Some approximation problems in semi-algebraic geometry

Friedland, Shmuel; Stawiska, Małgorzata

doi:10.4064/bc107-0-9

Cited by 13 publications

(25 citation statements)

References 14 publications

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“…In order to reduce the occurrance and the impact of spurious localizations, tracking algorithms are typically used [4], but for many real-time applications the implementation could result too cumbersome and computationally intensive. A relevant advantage of our approach to TDOA-based localization via the TDOA-space is that it gives new information and a better control over the optimization problem (27). In this section we show this fact in our minimal case of three sensors.…”

Section: The Estimation Of the Source Position In The Tdoa Spacementioning

confidence: 85%

“…We conclude this subsection by discussing on the number k of solutions of system (30). For the case Σ 2 = σ 2 I, this problem is known in the algebraic geometry literature (see [25,27]) as the computation of the Euclidean distance degree of a variety (the ellipse E in the present case). We remark that the knowledge of k is crucial for the correct functioning of any numerical algorithm used for solving system (30).…”

Section: 3mentioning

confidence: 99%

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On the statistical model of source localization based on Range Difference measurements

Compagnoni

Notari

Antonacci

et al. 2017

Journal of the Franklin Institute

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In this work we study the statistical model of source localization based on Range Difference measurements, under the assumption of Gaussian noise on the data. Our analysis is based on a previous work of the same authors concerning the localization in a noiseless scenario. We investigate the case of planar localization of a source using a minimal configuration of three non aligned receivers. We have four curved exponential families corresponding to four different, non disjoint, regions of the feasible set. For each family we solve Maximum Likelihood Estimation (MLE). This requires to find the projection of a point on a set of segments and arcs of ellipse. Then, we perform the analytic study of the localization accuracy. In particular, we give formulas for mean square error and bias of MLE, depending on the displacement vectors. We validate the results through Monte Carlo simulations, in a given setup of the receivers. As the set of feasible measurements is a semialgebraic variety, this investigation makes use of techniques from Algebraic Statistics and Information Geometry

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Section: The Estimation Of the Source Position In The Tdoa Spacementioning

confidence: 85%

Section: 3mentioning

confidence: 99%

On the statistical model of source localization based on Range Difference measurements

Compagnoni

Notari

Antonacci

et al. 2017

Journal of the Franklin Institute

View full text Add to dashboard Cite

show abstract

“…. Among the 16 tropes, only 4 contain no one of such points, namely the planes obtained by choosing always the plus sign in equations (31). This implies that these tropes do not contain any point in the first octant of R 3 and so they cannot intersect Im(T 3 ).…”

Section: The Image Of T 3 When the Receivers Are Not Collinearmentioning

confidence: 99%

“…For example, our study becomes instrumental for understanding and computing the MLE. As far as that is concerned, we can cite the recent definition of Euclidean Distance Degree [26,31] for algebraic statistical models, which is an indicator of the complexity of the MLE, and the continuous development of techniques for polynomial optimization over semialgebraic varieties [10].…”

Section: Introductionmentioning

confidence: 99%

The Algebro-geometric Study of Range Maps

et al. 2016

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In this paper, we study the combined mean field and homogenization limits for a system of weakly interacting diffusions moving in a two-scale, locally periodic confining potential, of the form considered in Duncan et al. (Brownian motion in an N-scale periodic potential, arXiv:1605.05854, 2016b). We show that, although the mean field and homogenization limits commute for finite times, they do not, in general, commute in the long time limit. In particular, the bifurcation diagrams for the stationary states can be different depending on the order with which we take the two limits. Furthermore, we construct the bifurcation diagram for the stationary McKeanVlasov equation in a two-scale potential, before passing to the homogenization limit, and we analyze the effect of the multiple local minima in the confining potential on the number and the stability of stationary solutions.

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“…. , i d ) ∈ Θ} of a hypermatrix A ∈ C n 1 ×···×n d are general, then there exists a hypermatrix B ∈ C n 1 ×···×n d of rank r that solves (18). Furthermore, although B may be not unique, its entries {b i 1 ···i d | (i 1 , .…”

Section: Applications To Other Approximation Problemsmentioning

confidence: 99%

Complex best r-term approximations almost always exist in finite dimensions

Yang

Michałek

Lim

2020

Applied and Computational Harmonic Analysis

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We show that in finite-dimensional nonlinear approximations, the best r-term approximant of a function f almost always exists over C but that the same is not true over R, i.e., the infimum inf f 1 ,...,fr ∈Y f − f1 − · · · − fr is almost always attainable by complex-valued functions f1, . . . , fr in Y , a set of functions that have some desired structures. Our result extends to functions that possess special properties like symmetry or skew-symmetry under permutations of arguments. For the case where Y is the set of separable functions, the problem becomes that of best rank-r tensor approximations. We show that over C, any tensor almost always has a unique best rank-r approximation. This extends to other notions of tensor ranks such as symmetric rank and alternating rank, to best r-block-terms approximations, and to best approximations by tensor networks. When applied to sparse-plus-low-rank approximations, we obtain that for any given r and k, a general tensor has a unique best approximation by a sum of a rank-r tensor and a k-sparse tensor with a fixed sparsity pattern; this arises in, for example, estimation of covariance matrices of a Gaussian hidden variable model with k observed variables conditionally independent given r hidden variables. The existential (but not the uniqueness) part of our result also applies to best approximations by a sum of a rank-r tensor and a k-sparse tensor with no fixed sparsity pattern, as well as to tensor completion problems.2010 Mathematics Subject Classification. 15A69, 41A50, 41A52, 41A65, 51M35.

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Some approximation problems in semi-algebraic geometry

Cited by 13 publications

References 14 publications

On the statistical model of source localization based on Range Difference measurements

On the statistical model of source localization based on Range Difference measurements

The Algebro-geometric Study of Range Maps

Complex best r-term approximations almost always exist in finite dimensions

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