相关论文: Algebraic Geometry of Error Amplification: the Pro…

We consider a reconstruction problem for ``spike-train'' signals $F$ 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 $m_k(F)$,…

The paper is devoted to the characterization of the geometry of Prony curves arising from spike-train signals. We give a sufficient condition which guarantees the blowing up of the amplitudes of a Prony curve S in case where some of its…

This is a survey paper discussing one specific (and classical) system of algebraic equations - the so called "Prony system". We provide a short overview of its unusually wide connections with many different fields of Mathematics, stressing…

We start a systematic study of the topology, geometry and singularities of the Prony varieties $S_q(\mu)$, defined by the first $q+1$ equations of the classical Prony system $$\sum_{j=1}^d a_j x_j^k = \mu_k, \ k= 0,1,\ldots \ .$$ Prony…

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…

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…

Sum-Product Networks with complex probability distribution at the leaves have been shown to be powerful tractable-inference probabilistic models. However, while learning the internal parameters has been amply studied, learning complex leaf…

The stable profile of the boundary of a plant's leaf fluctuating in the direction transversal to the leaf's surface is described in the framework of a model called a "surface \`a godets". It is shown that the information on the profile is…

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…

We consider a signal reconstruction problem for signals $F$ of the 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 $m_k(F)$ to be known for $k=0,1,\ldots,N,$ with an absolute…

We prove new explicit inapproximability results for the Vertex Cover Problem on the Power Law Graphs and some functional generalizations of that class of graphs. Our results depend on special bounded degree amplifier constructions for those…

Polynomial convergence bounds are considered for left, right, and split preconditioned GMRES. They include the cases of Weighted and Deflated GMRES for a linear system Ax = b. In particular, the case of positive definite A is considered.…

This paper provides fascinating connections between several mathematical problems which lie on the intersection of several mathematics subjects, namely algebraic geometry, approximation theory, complex-harmonic analysis and high dimensional…

We propose an algebraic framework generalizing several variants of Prony's method and explaining their relations. This includes Hankel and Toeplitz variants of Prony's method for the decomposition of multivariate exponential sums,…

We study decision rule approximations for generic multi-stage robust linear optimization problems. We consider linear decision rules for the case when the objective coefficients, the recourse matrices, and the right-hand sides are…

A method is created to automatically increase the threshold projection parameter in three-field density-based topology optimization to achieve a near binary design. The parameter increase each iteration is based on an exponential growth…

We develop the rudiments of a theory of parametrized $\infty$-operads, including parametrized generalizations of monoidal envelopes, Day convolution, operadic left Kan extensions, results on limits and colimits of algebras, and the…

We investigate straight-line drawings of topological graphs that consist of a planar graph plus one edge, also called almost-planar graphs. We present a characterization of such graphs that admit a straight-line drawing. The…

We propose an adaptive finite element algorithm to approximate solutions of elliptic problems whose forcing data is locally defined and is approximated by regularization (or mollification). We show that the energy error decay is…

We introduce the notion of central extension of gerbes on a topological space. We then show that there are obstruction classes to lifting objects and isomorphisms in a central extension. We also discuss pronilpotent gerbes. These results…