[Paper Review] A reduced fast component-by-component construction of lattice point sets with small weighted star discrepancy

Ralph Kritzinger, Helene Laimer|arXiv (Cornell University)|Jan 28, 2015

Mathematical Approximation and Integration16 references3 citations

TL;DR

This paper presents a reduced fast component-by-component (CBC) algorithm for constructing lattice point sets with small weighted star discrepancy, significantly lowering construction cost when weights decay rapidly. By exploiting structured generating vectors and optimizing component selection, the method achieves strong polynomial tractability under summability conditions on weights and shift parameters, enabling efficient quasi-Monte Carlo integration in high dimensions.

ABSTRACT

The weighted star discrepancy of point sets appears in the weighted Koksma-Hlawka inequality and thus is a measure for the quality of point sets with respect to their performance in quasi-Monte Carlo algorithms. A special choice of point sets are lattice point sets whose generating vector can be obtained one component at a time such that the resulting lattice point set has a small weighted star discrepancy. In this paper we consider a reduced fast component-by-component algorithm which significantly reduces the construction cost for such generating vectors provided that the weights decrease fast enough.

Motivation & Objective

• To reduce the computational cost of constructing lattice point sets with small weighted star discrepancy for quasi-Monte Carlo integration.
• To develop a component-by-component algorithm that scales efficiently with dimension and sample size by exploiting structured generating vectors.
• To establish conditions under which the resulting lattice rules achieve strong polynomial tractability in high-dimensional integration problems.
• To improve upon existing CBC constructions by reducing the number of operations required while maintaining optimal discrepancy bounds.

Proposed method

• The method uses a reduced fast CBC algorithm that constructs generating vectors component by component, focusing only on relevant components based on a weight sequence and shift parameters.
• It introduces a transformation of the generating vector using base-$b$ digits and shift parameters $w_j$, leading to a more efficient computation of the weighted star discrepancy.
• The algorithm leverages the identity $ R_{N,oldsymbol{ u}}^s( extbf{z}) = \frac{1}{N} \sum_{k=0}^{N-1} \eta_s(k) $ to compute the discrepancy-related term efficiently.
• It applies bounds on harmonic sums $ S_N \leq 4\log N $ to control error terms in the discrepancy estimate.
• The construction cost is reduced to $ O(N\log N + \min\{s,t\}N + N\sum_{d=1}^{\min\{s,t\}} (m - w_d)b^{-w_d}) $, significantly less than the standard $ O(sN\log N) $ CBC cost.
• It uses product weights $ \gamma_{\mathfrak{u}} = \prod_{j\in\mathfrak{u}} \gamma_j $ and assumes $ \gamma_j \leq 1 $, non-increasing, and $ \sum_j \gamma_j b^{w_j} < \infty $.

Experimental results

Research questions

• RQ1 Can the construction cost of lattice point sets with small weighted star discrepancy be reduced while preserving low discrepancy?
• RQ2 Under what conditions on the weights and shift parameters does the reduced CBC algorithm achieve strong polynomial tractability?
• RQ3 How does the reduced CBC algorithm compare in efficiency to the standard CBC algorithm in terms of computational complexity?
• RQ4 What is the impact of the shift parameters $ w_j $ on the discrepancy and construction cost?
• RQ5 Can the discrepancy bound be tightly controlled using the transformed generating vector and harmonic sum estimates?

Key findings

• The reduced fast CBC algorithm reduces construction cost to $ O(N\log N + \min\{s,t\}N + N\sum_{d=1}^{\min\{s,t\}} (m - w_d)b^{-w_d}) $, which is significantly cheaper than the standard $ O(sN\log N) $ CBC method.
• Under the condition $ \sum_{j=1}^\infty \gamma_j b^{w_j} < \infty $, the method achieves strong polynomial tractability, with $ R_{N,\boldsymbol{\gamma}}^s(\mathbf{z}) \leq c_{\boldsymbol{\gamma},\delta} N^{\delta - 1} $ for any $ \delta > 0 $.
• The weighted star discrepancy satisfies $ D_{N,\boldsymbol{\gamma}}^*(\mathbf{z}) \leq O(1/N) + \frac{1}{2} R_{N,\boldsymbol{\gamma}}^s(\mathbf{z}) $, ensuring convergence proportional to $ 1/N $ under summability.
• The method guarantees that $ R_{N,\boldsymbol{\gamma}}^s(\mathbf{z}) \leq \varepsilon $ when $ N \geq (c_{\boldsymbol{\gamma},\delta} \varepsilon^{-1})^{1/(1 - \delta)} $, confirming strong polynomial tractability.
• The bound $ \prod_{j=1}^s (1 + 13\gamma_j b^{w_j} \log N) \leq (1 + \sigma_d^{-1})^d N^{(\sigma_0 + 1)\sigma_d} $ enables the derivation of tractability results under summable weights.
• The theoretical framework confirms that the reduced CBC algorithm maintains optimal discrepancy bounds while drastically reducing computational effort, especially when weights decay rapidly.