Uniform projection designs under the stratified $L_2$-discrepancy

Sixu Liu; Yaping Wang

arxiv: 2605.19900 · v1 · pith:6467LRHXnew · submitted 2026-05-19 · 🧮 math.ST · stat.TH

Uniform projection designs under the stratified L₂-discrepancy

Sixu Liu , Yaping Wang This is my paper

Pith reviewed 2026-05-20 01:22 UTC · model grok-4.3

classification 🧮 math.ST stat.TH

keywords uniform projection designsstratified L2-discrepancyU-type designsspace-filling designsprojection uniformitydiscrepancy boundscomputer experiments

0 comments

The pith

For U-type designs, the average squared stratified L2-discrepancy over two-dimensional projections reduces to an explicit formula in row-pairwise weighted hierarchical distances with sharp bounds attained by many known optimal constructions

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a criterion called Φ_SD that measures how uniformly a space-filling design projects onto every pair of dimensions by averaging the squared stratified L2-discrepancy across those projections. For U-type (n, m, s^p) designs this average simplifies to a direct sum over pairs of rows using their weighted hierarchical distances. The reduction matters because practitioners need designs that remain uniform in low dimensions even when the full space is high-dimensional, as occurs in computer experiments and simulations. The authors derive the exact expression, prove matching lower and upper bounds together with equality cases, and observe that many existing optimal designs already reach the lower bound. They further prove that any design optimal for the complete stratified L2-discrepancy is automatically optimal for Φ_SD, and they note the criterion runs in O(n² m) time.

Core claim

For U-type (n, m, s^p) designs, Φ_SD admits an explicit expression in terms of row-pairwise weighted hierarchical distances, together with sharp lower and upper bounds and their equality conditions; many known optimal constructions achieve the lower bound of Φ_SD, and designs that attain the lower bound of the full stratified L2-discrepancy also attain the lower bound of Φ_SD.

What carries the argument

Φ_SD, defined as the average squared stratified L2-discrepancy over all two-dimensional projections and reduced to a sum over row-pairwise weighted hierarchical distances

If this is right

The criterion can be evaluated in O(n² m) time, offering a modest reduction in operations compared with direct projection-wise evaluation.
Designs attaining the lower bound of the full stratified L2-discrepancy also attain the lower bound of Φ_SD.
Many known optimal constructions for other uniformity criteria already achieve the lower bound of Φ_SD.
The explicit bounds supply a concrete benchmark for selecting or constructing designs with good projection uniformity.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same pairwise-distance reduction may extend to averages over three-dimensional projections, allowing similar fast criteria for higher-order uniformity.
Optimization algorithms could directly minimize the pairwise expression to search for new designs that improve projection behavior.
The approach may unify bounds across different discrepancy measures used in experimental design.
Users of existing space-filling designs can quickly re-evaluate them for low-dimensional performance without recomputing every projection.

Load-bearing premise

The stratified L2-discrepancy is defined so that its average over all two-dimensional projections reduces directly to pairwise distances between rows without extra fitting or adjustments.

What would settle it

Select a concrete U-type design known to be optimal under another criterion, compute Φ_SD both by direct averaging over its two-dimensional projections and by the derived pairwise-distance formula, and check whether the two values coincide and equal the stated lower bound.

read the original abstract

This paper studies a uniform projection criterion for space-filling designs under the stratified $L_2$-discrepancy. The criterion, denoted by $\Phi_{SD}$, is the average squared stratified $L_2$-discrepancy over all two-dimensional projections. For U-type $(n,m,s^p)$ designs, we derive an explicit formula for $\Phi_{SD}$ in terms of row-pairwise weighted hierarchical distances, and we establish sharp lower and upper bounds with equality conditions. We further show that many known optimal constructions attain the lower bound of $\Phi_{SD}$, and that designs attaining the lower bound of the full stratified $L_2$-discrepancy also attain the lower bound of $\Phi_{SD}$. The criterion can be evaluated in $O(n^2m)$ time, with a modest reduction in arithmetic operations compared with direct projection-wise evaluation. Numerical studies illustrate the theoretical results and show that $\Phi_{SD}$ is effective for assessing low-dimensional projection uniformity.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives a closed-form reduction of average 2D projection stratified L2-discrepancy to pairwise distances for U-type designs, plus sharp bounds that known constructions meet.

