Tusnády’s Problem, the Transference Principle, and Non-uniform QMC Sampling

Aistleitner, Christoph; Bilyk, Dmytro Olexandrovych; Nikolov, Aleksandar

doi:10.1007/978-3-319-91436-7_8

Cited by 8 publications

(15 citation statements)

References 32 publications

(63 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Ì ÓÖ Ñ 1.3 (Aistleitner, Bilyk, Nikolov, [1])º For every d ≥ 1, there exists a constant c d (depending only on d) such that the following holds. For every N ≥ 2 and every normalized Borel measure μ on [0, 1] d there exist points…”

Section: Star-discrepancymentioning

confidence: 99%

“…Hence this case can be treated by generalizing the arguments given in [9,10]. 1 Ê Ñ Ö 1.4º Recall that in dimension d = 1, Lebesgue's decomposition theorem states any Borel measure μ can be written as…”

Section: Dimensionmentioning

confidence: 99%

“…Let r k = r e k r e where 0 < r < 1 2 . Then for each N, there exists a set (x N i ) N i=1 and a constant c r so that the associated measure ν N as in (1.…”

Section: ü ñôð 27ºmentioning

confidence: 99%

“…In [1], the authors ask whether the Lebesgue measure is the hardest measure to approximate by finite sets. They guess that the answer is yes and justify the conjecture because the Lebesgue measure is spread throughout the entire cube [0, 1] d and treats all points the same.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Families of Well Approximable Measures

Fairchild

Goering

Weiß

2021

Uniform Distribution Theory

View full text Add to dashboard Cite

We provide an algorithm to approximate a finitely supported discrete measure μ by a measure νN corresponding to a set of N points so that the total variation between μ and νN has an upper bound. As a consequence if μ is a (finite or infinitely supported) discrete probability measure on [0, 1] d with a sufficient decay rate on the weights of each point, then μ can be approximated by νN with total variation, and hence star-discrepancy, bounded above by (log N)N− 1. Our result improves, in the discrete case, recent work by Aistleitner, Bilyk, and Nikolov who show that for any normalized Borel measure μ, there exist finite sets whose star-discrepancy with respect to μ is at most ( log N ) d − 1 2 N − 1 {\left( {\log \,N} \right)^{d - {1 \over 2}}}{N^{ - 1}} . Moreover, we close a gap in the literature for discrepancy in the case d =1 showing both that Lebesgue is indeed the hardest measure to approximate by finite sets and also that all measures without discrete components have the same order of discrepancy as the Lebesgue measure.

show abstract

Section: Star-discrepancymentioning

confidence: 99%

Section: Dimensionmentioning

confidence: 99%

“…Let r k = r e k r e where 0 < r < 1 2 . Then for each N, there exists a set (x N i ) N i=1 and a constant c r so that the associated measure ν N as in (1.…”

Section: ü ñôð 27ºmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Families of Well Approximable Measures

Fairchild

Goering

Weiß

2021

Uniform Distribution Theory

View full text Add to dashboard Cite

show abstract

“…Integration with respect to measures other than the Lebesgue measure arises often, and a constructive proof of our upper bound could have significant impact in practice. We refer to the note [1] for an exposition of these connections.…”

Section: Lemma 4 and Theorem 2 Imply Thatmentioning

confidence: 99%

Tighter Bounds for the Discrepancy of Boxes and Polytopes

Nikolov

2017

Mathematika

Self Cite

View full text Add to dashboard Cite

Abstract. Combinatorial discrepancy is a complexity measure of a collection of sets which quantifies how well the sets in the collection can be simultaneously balanced. More precisely, we are given an n-point set P, and a collection F = {F 1 , . . . , F m } of subsets of P, and our goal is color P with two colors, red and blue, so that the maximum over the F i of the absolute difference between the number of red elements and the number of blue elements (the discrepancy) is minimized. Combinatorial discrepancy has many applications in mathematics and computer science, including constructions of uniformly distributed point sets, and lower bounds for data structures and private data analysis algorithms. We investigate the combinatorial discrepancy of geometrically defined systems, in which P is an n-point set in d-dimensional space, and F is the collection of subsets of P induced by dilations and translations of a fixed convex polytope B. Such set systems include systems of sets induced by axis-aligned boxes, whose discrepancy is the subject of the well-known Tusnády problem. We prove new discrepancy upper and lower bounds for such set systems by extending the approach based on factorization norms previously used by the author, Matoušek, and Talwar. We also outline applications of our results to geometric discrepancy, data structure lower bounds, and differential privacy.

show abstract

Approximation of Discrete Measures by Finite Point Sets

Weiß

2023

Uniform Distribution Theory

View full text Add to dashboard Cite

For a probability measure μ on [0, 1] without discrete component, the best possible order of approximation by a finite point set in terms of the star-discrepancy is &inline as has been proven relatively recently. However, if μ contains a discrete component no non-trivial lower bound holds in general because it is straightforward to construct examples without any approximation error in this case. This might explain, why the approximation of discrete measures on [0, 1] by finite point sets has so far not been completely covered in the existing literature. In this note, we close the gap by giving a complete description for discrete measures. Most importantly, we prove that for any discrete measures (not supported on one point only) the best possible order of approximation is for infinitely many N bounded from below by &inline for some constant 6 ≥ c> 2 which depends on the measure. This implies, that for a finitely supported discrete measure on [0, 1]d the known possible order of approximation &inline is indeed the optimal one.

show abstract

Tusnády’s Problem, the Transference Principle, and Non-uniform QMC Sampling

Cited by 8 publications

References 32 publications

Families of Well Approximable Measures

Families of Well Approximable Measures

Tighter Bounds for the Discrepancy of Boxes and Polytopes

Approximation of Discrete Measures by Finite Point Sets

Contact Info

Product

Resources

About