Low-discrepancy Point Research Articles

Message-Passing Monte Carlo: Generating low-discrepancy point sets via graph neural networks

Discrepancy is a well-known measure for the irregularity of the distribution of a point set. Point sets with small discrepancy are called low discrepancy and are known to efficiently fill the space in a uniform manner. Low-discrepancy points play a central role in many problems in science and engineering, including numerical integration, computer vision, machine perception, computer graphics, machine learning, and simulation. In this work, we present a machine learning approach to generate a new class of low-discrepancy point sets named Message-Passing Monte Carlo (MPMC) points. Motivated by the geometric nature of generating low-discrepancy point sets, we leverage tools from Geometric Deep Learning and base our model on graph neural networks. We further provide an extension of our framework to higher dimensions, which flexibly allows the generation of custom-made points that emphasize the uniformity in specific dimensions that are primarily important for the particular problem at hand. Finally, we demonstrate that our proposed model achieves state-of-the-art performance superior to previous methods by a significant margin. In fact, MPMC points are empirically shown to be either optimal or near-optimal with respect to the discrepancy for low dimension and small number of points, i.e., for which the optimal discrepancy can be determined.

Heuristic approaches to obtain low-discrepancy point sets via subset selection

Building upon the exact methods presented in our earlier work (2022) [5], we introduce a heuristic approach for the star discrepancy subset selection problem. The heuristic gradually improves the current-best subset by replacing one of its elements at a time. While it does not necessarily return an optimal solution, we obtain promising results for all tested dimensions. For example, for moderate sizes 30≤n≤240, we obtain point sets in dimension 6 with L∞ star discrepancy up to 35% better than that of the first n points of the Sobol' sequence. Our heuristic works in all dimensions, the main limitation being the precision of the discrepancy calculation algorithms. We provide a comparison with an energy functional introduced by Steinerberger (2019) [31], showing that our heuristic performs better on all tested instances. Finally, our results give further empirical information on inverse star discrepancy conjectures.

Global Likelihood Sampler for Multimodal Distributions

Drawing samples from a target distribution is essential for statistical computations when the analytical solution is infeasible. Many existing sampling methods may be easy to fall into the local mode or strongly depend on the proposal distribution when the target distribution is complicated. In this article, the Global Likelihood Sampler (GLS) is proposed to tackle these problems and the GL bootstrap is used to assess the Monte Carlo error. GLS takes the advantage of the randomly shifted low-discrepancy point set to sufficiently explore the structure of the target distribution. It is efficient for multimodal and high-dimensional distributions and easy to implement. It is shown that the empirical cumulative distribution function of the samples uniformly converges to the target distribution under some conditions. The convergence for the approximate sampling distribution of the sample mean based on the GL bootstrap is also obtained. Moreover, numerical experiments and a real application are conducted to show the effectiveness, robustness, and speediness of GLS compared with some common methods. It illustrates that GLS can be a competitive alternative to existing sampling methods. Supplementary materials for this article are available online.

A multi-body dynamical evolution model for generating the point set with best uniformity

Generating the low-discrepancy point sets in high-dimensional space is an optimization problem which involves two issues: how to define the objective function of optimization, and how to optimize this optimization problem with tens of thousands of variables. Inspired by natural phenomena, we make two assumptions: the first is that the static solution to the multi-body problem is a low-discrepancy point set, and the second is that the discrepancy of bodies is the lowest when the potential energy is the smallest. Under these assumptions, the objective function is defined as the potential energy of the point set. A dynamical evolutionary model (DEM) based on the minimum potential energy principle is established to construct low-discrepancy point sets. The central difference algorithm is adopted to solve the DEM and the selection of coefficients to ensure the convergence is discussed in detail. Numerical examples confirm the assumption that there is a significant positive correlation between the potential energy and the discrepancy. We also combine the DEM with the restarting technique to generate a series of low-discrepancy point sets. These point sets are unbiased and perform better than other low-discrepancy point sets in terms of the discrepancy, the potential energy, integrating eight test functions and computing the statistical moments for two practical stochastic problems. Numerical examples also show that the DEM can generate uniformly distributed point sets in non-cubes. More interestingly, it is observed that the DEM point sets can greatly improve the convergence speed of the heterogeneous comprehensive learning particle swarm optimization.

On approximating the shape of one-dimensional posterior marginals using the low discrepancy points

A method to approximate Bayesian posterior by evaluating it on a low discrepancy sequence (LDS) point set has recently been proposed. However, this method does not focus on finding the posterior marginals. Finding posterior marginals when the posterior approximation is obtained using LDS is not straightforward, and as yet, there is no method to approximate one dimensional marginals using an LDS. We propose an approximation method for this problem. This method is based on an s-dimensional integration rule together with fitting a polynomial smoothing function. We state and prove results showing conditions under which this polynomial smoothing function will converge to the true one-dimensional function. We also demonstrate the computational efficiency of the new approach compared to a grid based approach.