read the letter

The main point is that the authors derive an explicit formula for Φ_SD, the average squared stratified L2-discrepancy over all two-dimensional projections, and express it directly as a sum of row-pairwise weighted hierarchical distances for U-type (n,m,s^p) designs. They also prove sharp lower and upper bounds with equality conditions and note that many existing optimal designs hit the lower bound. In addition, optimality for the full stratified L2-discrepancy carries over to this projection version. The computation runs in O(n²m) time, which is a modest improvement over evaluating each projection separately. Numerical checks confirm the formula and show the criterion picks up low-dimensional uniformity issues. These pieces look internally consistent and follow from the discrepancy definition without extra fitting terms. The work is narrow but clean: it stays inside space-filling design theory and focuses on 2D projections of U-type designs. A clear limitation is that the reduction and bounds are stated only for this setting; nothing is shown for higher-dimensional projections or non-U-type constructions, so users would need to adapt the approach themselves. The efficiency gain is described as modest, which matches the arithmetic count. This paper is aimed at people who already work with discrepancy criteria for experimental designs and want a practical way to check projection uniformity. A reader looking for explicit formulas and optimality transfer results will find usable material here. The claims are specific enough and the derivations appear verifiable, so the paper deserves a serious referee rather than a desk reject. I would send it out for review.

Referee Report

0 major / 3 minor

Summary. The paper introduces the uniform projection criterion Φ_SD, defined as the average squared stratified L2-discrepancy over all two-dimensional projections of a design. For U-type (n,m,s^p) designs, it derives an explicit closed-form expression for Φ_SD in terms of row-pairwise weighted hierarchical distances, establishes sharp lower and upper bounds together with equality cases, shows that many known optimal constructions attain the lower bound, and proves that designs attaining the lower bound for the full stratified L2-discrepancy also attain it for Φ_SD. The criterion admits an O(n²m)-time evaluation algorithm.

Significance. If the central derivations hold, the work supplies a theoretically grounded, computationally efficient projection-uniformity criterion with explicit bounds and equality conditions. The reduction to pairwise distances and the link between full and projected discrepancy are useful for both analysis and construction of space-filling designs. The result strengthens the toolkit for assessing low-dimensional uniformity without exhaustive projection-wise computation.

minor comments (3)

The abstract states that the formula is derived directly from the discrepancy definition; the manuscript should include a brief remark in §2 or §3 confirming that no auxiliary fitting constants are introduced in the reduction to weighted hierarchical distances.
Equality conditions for the lower bound are asserted for known constructions; a short table or explicit list in §4 identifying which constructions achieve equality would improve readability.
The O(n²m) complexity claim is stated without a detailed operation count; adding one sentence comparing the arithmetic operations to direct projection-wise evaluation would clarify the modest reduction mentioned.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our manuscript, as well as for the recommendation of minor revision. We are pleased that the referee recognizes the theoretical contributions, including the explicit formula for Φ_SD, the sharp bounds, the optimality results, and the computational efficiency. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity: explicit algebraic reduction from discrepancy definition

full rationale

The paper derives an explicit closed-form expression for Φ_SD (average squared stratified L2-discrepancy over 2D projections) directly from the definition of the stratified discrepancy for U-type designs, expressing it as a sum over row-pairwise weighted hierarchical distances. This is an algebraic identity obtained by expanding the discrepancy integral or sum under the given stratification, without parameter fitting, post-hoc adjustments, or re-expression of the target quantity as an input. No self-citations are invoked as load-bearing for the central formula or bounds; the lower/upper bounds follow from the derived expression with equality cases verified against known constructions. The derivation chain is self-contained against the discrepancy definition and does not reduce to a tautology or fitted input renamed as prediction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The derivation rests on the standard definition of stratified L2-discrepancy and the structure of U-type designs; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption Stratified L2-discrepancy is a well-defined uniformity measure whose squared value on projections can be expressed via pairwise weighted hierarchical distances.
Invoked to obtain the explicit formula for Φ_SD.

