Approximation by finitely supported measures

Kloeckner, Benoît

doi:10.1051/cocv/2010100

Cited by 40 publications

(31 citation statements)

References 16 publications

(23 reference statements)

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“…Since d r (δ •,n • , µ) ≤ d r (δ un • , µ) for every µ ∈ P r and n ∈ N, clearly lim n→∞ d r (δ •,n • , µ) = 0. The rate of convergence of d r (δ •,n • , µ) has been, and continues to be, studied extensively; see, e.g., [17,19,20,21,23,27] and the references therein. Arguably the simplest situation occurs if µ ∈ P r has a non-trivial absolutely continuous part and satisfies a mild moment condition.…”

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confidence: 99%

Best finite constrained approximations of one-dimensional probabilities

Berger

2019

Journal of Approximation Theory

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This paper studies best finitely supported approximations of one-dimensional probability measures with respect to the L r -Kantorovich (or transport) distance, where either the locations or the weights of the approximations' atoms are prescribed. Necessary and sufficient optimality conditions are established, and the rate of convergence (as the number of atoms goes to infinity) is discussed. In view of emerging mathematical and statistical applications, special attention is given to the case of best uniform approximations (i.e., all atoms having equal weight). The approach developed in this paper is elementary; it is based on best approximations of (monotone) L r -functions by step functions, and thus different from, yet naturally complementary to, the classical Voronoi partition approach.Keywords. Constrained approximation, best uniform approximation, asymptotically best approximation, Kantorovich distance, balanced function, quantile function.When endowed with the metric d r , the space P r is separable and complete, and d r (µ n , µ) → 0 implies that µ n → µ weakly. Note that P r ⊃ P s and d r ≤ d s whenever r < s. On P s , the metrics d r and d s are not equivalent, as the example of µ n = (1 − n −s )δ 0 + n −s δ n shows, for which d s (µ n , δ 0 ) ≡ 1, and yet lim n→∞ d r (µ n , δ 0 ) = 0 for all r < s, and hence µ n → δ 0 weakly. The reader is referred to [12,32] for details on the mathematical background of the Kantorovich distance, and to [17,32] for a discussion of its usefulness in the study of mass transportation and quantization problems.

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confidence: 99%

Best finite constrained approximations of one-dimensional probabilities

Berger

2019

Journal of Approximation Theory

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“…Typically ρ is in this context compactly supported and the metric is the Wasserstein distance. In this case the best approximation can be constructed by covering the support of ρ with appropriate balls and using the Voronoi tessellation generated by their centres, and rates of convergence as N → ∞ can be obtained under suitable regularity of ρ; see [33,35]. The empirical approximation constructed in this paper is specific to our problem-we are not concerned with its optimality in approaching ρ but with the fact that it also has to preserve the energy as N → ∞; see Lemma 6.6.…”

Section: Limsup Inequalitymentioning

confidence: 99%

Discrete minimisers are close to continuum minimisers for the interaction energy

Cañizo¹,

Patacchini²

2018

Calc. Var.

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Abstract. Under suitable technical conditions we show that minimisers of the discrete interaction energy for attractive-repulsive potentials converge to minimisers of the corresponding continuum energy as the number of particles goes to infinity. We prove that the discrete interaction energy Γ-converges in the narrow topology to the continuum interaction energy. As an important part of the proof we study support and regularity properties of discrete minimisers: we show that continuum minimisers belong to suitable Morrey spaces and we introduce the set of empirical Morrey measures as a natural discrete analogue containing all the discrete minimisers.

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“…Its proof follows from a classical covering argument, that can be found e.g. in Proposition 4.1 of [15]. (ii) for every positive ε, N µ (ε) ≤ α µ 5 ℓ /ε ℓ .…”

Section: Convergence Under Empirical Samplingmentioning

confidence: 99%

Witnessed k-distance

Guibas

Mérigot

Morozov

2011

Proceedings of the Twenty-Seventh Annual Symposium on Computational Geometry

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International audienceDistance function to a compact set plays a central role in several areas of computational geometry. Methods that rely on it are robust to the perturbations of the data by the Hausdorff noise, but fail in the presence of outliers. The recently introduced distance to a measure offers a solution by extending the distance function framework to reasoning about the geometry of probability measures, while maintaining theoretical guarantees about the quality of the inferred information. A combinatorial explosion hinders working with distance to a measure as an ordinary power distance function. In this paper, we analyze an approximation scheme that keeps the representation linear in the size of the input, while maintaining the guarantees on the inference quality close to those for the exact but costly representation

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Approximation by finitely supported measures

Cited by 40 publications

References 16 publications

Best finite constrained approximations of one-dimensional probabilities

Best finite constrained approximations of one-dimensional probabilities

Discrete minimisers are close to continuum minimisers for the interaction energy

Witnessed k-distance

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