Dmitry Batenkov

Dmitry Batenkov, Laurent Demanet, Hrushikesh N. Mhaskar

Soft extrapolation refers to the problem of recovering a function from its samples, multiplied by a fast-decaying window and perturbed by an additive noise, over an interval which is potentially larger than the essential support of the window. A core theoretical question is to provide bounds on the possible amount of extrapolation, depending on the sample perturbation level and the function prior. In this paper we consider soft extrapolation of entire functions of finite order and type (containing the class of bandlimited functions as a special case), multiplied by a super-exponentially decaying window (such as a Gaussian). We consider a weighted least-squares polynomial approximation with judiciously chosen number of terms and a number of samples which scales linearly with the degree of approximation. It is shown that this simple procedure provides stable recovery with an extrapolation factor which scales logarithmically with the perturbation level and is inversely proportional to the characteristic lengthscale of the function. The pointwise extrapolation error exhibits a Hölder-type continuity with an exponent derived from weighted potential theory, which changes from 1 near the available samples, to 0 when the extrapolation distance reaches the characteristic smoothness length scale of the function. The algorithm is asymptotically minimax, in the sense that there is essentially no better algorithm yielding meaningfully lower error over the same smoothness class. When viewed in the dual domain, the above problem corresponds to (stable) simultaneous de-convolution and super-resolution for objects of small space/time extent. Our results then show that the amount of achievable super-resolution is inversely proportional to the object size, and therefore can be significant for small objects.

Adar Kahana, Symeon Papadimitropoulos, Eli Turkel et al.

Inverse source problems are central to many applications in acoustics, geophysics, non-destructive testing, and more. Traditional imaging methods suffer from the resolution limit, preventing distinction of sources separated by less than the emitted wavelength. In this work we propose a method based on physically-informed neural-networks for solving the source refocusing problem, constructing a novel loss term which promotes super-resolving capabilities of the network and is based on the physics of wave propagation. We demonstrate the approach in the setup of imaging an a-priori unknown number of point sources in a two-dimensional rectangular waveguide from measurements of wavefield recordings along a vertical cross-section. The results show the ability of the method to approximate the locations of sources with high accuracy, even when placed close to each other.

Dmitry Batenkov, Yosef Yomdin

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.

Dmitry Batenkov

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.

Dmitry Batenkov

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.

Dmitry Batenkov

We consider polynomial systems of Prony type, appearing in many areas of mathematics. Their robust numerical solution is considered to be difficult, especially in "near-colliding" situations. We consider a case when the structure of the system is a-priori fixed. We transform the nonlinear part of the Prony system into a Hankel-type polynomial system. Combining this representation with a recently discovered "decimation" technique, we present an algorithm which applies homotopy continuation to an appropriately chosen Hankel-type system as above. In this way, we are able to solve for the nonlinear variables of the original system with high accuracy when the data is perturbed.

Dmitry Batenkov, Yosef Yomdin

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.

Dmitry Batenkov

In this note we present a simple upper bound for the row-wise norm of the inverses of general confluent Vandermonde matrices.

Dmitry Batenkov, Yaniv Romano, Michael Elad

The traditional sparse modeling approach, when applied to inverse problems with large data such as images, essentially assumes a sparse model for small overlapping data patches. While producing state-of-the-art results, this methodology is suboptimal, as it does not attempt to model the entire global signal in any meaningful way - a nontrivial task by itself. In this paper we propose a way to bridge this theoretical gap by constructing a global model from the bottom up. Given local sparsity assumptions in a dictionary, we show that the global signal representation must satisfy a constrained underdetermined system of linear equations, which can be solved efficiently by modern optimization methods such as Alternating Direction Method of Multipliers (ADMM). We investigate conditions for unique and stable recovery, and provide numerical evidence corroborating the theory.