pith-pipeline@v0.9.0 · 5697 in / 1179 out tokens · 37695 ms · 2026-05-20T01:22:34.705826+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1: Φ_SD(D) = 1/(n² m (m-1)) ∑_a ∑_b d_ab² + C_SD(m,s,p) where d_ab = ∑_k ∑_i ω(i)/s^i (1-δ_i(z_ak,z_bk))
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Corollary 1: NRT-distance representation d_ab = ∑_k (A0 - A_(p-ρ(x_ak,x_bk))^(0))

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

31 extracted references · 31 canonical work pages

[1]

Bierbrauer, J., Edel, Y., and Schmid, W. C. (2002). Coding-theoretic constructions for (t, m, s)-nets and ordered orthogonal arrays.Journal of Combinatorial Designs, 10:403–418

work page 2002
[2]

and Yang, X

Chen, H., Zhang, Y. and Yang, X. (2021). Uniform projection nested Latin hypercube designs.Statistical Papers62, 2031–2045

work page 2021
[3]

T., Li, R., and Sudjianto, A

Fang, K. T., Li, R., and Sudjianto, A. (2006).Design and modeling for computer experiments. CRC Press, New York

work page 2006
[4]

T., Liu, M

Fang, K. T., Liu, M. Q., Qin, H., and Zhou, Y. D. (2018).Theory and Application of Uniform Experimental Designs. Springer, Singapore

work page 2018
[5]

Fang, K. T. and Qin, H. (2005). Uniformity pattern and related criteria for two-level factorials.Sci. China Ser. A48, 1–11

work page 2005
[6]

He, Y., Cheng, C.-S., and Tang, B. (2018). Strong orthogonal arrays of strength two plus.The Annals of Statistics46, 457–468

work page 2018
[7]

and Tang, B

He, Y. and Tang, B. (2013). Strong orthogonal arrays and associated Latin hypercubes for computer experiments.Biometrika100, 254–260

work page 2013
[8]

S., Sloane, N

Hedayat, A. S., Sloane, N. J. and Stufken, J. (1999).Orthogonal Arrays: Theory and Applications. Springer, New York

work page 1999
[9]

Hickernell, F. J. and Liu, M. (2002). Uniform designs limit aliasing.Biometrika89, 893–904

work page 2002
[10]

Jiang, B., Gao, Q., and Wang, Y. (2025). Construction of optimal designs under the minimum aberration-type space-filling criterion. Preprint

work page 2025
[11]

E., Moore, L

Johnson, M. E., Moore, L. M. and Ylvisaker, D. (1990). Minimax and maximin distance designs.J. Statist. Plan. Infer.26, 131–148

work page 1990
[12]

R., Gul, E

Joseph, V. R., Gul, E. and Ba, S.. (2015). Maximum projection designs for computer experiments.Biometrika,102, 371–380

work page 2015
[13]

Li, W., Liu, M. Q. and Tang, B. (2020). A method of constructing maximin distance designs.Biometrika108, 845–855. 19

work page 2020
[14]

and Sun, F

Liu, S., Wang, Y. and Sun, F. (2023). Two-dimensional projection uniformity for space-filling designs.Canad. J. Statist.51, 293–311

work page 2023
[15]

W., Olkin, I

Marshall, A. W., Olkin, I. and Arnold, B. C. (2009).Inequalities: Theory of Majorization and Its Applications (2nd). Springer, New York

work page 2009
[16]

D., Beckman, R

McKay, M. D., Beckman, R. J. and Conover, W. J. (1979). A comparison of three methods for selecting values of input variables in the analysis of output from a computer code.Technometrics21, 239–245

work page 1979
[17]

and Santner, T

Moon, H., Dean, A. and Santner, T. (2012). Two-stage sensitivity-based group screening in computer experiments.Technometrics,54, 376–387

work page 2012
[18]

and Xu, H

Onyambu, S. and Xu, H. (2025). Tuning differential evolution algorithm for constructing uniform projection designs.J. Statist. Plan. Infer., 106356

work page 2025
[19]

