of these discontinuities remains an open question. In this paper we prove that the locations of the jumps (and subsequently the pointwise values of the function) can be reconstructed with at least "half the classical accuracy". In particular, we develop a constructive approximation procedure which, given the first $k$ Fourier coefficients of a piecewise-$C^{2d+1}$ function, recovers the locations of the jumps with accuracy $\sim k^{-(d+2)}$, and the values of the function between the jumps with accuracy $\sim k^{-(d+1)}$ (similar estimates are obtained for the associated jump magnitudes). A key ingredient of the algorithm is to start with the case of a single discontinuity, where a modified version of one of the existing algebraic methods (due to K.Eckhoff) may be applied. It turns out that the additional orders of smoothness produce a highly correlated error terms in the Fourier coefficients, which eventually cancel out in the corresponding algebraic equations. To handle more than one jump, we propose to apply a localization procedure via a convolution in the Fourier domain.

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