On the discrepancy of jittered sampling

Pausinger, Florian; Steinerberger, Stefan

doi:10.1016/j.jco.2015.11.003

“…It is generally believed that standard low-discrepancy sets, while providing good bounds with respect to the number of points N, yield very bad, often exponential, dependence on the dimension. However, as we shall see, it appears that for jittered sampling this behavior is quite reasonable (see also [29] for a discussion of a similar effect). This is consistent with the fact that this construction is intermediate between purely random and deterministic sets.…”

Section: 2mentioning

confidence: 56%

One-bit sensing, discrepancy and Stolarsky's principle

Bilyk¹,

Lacey²

2017

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A sign-linear one bit map from the d-dimensional sphere S d to the N -dimensional Hamming cube H N = {−1, +1} n is given by, the smallest integer N so that there is a sign-linear map which has the δ-restricted isometric property, where we impose normalized geodesic distance on S d and Hamming metric on H N . Up to a polylogarithmic factor, N (d, δ) ≈ δ −2+ 2 d+1 , which has a dimensional correction in the power of δ. This is a question that arises from the one bit sensing literature, and the method of proof follows from geometric discrepancy theory. We also obtain an analogue of the Stolarsky invariance principle for this situation, which implies that minimizing the L 2 average of the embedding error is equivalent to minimizing the discrete energy i,jResearch supported in part by NSF grants DMS 1101519 and DMS 1265570.

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“…A result of the first and third author [11] shows that any decomposition into N sets of equal measure always yields a smaller expected squared L 2 −discrepancy than N completely randomly chosen points: even the most primitive Jittered Sampling construction is better than Monte Carlo. However, currently known quantitative bounds [11] do not imply any effective improvement for N (2d) 2d points. The motivation of our paper is to gain a better understanding of Jittered Sampling.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Optimal Jittered Sampling for two points in the unit square

Pausinger¹,

Rachh

²

,

Steinerberger

³

2018

Statistics & Probability Letters

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Jittered Sampling is a refinement of the classical Monte Carlo sampling method. Instead of picking n points randomly from [0, 1] 2 , one partitions the unit square into n regions of equal measure and then chooses a point randomly from each partition. Currently, no good rules for how to partition the space are available. In this paper, we present a solution for the special case of subdividing the unit square by a decreasing function into two regions so as to minimize the expected squared L 2 −discrepancy. The optimal partitions are given by a highly nonlinear integral equation for which we determine an approximate solution. In particular, there is a break of symmetry and the optimal partition is not into two sets of equal measure. We hope this stimulates further interest in the construction of good partitions.2010 Mathematics Subject Classification. 65C05 (primary) and 93E20 (secondary).

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“…Jittered sampling is sometimes referred to as 'stratified sampling' in the literature, but we will use the term 'stratified sampling' in a more broad sense as outlined below. Motivated by recent progress [28,29] the aim of this paper is to take a systematic look at jittered sampling and its extension based on more general partitions = ( 1 , . .…”

Section: Setting and Main Questionsmentioning

confidence: 99%

“…It was shown in [28] that the asymptotic order of the star-discrepancy of a point set obtained from jittered sampling is O(N − 1 2 − 1 2d ). Thus, taking partitions can significantly improve the expected discrepancy of (random) point sets in small dimensions d ≥ 2.…”

Section: Setting and Main Questionsmentioning

confidence: 99%

Discrepancy of stratified samples from partitions of the unit cube

Kiderlen

¹

,

Pausinger

²

2021

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We extend the notion of jittered sampling to arbitrary partitions and study the discrepancy of the related point sets. Let $${\varvec{\Omega }}=(\Omega _1,\ldots ,\Omega _N)$$ Ω = ( Ω 1 , … , Ω N ) be a partition of $$[0,1]^d$$ [ 0 , 1 ] d and let the ith point in $${{\mathcal {P}}}$$ P be chosen uniformly in the ith set of the partition (and stochastically independent of the other points), $$i=1,\ldots ,N$$ i = 1 , … , N . For the study of such sets we introduce the concept of a uniformly distributed triangular array and compare this notion to related notions in the literature. We prove that the expected $${{{\mathcal {L}}}_p}$$ L p -discrepancy, $${{\mathbb {E}}}{{{\mathcal {L}}}_p}({{\mathcal {P}}}_{\varvec{\Omega }})^p$$ E L p ( P Ω ) p , of a point set $${{\mathcal {P}}}_{\varvec{\Omega }}$$ P Ω generated from any equivolume partition $${\varvec{\Omega }}$$ Ω is always strictly smaller than the expected $${{{\mathcal {L}}}_p}$$ L p -discrepancy of a set of N uniform random samples for $$p>1$$ p > 1 . For fixed N we consider classes of stratified samples based on equivolume partitions of the unit cube into convex sets or into sets with a uniform positive lower bound on their reach. It is shown that these classes contain at least one minimizer of the expected $${{{\mathcal {L}}}_p}$$ L p -discrepancy. We illustrate our results with explicit constructions for small N. In addition, we present a family of partitions that seems to improve the expected discrepancy of Monte Carlo sampling by a factor of 2 for every N.

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On the discrepancy of jittered sampling

Cited by 19 publications

References 22 publications

One-bit sensing, discrepancy and Stolarsky's principle

One-bit sensing, discrepancy and Stolarsky's principle

Optimal Jittered Sampling for two points in the unit square

Discrepancy of stratified samples from partitions of the unit cube

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