J., Williams, B

Santner, T. J., Williams, B. J., and Notz, W. I. (2003).The Design and Analysis of Computer Experiments.New York: Springer

work page 2003
[20]

and Tang, B

Sun, C.-Y. and Tang, B. (2023). Uniform projection designs and strong orthogonal arrays.J. Amer. Statist. Assoc.118, 417–423

work page 2023
[21]

and Xu, H

Sun, F., Wang, Y. and Xu, H. (2019). Uniform projection designs.The Annals of Statistics47, 641–661

work page 2019
[22]

and Xu, H

Tian, Y. and Xu, H. (2022). A minimum aberration-type criterion for selecting space- filling designs.Biometrika109, 489–501

work page 2022
[23]

and Xu, H

Tian, Y. and Xu, H. (2024). Stratification pattern enumerator and its applications. J. R. Stat. Soc. Ser. B. Stat. Methodol.86, 364–385

work page 2024
[24]

and Xu, H

Tian, Y. and Xu, H. (2025). A stratifiedL 2-discrepancy with application to space- filling designs.J. R. Stat. Soc. Ser. B. Stat. Methodol.: qkaf055

work page 2025
[25]

and Xu, H

Wang, L., Xiao, Q. and Xu, H. (2018). Optimal maximinL 1-distance Latin hypercube designs based on good lattice point designs.The Annals of Statistics46, 3741–3766

work page 2018
[26]

and Xu, H

Wang, Y., Sun, F. and Xu, H. (2022). On design orthogonality, maximin distance, and projection uniformity for computer experiments.J. Amer. Statist. Assoc.117, 375–385

work page 2022
[27]

and Sun, F

Zhang, X., Wang, Y. and Sun, F. (2025). Theory and applications of stratification criteria based on space-filling pattern and projection pattern.Metrika88, no. 4, 445–468

work page 2025
[28]

(2023).Construction and properties of uniform projection designs

Zhou, Y. (2023).Construction and properties of uniform projection designs. Disser- tation, Northeast Normal University. 20

work page 2023
[29]

D., Fang, K

Zhou, Y. D., Fang, K. T. and Ning, J. (2013). Mixture discrepancy for quasi-random point sets.J. Complexity29, 283–301

work page 2013
[30]

and Sun, F

Zhou, Y., Xiao, Q. and Sun, F. (2023). Construction of uniform projection designs via level permutation and expansion.J. Statist. Plan. Infer.222, 209–225

work page 2023
[31]

Zhou, Y. D. and Xu, H. (2014). Space-filling fractional factorial designs.J. Am. Statist. Assoc.109, 1134–44. 21

work page 2014

[1] [1]

Bierbrauer, J., Edel, Y., and Schmid, W. C. (2002). Coding-theoretic constructions for (t, m, s)-nets and ordered orthogonal arrays.Journal of Combinatorial Designs, 10:403–418

work page 2002

[2] [2]

and Yang, X

Chen, H., Zhang, Y. and Yang, X. (2021). Uniform projection nested Latin hypercube designs.Statistical Papers62, 2031–2045

work page 2021

[3] [3]

T., Li, R., and Sudjianto, A

Fang, K. T., Li, R., and Sudjianto, A. (2006).Design and modeling for computer experiments. CRC Press, New York

work page 2006

[4] [4]

T., Liu, M

Fang, K. T., Liu, M. Q., Qin, H., and Zhou, Y. D. (2018).Theory and Application of Uniform Experimental Designs. Springer, Singapore

work page 2018

[5] [5]

Fang, K. T. and Qin, H. (2005). Uniformity pattern and related criteria for two-level factorials.Sci. China Ser. A48, 1–11

work page 2005

[6] [6]

He, Y., Cheng, C.-S., and Tang, B. (2018). Strong orthogonal arrays of strength two plus.The Annals of Statistics46, 457–468

work page 2018

[7] [7]

and Tang, B

He, Y. and Tang, B. (2013). Strong orthogonal arrays and associated Latin hypercubes for computer experiments.Biometrika100, 254–260

work page 2013

[8] [8]

S., Sloane, N

Hedayat, A. S., Sloane, N. J. and Stufken, J. (1999).Orthogonal Arrays: Theory and Applications. Springer, New York

work page 1999

[9] [9]

Hickernell, F. J. and Liu, M. (2002). Uniform designs limit aliasing.Biometrika89, 893–904

work page 2002

[10] [10]

Jiang, B., Gao, Q., and Wang, Y. (2025). Construction of optimal designs under the minimum aberration-type space-filling criterion. Preprint

work page 2025

[11] [11]