Quasi-Monte Carlo method for calculating X-ray scatter in CT

In this paper we transform the trajectories of X-ray as it interacts with a phantom into a high-dimensional integration problem and give the integral formula for the probability of photons emitted from the X-ray source through the phantom to reach the detector. We propose a superior algorithm called gQMCFRD, which combines GPU-based quasi-Monte Carlo (gQMC) method with forced random detection (FRD) technique to simulate this integral. QMC simulation is deterministic versions of Monte Carlo (MC) simulation, which uses deterministic low discrepancy points (such as Sobol' points) instead of the random points. By using the QMC and FRD technique, the gQMCFRD greatly increases the simulation convergence rate and efficiency. We benchmark gQMCFRD, GPU based MC tool (gMCDRR), which performs conventional simulations, a GPU-based Metropolis MC tool (gMMC), which uses the Metropolis-Hasting algorithm to sample the entire photon path from the X-ray source to the detector and gMCFRD, that uses random points for sampling against PENELOPE subroutines: MC-GPU. The results are in excellent agreement and the Efficiency Improvement Factor range 27 ∼ 37 (or 1.09 ∼ 1.16, or 0.12 ∼ 0.15, or 3.62 ∼ 3.70) by gQMCFRD (or gMCDRR, or gMMC, or gMCFRD) with comparison to MC-GPU in all cases. It shows that gQMCFRD is more effective in these cases.

Dispersion of digital (0, m, 2)-nets

We study the dispersion of digital (0, m, 2)-nets; i.e. the size of the largest axes-parallel box within such point sets. Digital nets are an important class of low-discrepancy point sets. We prove tight lower and upper bounds for certain subclasses of digital nets where the generating matrices are of triangular form and compute the dispersion of special nets such as the Hammersley point set exactly.

Quantum Discrepancy: A Non-Commutative Version of Combinatorial Discrepancy

In this paper, we introduce a notion of quantum discrepancy, a non-commutative version of combinatorial discrepancy which is defined for projection systems, i.e. finite sets of orthogonal projections, as non-commutative counterparts of set systems. We show that besides its natural algebraic formulation, quantum discrepancy, when restricted to set systems, has a probabilistic interpretation in terms of determinantal processes. Determinantal processes are a family of point processes with a rich algebraic structure. A common feature of this family is the local repulsive behavior of points. Alishahi and Zamani (2015) exploit this repelling property to construct low-discrepancy point configurations on the sphere. 
 We give an upper bound for quantum discrepancy in terms of $N$, the dimension of the space, and $M$, the size of the projection system, which is tight in a wide range of parameters $N$ and $M$. Then we investigate the relation of these two kinds of discrepancies, i.e. combinatorial and quantum, when restricted to set systems, and bound them in terms of each other.

An algorithm to compute the [formula omitted]-value of a digital net and of its projections

Digital nets are among the most successful methods to construct low-discrepancy point sets for quasi-Monte Carlo integration. Their quality is traditionally assessed by a measure called the t-value. A refinement computes the t-value of the projections over subsets of coordinates and takes a weighted average (or some other function) of these values. It is also of interest to compute the t-values of embedded nets obtained by taking subsets of the points. In this paper, we propose an efficient algorithm to compute such measures and we compare our approach with previously proposed methods both empirically and in terms of computational complexity.

Theory of generalized discrepancies on a ball of arbitrary finite dimensions and algorithms for finding low-discrepancy point sets

One of the objectives of this paper is to extend the idea of the generalized discrepancy to the ball of arbitrary finite dimensions and to study its properties. We first construct orthonormal systems in higher dimensions, Sobolev spaces as well as particular differential operators. The discrepancy is then derived from an error estimate for numerical integration on a ball. We also present some new statistical and numerical properties of the generalized discrepancy which have also been unknown in three dimensions before. In addition, this paper focuses on constructing and implementing novel algorithms in order to obtain low-discrepancy point grids on the ball. Such low-discrepancy grids are uniformly distributed and are appropriate for quadrature rules (sometimes also called cubature rules) and as centres of localized trial functions in tomographic inverse problems on the ball. For this purpose, we also consider the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm and examine different BFGS updates and line search methods.

