In [Mas82] and [Vee78] it was proved independently that almost every interval exchange transformation is uniquely ergodic. The Birkhoff ergodic theorem implies that these maps mainly have uniformly distributed orbits. This raises the question under which conditions the orbits yield low-discrepancy sequences. The case o…
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22 results for “Low-discrepancy”
MPMC generates low-discrepancy points using graph neural networks.
problem Generating efficient low-discrepancy point sets.
method Leveraging Graph Neural Networks to model geometric properties.
result Achieves state-of-the-art performance in generating low-discrepancy points.
Using Low-Discrepancy Points for Data Compression in Machine Learning: An Experimental Comparisonstat.ML
Two methods using low-discrepancy points improve data compression for neural networks.
problem Efficiently compress large datasets for neural network training.
method Two methods based on low-discrepancy points: digital nets with averaging and clustering.
result Second method outperforms supercompress in compression error and neural network accuracy.
Generation of pseudorandom numbers from different probability distributions has been studied extensively in the Monte Carlo simulation literature. Two standard generation techniques are the acceptance-rejection and inverse transformation methods. An alternative approach to Monte Carlo simulation is the quasi-Monte Carl…
New deep learning method outperforms random training data.
problem Improving accuracy of deep learning algorithms in high dimensions.
method Training with low-discrepancy sequences instead of random data.
result Significantly outperforms standard deep learning algorithms.
Langevin Quasi-Monte Carlostat.CO
LMC improves sampling from complex distributions using quasi-random sequences.
problem Sampling from complex high-dimensional distributions with high accuracy.
method Using completely uniformly distributed (CUD) sequences in Langevin Monte Carlo (LMC) to generate Gaussian perturbations.
result LMC with low-discrepancy CUD sequences achieves smaller estimation error than standard LMC.
This paper defines the notion of class discrepancy for families of functions. It shows that low discrepancy classes admit small offline and streaming coresets. We provide general techniques for bounding the class discrepancy of machine learning problems. As corollaries of the general technique we bound the discrepancy …
We consider the problem of solving a large-scale Quadratically Constrained Quadratic Program. Such problems occur naturally in many scientific and web applications. Although there are efficient methods which tackle this problem, they are mostly not scalable. In this paper, we develop a method that transforms the quadra…
Comparative Study of Monte Carlo and Quasi-Monte Carlo Techniques for Enhanced Derivative Pricingq-fin.PR
This study compares MC and QMC methods for derivative pricing, showing QMC's superior convergence rates.
problem Improving derivative pricing accuracy and efficiency in high-dimensional settings.
method Compared Monte Carlo and quasi-Monte Carlo techniques, focusing on convergence rates and low-discrepancy sequences.
result Quasi-Monte Carlo methods achieve superior convergence rates and reduce root mean square error in derivative pricing.
We consider the problem of adaptive stratified sampling for Monte Carlo integration of a differentiable function given a finite number of evaluations to the function. We construct a sampling scheme that samples more often in regions where the function oscillates more, while allocating the samples such that they are wel…
Improved algorithm for low-discrepancy colorings with practical time complexity.
problem Finding near-optimal colorings for set systems with low discrepancy.
method Randomized algorithm using primal-dual reweighing and matchings with low crossing number.
result Improved time complexity for constructing colorings and approximations.
RQMC improves QMC by providing practical error bounds for financial applications.
problem Lack of practical error estimates in QMC methods.
method Combines Sobol LDS with randomized scrambling methods.
result RQMC outperforms standard QMC in convergence rates and provides error bounds.
Optimizes kernel discrepancies by selecting subsets efficiently.
problem Improving kernel discrepancies for QMC methods.
method Introduces a novel subset selection algorithm for kernel discrepancies.
result Efficiently generates low-discrepancy samples from various distributions.
We consider the problem of improving the efficiency of randomized Fourier feature maps to accelerate training and testing speed of kernel methods on large datasets. These approximate feature maps arise as Monte Carlo approximations to integral representations of shift-invariant kernel functions (e.g., Gaussian kernel).…
Driven by the need for parallelizable hyperparameter optimization methods, this paper studies \emph{open loop} search methods: sequences that are predetermined and can be generated before a single configuration is evaluated. Examples include grid search, uniform random search, low discrepancy sequences, and other sampl…
Pricing and Risk Management with High-Dimensional Quasi Monte Carlo and Global Sensitivity Analysisq-fin.RM
We review and apply Quasi Monte Carlo (QMC) and Global Sensitivity Analysis (GSA) techniques to pricing and risk management (greeks) of representative financial instruments of increasing complexity. We compare QMC vs standard Monte Carlo (MC) results in great detail, using high-dimensional Sobol' low discrepancy sequen…
Samplets and multiwavelets constructed from scattered data converge to specific densities in the limit.
problem Constructing data-adapted multiresolution analyses and multiwavelets with flexible vanishing moments.
method Probabilistic framework for samplet construction; convergence to multiwavelets with broken polynomial densities.
result Samplet construction converges to multiwavelets in the infinite data limit.
Transport Quasi-Monte Carlomath.NA
Improves QMC for complex distributions using transport maps.
problem Challenges in applying QMC to general target distributions.
method Train a transport map to approximate target distributions, ensuring RQMC achieves superior error rates.
result Transport QMC achieves faster convergence rates than standard Monte Carlo under mild conditions.
This thesis advances algorithms and software for QMC, GP, and sciML.
problem Efficient high-dimensional integration, interpolation, and PDE modeling.
method Developed new algorithms and software for QMC, GP, and sciML.
result Efficient and accurate methods for high-dimensional problems.
Application of Quasi Monte Carlo and Global Sensitivity Analysis to Option Pricing and Greeksq-fin.CP
QMC and GSA improve option pricing and risk measures efficiency.
problem Efficiently pricing and hedging complex financial instruments.
method Application of QMC and GSA techniques for financial instrument pricing and hedging, comparing MC vs QMC and analyzing greeks computation.
result QMC outperforms MC in most cases, especially in high-dimensional simulations, leading to faster and more stable convergence.
This study compares MC and QMC methods for likelihood functions.
problem Approximating the normalizing constant of posterior distributions and marginal likelihoods.
method Characterizes the integration error of MC and QMC methods for likelihood functions.
result QMC outperforms MC under certain conditions, especially in high dimensions.
Statistical machine learning models should be evaluated and validated before putting to work. Conventional k-fold Monte Carlo Cross-Validation (MCCV) procedure uses a pseudo-random sequence to partition instances into k subsets, which usually causes subsampling bias, inflates generalization errors and jeopardizes the r…