E., Moore, L

Johnson, M. E., Moore, L. M. and Ylvisaker, D. (1990). Minimax and maximin distance designs.J. Statist. Plan. Infer.26, 131–148

work page 1990

[12] [12]

R., Gul, E

Joseph, V. R., Gul, E. and Ba, S.. (2015). Maximum projection designs for computer experiments.Biometrika,102, 371–380

work page 2015

[13] [13]

Li, W., Liu, M. Q. and Tang, B. (2020). A method of constructing maximin distance designs.Biometrika108, 845–855. 19

work page 2020

[14] [14]

and Sun, F

Liu, S., Wang, Y. and Sun, F. (2023). Two-dimensional projection uniformity for space-filling designs.Canad. J. Statist.51, 293–311

work page 2023

[15] [15]

W., Olkin, I

Marshall, A. W., Olkin, I. and Arnold, B. C. (2009).Inequalities: Theory of Majorization and Its Applications (2nd). Springer, New York

work page 2009

[16] [16]

D., Beckman, R

McKay, M. D., Beckman, R. J. and Conover, W. J. (1979). A comparison of three methods for selecting values of input variables in the analysis of output from a computer code.Technometrics21, 239–245

work page 1979

[17] [17]

and Santner, T

Moon, H., Dean, A. and Santner, T. (2012). Two-stage sensitivity-based group screening in computer experiments.Technometrics,54, 376–387

work page 2012

[18] [18]

and Xu, H

Onyambu, S. and Xu, H. (2025). Tuning differential evolution algorithm for constructing uniform projection designs.J. Statist. Plan. Infer., 106356

work page 2025

[19] [19]

J., Williams, B

Santner, T. J., Williams, B. J., and Notz, W. I. (2003).The Design and Analysis of Computer Experiments.New York: Springer

work page 2003

[20] [20]

and Tang, B

Sun, C.-Y. and Tang, B. (2023). Uniform projection designs and strong orthogonal arrays.J. Amer. Statist. Assoc.118, 417–423

work page 2023

[21] [21]

and Xu, H

Sun, F., Wang, Y. and Xu, H. (2019). Uniform projection designs.The Annals of Statistics47, 641–661

work page 2019

[22] [22]

and Xu, H

Tian, Y. and Xu, H. (2022). A minimum aberration-type criterion for selecting space- filling designs.Biometrika109, 489–501

work page 2022

[23] [23]

and Xu, H

Tian, Y. and Xu, H. (2024). Stratification pattern enumerator and its applications. J. R. Stat. Soc. Ser. B. Stat. Methodol.86, 364–385

work page 2024

[24] [24]

and Xu, H

Tian, Y. and Xu, H. (2025). A stratifiedL 2-discrepancy with application to space- filling designs.J. R. Stat. Soc. Ser. B. Stat. Methodol.: qkaf055

work page 2025

[25] [25]

and Xu, H

Wang, L., Xiao, Q. and Xu, H. (2018). Optimal maximinL 1-distance Latin hypercube designs based on good lattice point designs.The Annals of Statistics46, 3741–3766

work page 2018

[26] [26]

and Xu, H

Wang, Y., Sun, F. and Xu, H. (2022). On design orthogonality, maximin distance, and projection uniformity for computer experiments.J. Amer. Statist. Assoc.117, 375–385

work page 2022

[27] [27]

and Sun, F

Zhang, X., Wang, Y. and Sun, F. (2025). Theory and applications of stratification criteria based on space-filling pattern and projection pattern.Metrika88, no. 4, 445–468

work page 2025

[28] [28]

(2023).Construction and properties of uniform projection designs

Zhou, Y. (2023).Construction and properties of uniform projection designs. Disser- tation, Northeast Normal University. 20

work page 2023

[29] [29]

D., Fang, K

Zhou, Y. D., Fang, K. T. and Ning, J. (2013). Mixture discrepancy for quasi-random point sets.J. Complexity29, 283–301

work page 2013

[30] [30]

and Sun, F

Zhou, Y., Xiao, Q. and Sun, F. (2023). Construction of uniform projection designs via level permutation and expansion.J. Statist. Plan. Infer.222, 209–225

work page 2023

[31] [31]

Zhou, Y. D. and Xu, H. (2014). Space-filling fractional factorial designs.J. Am. Statist. Assoc.109, 1134–44. 21

work page 2014