A nonlocal functional promoting low-discrepancy point sets

Let X=x1,…,xN⊂Td≅[0,1]d be a set of N points in the d−dimensional torus that we want to arrange as regularly possible. The purpose of this paper is to introduce the energy functional E(X)=∑1≤m,n≤Nm≠n∏k=1d(1−log2sinπ|xm,k−xn,k|) and to suggest that moving a set X into the direction −∇E(X) may have the effect of increasing regularity of the set in the sense of decreasing discrepancy. We numerically demonstrate the effect for Halton, Hammersley, Kronecker, Niederreiter and Sobol sets. Lattices in d=2 are critical points of the energy functional, some (possibly all) are strict local minima.

Quasi-Monte Carlo integration on manifolds with mapped low-discrepancy points and greedy minimal Riesz s-energy points

In this paper we consider two sets of points for Quasi-Monte Carlo integration on two-dimensional manifolds. The first is the set of mapped low-discrepancy sequence by a measure preserving map, from a rectangle U⊂R2 to the manifold. The second is the greedy minimal Riesz s-energy points extracted from a suitable discretization of the manifold. Thanks to the Poppy-seed Bagel Theorem we know that the classes of points with minimal Riesz s-energy, under suitable assumptions, are asymptotically uniformly distributed with respect to the normalized Hausdorff measure. They can then be considered as quadrature points on manifolds via the Quasi-Monte Carlo (QMC) method. On the other hand, we do not know if the greedy minimal Riesz s-energy points are a good choice to integrate functions with the QMC method on manifolds. Through theoretical considerations, by showing some properties of these points and by numerical experiments, we attempt to answer to these questions.

PB 11 Phase-coupling optimization (PCO) – a new algorithm for the analysis and localization of phase-coupling in multivariate data

Introduction Phase-coupling between neuronal oscillations in the brain has been hypothesized as a mechanism for interactions between brain areas; moreover, phases of neuronal oscillations were found related to performance in memory and perception tasks. However, in multivariate data, such as multi-channel EEG or MEG, the analysis of phase-coupling remains challenging. Sensor-space analysis is suboptimal regarding the signal-to-noise ratio (SNR) and might fail to localize the sources of phase-coupling appropriately. Here, we introduce phase-coupling optimization (PCO), an algorithm seeking spatial filters that maximize the coupling of oscillatory phases to an independent variable (e.g., detected vs. undetected events in a perception task). The resulting spatial filters/patterns can then be used for inverse source modeling. Methods Due to its simplicity, the “mean vector length” measure was chosen to quantify the degree of phase-coupling and is being maximized using the quasi-Newton Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm. To avoid local minima, the optimization is restarted several times from low-discrepancy pseudo-random initial starting points. As regularization, spatio-spectral decomposition (SSD) reduces the search-space while maximizing the SNR in the selected frequency band. The performance of the algorithm was verified using realistic forward-model EEG simulations. Results Simulations demonstrated that PCO, together with SSD as pre-processing, can reliably localize the sources of phase-coupling in multi-channel EEG and characterize the coupling relation down to a SNR of −10 dB (i.e., a power ratio of 0.1) using 500 simulated task repetitions. Additionally, PCO is superior to a multiple regression approach and by a large margin outperforms sensor-space analysis based on a current source density estimation. Discussion The analysis of phase-coupling in sensor space is suboptimal due to the SNR of the signal and might even lead to erroneous source localization. Techniques such as beamforming on the other hand, require a well-founded a priori assumption on the spatial origin of phase-coupling and complex inverse problem solvers. PCO seeks the optimal spatial projection for the analysis of phase-coupling and demonstrated its reliability even for very low SNR. Hence, we regard it as a promising tool for the analysis of neuronal oscillations. Significance PCO increases the sensitivity and reliability of phase-coupling analyses. The obtained spatial patterns can be used for inverse source modeling.

Convergence of sequential quasi-Monte Carlo smoothing algorithms

Gerber and Chopin [J. R. Stat. Soc. Ser. B. Stat. Methodol. 77 (2015) 509–579] recently introduced Sequential quasi-Monte Carlo (SQMC) algorithms as an efficient way to perform filtering in state–space models. The basic idea is to replace random variables with low-discrepancy point sets, so as to obtain faster convergence than with standard particle filtering. Gerber and Chopin (2015) describe briefly several ways to extend SQMC to smoothing, but do not provide supporting theory for this extension. We discuss more thoroughly how smoothing may be performed within SQMC, and derive convergence results for the so-obtained smoothing algorithms. We consider in particular SQMC equivalents of forward smoothing and forward filtering backward sampling, which are the most well-known smoothing techniques. As a preliminary step, we provide a generalization of the classical result of Hlawka and Muck [Computing (Arch. Elektron. Rechnen) 9 (1972) 127–138] on the transformation of QMC point sets into low discrepancy point sets with respect to non uniform distributions. As a corollary of the latter, we note that we can slightly weaken the assumptions to prove the consistency of SQMC.

Stochastic Collocation vial1-Minimisation on Low Discrepancy Point Sets with Application to Uncertainty Quantification

AbstractVarious numerical methods have been developed in order to solve complex systems with uncertainties, and the stochastic collocation method usingl1-minimisation on low discrepancy point sets is investigated here. Halton and Sobol' sequences are considered, and low discrepancy point sets and random points are compared. The tests discussed involve a given target function in polynomial form, high-dimensional functions and a random ODE model. Our numerical results show that the low discrepancy point sets perform as well or better than random sampling for stochastic collocation vial1-minimisation.

A Method for Generating Low Discrepancy Samples for General Risk Management Simulations

This article will describe a new way of producing low discrepancy point sets in a very distinct way from that of standard quasi-random methods. Standard quasi-random point sets, such as Sobol sequences, utilize a number theoretical approach; whereas the method in this article uses quite a different approach which can be customized to the important case where different variables have different importance to the estimation of interest. First, the basic approach will be described and then a way to generalize the approach to the case of simulating under spectral decomposition will be detailed.

Low-Discrepancy Points for Deterministic Assignment of Hidden Weights in Extreme Learning Machines.

The traditional extreme learning machine (ELM) approach is based on a random assignment of the hidden weight values, while the linear coefficients of the output layer are determined analytically. This brief presents an analysis based on geometric properties of the sampling points used to assign the weight values, investigating the replacement of random generation of such values with low-discrepancy sequences (LDSs). Such sequences are a family of sampling methods commonly employed for numerical integration, yielding a more efficient covering of multidimensional sets with respect to random sequences, without the need for any computationally intensive procedure. In particular, we prove that the universal approximation property of the ELM is guaranteed when LDSs are employed, and how an efficient covering affects the convergence positively. Furthermore, since LDSs are generated deterministically, the results do not have a probabilistic nature. Simulation results confirm, in practice, the good theoretical properties given by the combination of ELM with LDSs.

Analysis of discrete least squares on multivariate polynomial spaces with evaluations at low-discrepancy point sets

We analyze the stability and accuracy of discrete least squares on multivariate polynomial spaces to approximate a given function depending on a multivariate random variable uniformly distributed on a hypercube. The polynomial approximation is calculated starting from pointwise noise-free evaluations of the target function at low-discrepancy point sets. We prove that the discrete least-squares approximation, in a multivariate anisotropic tensor product polynomial space and with evaluations at low-discrepancy point sets, is stable and accurate under the condition that the number of evaluations is proportional to the square of the dimension of the polynomial space, up to logarithmic factors. This result is analogous to those obtained in Cohen et al. (2013), Migliorati et al. (2014), Migliorati (2013) and Chkifa et al. (in press) for discrete least squares with random point sets, however it holds with certainty instead of just with high probability. The result is further generalized to arbitrary polynomial spaces associated with downward closed multi-index sets, but with a more demanding (and probably nonoptimal) proportionality between the number of evaluation points and the dimension of the polynomial space.

Low-discrepancy point sets for non-uniform measures

In the present paper we prove several results concerning the existence of low-discrepancy point sets with respect to an arbitrary non-uniform measure $\mu$ on the $d$-dimensional unit cube. We improve a theorem of Beck, by showing that for any $d \geq 1$, $N \geq 1,$ and any non-negative, normalized Borel measure $\mu$ on $[0,1]^d$ there exists a point set $x_1, \dots, x_N \in [0,1]^d$ whose star-discrepancy with respect to $\mu$ is of order $$ D_N^*(x_1, \dots, x_N; \mu) \ll \frac{(\log N)^{(3d+1)/2}}{N}. $$ For the proof we use a theorem of Banaszczyk concerning the balancing of vectors, which implies an upper bound for the linear discrepancy of hypergraphs. Furthermore, the theory of large deviation bounds for empirical processes indexed by sets is discussed, and we prove a numerically explicit upper bound for the inverse of the discrepancy for Vapnik--Cervonenkis classes. Finally, using a recent version of the Koksma--Hlawka inequality due to Brandolini, Colzani, Gigante and Travaglini, we show that our results imply the existence of cubature rules yielding fast convergence rates for the numerical integration of functions having discontinuities of a certain form.

A direct inversion method for non-uniform quasi-random point sequences

The inversion method is an effective approach for transforming uniform random points according to a given probability density function. In two dimensions, horizontal and vertical displacements are computed successively using a marginal and then all conditional density functions. When quasi-random low-discrepancy points are provided as input, spurious artifacts might appear if the density function is not separable. Therefore, this paper relies on combining intrinsic properties of the golden ratio sequence and the Hilbert space filling curve for generating non-uniform point sequences using a single step inversion method. Experiments show that this approach improves efficiency while avoiding artifacts for general discrete probability density functions.